I like the explanation that there are other possible rules for multiplying negative numbers, but that mathematicians found this rule to be more useful, so that's what we most often use.
yep, best answer that summarizes my intention as to why even introduce rings in the first place!! Since this video may have not been too explicit about it, you get a pin
Yes. I think this video explains why they are useful very well. If we want to define addition and multiplication as we do in a Ring, a negative times a negative is positive. He even goes through a series of proofs to show why this must be the case. I guess EpsilonDelta could have spelled it out a bit more explicitly at the end, though.
Interesting take. Makes me think of what are necessary and sufficient conditions for an argument to be considered explanatory as opposed to illustrative. This dovetails into Godel’s Inconcompleteness theorem does it not?
I liked this introduction to rings, but this doesn't explain "the real reason why", it just shows that it must follow from several other properties we choose. This begs the question - what is the reason we choose these properties (distributivity, etc)? The answer is that it's properties we like as they describe things we want to describe, but the same reason justifies just choosing negative*negative=positive as an axiom on its own (with some other stuff) and then proving distributivity (or some other ring axiom we didn't include in this axiom set). Therefore the "real reason" is the car example at the start - because it describes natural things that we want to describe.
I think the only way to explain why a negative times a negative is positive is from some set of axioms. This answers "the reason why", but you say you want to know "the *real* reason why". That just results in trying to ask "why are the ring axioms the way they are", and you could try and find some even more fundamental axioms to prove the ring axioms from, but then you are still left with the question of "why are *those* axioms the way they are?" Your line of questioning is a never-ending regression of "what axioms can I use to prove the axioms I already have?". All you care about in math is the consequences of a set of axioms, not the reason they were chosen, although you may care to choose good axioms.
Okay that makes more sense. I was confused why this would be a better "why". That said, I always get suspicious when someone claims that there is a "why" in math - "why" requires an alternative, another way it could have been, some other axiom that could have been used, and a coherent reason to believe that the axiom really is the root and that replacing it should be assigned logical causality. the only "why" that makes sense to me is the evaluation order of the person or computer working through the problem - since math is made of equivalences and nonequivalences you can always go from anywhere to anywhere, afaict?
I asked this question to my maths teacher in 5th grade and she probably couldn't understand that I was trying to think abstract and she thought I was dumb
You can also prove it using Peano's arithmetic. The natural numbers are defined as a set that contains an initial element that we call 0. And there's a function S (for "successor"), where for every natural number *n* , *S(n)* is also natural; and *S(a) = S(b)* if, and only if, *a = b* ; also, no natural number satisfies *S(n) = 0* . Given that, the natural numbers are {0, S(0), S(S(0)), S(S(S(0))), ...}, also known as {0, 1, 2, 3, ...}. Note that every natural number is either zero or a successor of another natural number, so we can use that to define the possible operations on this set. Addition can be defined as: a + 0 = a a + S(b) = S(a + b) And multiplication can be defined as: a * 0 = 0 a * S(b) = a + (a * b) Examples: a + 1 = a + S(0) = S(a + 0) = S(a) a * 1 = a * S(0) = a + a * 0 = a + 0 = a By those definitions you can prove commutativity, associativity and distributivity, which will be needed for this proof. But it would be pretty verbose so I'm going to let it out of the comment (you can search it, though). Also, you can define subtraction as simply as: *a - b = c* , if and only if *c + b = a* However, Peano's arithmetic defines only natural numbers, if we want to extend it for negative integers, we can create an "imaginary" unit *w* (spoiler: we usually call it "-1") that by definition holds the property: S(w) = 0 With that, we can simply apply the operations definitions: a + S(w) = a + 0 S(a + w) = a (a + w) + 1 = a a + w = a - 1 a * S(w) = a * 0 a + (a * w) = 0 a * w = 0 - a S(w) = 0 w + 1 = 0 w = 0 - 1 What would happen, though, when multiplying *w* by *w* ? w * S(w) = w * 0 w + (w * w) = 0 w + (w * w) = S(w) w + (w * w) = w + 1 w * w = (w + 1) - w w * w = 1 Well, that's interesting, we just found another important property of *w* . Now, we're finally ready to prove that negatives cancel out on multiplication: (0 - a) * (0 - b) = (a * w) * (b * w) = (a * b) * (w * w) = (a * b) * 1 = a * b Remember the spoiler I gave you earlier? So, we can use a simpler notation for "0 - n": we can simply write *-n* . So, we can write *w* , or "0 - 1" as simply *-1* . And concluding my proof, we discovered that: -a * -b = a * b Thank you if you've read this far, if possible tell me what you think about this proof.
Ahhh yes, it is definitely cool system to consider, as there is a structured natural total ordering on N, and properties of the semirings are now theorems in the peano axioms. Only problem I have with using peano axioms to prove negative times negative is positive is, well you need a video this long just to prove that 1+1=2 first in peano axiom, then extend the ordering to allow subtraction for all elements which is another video, and lastly, its use is exclusive to a totally ordered sets
I have a Mathematics Degree with a Major in Statistics - so I shouldn't be commenting on this subject. That said, I would offer that as soon as we accept that a number like 2 exists, and it represents some distance from 0, it should follow that there are other directions away from 0 that must also exist. Humans have always had an innate understanding of rotations - we look left & right and we look behind us to see what might be trying to eat us - so the idea of rotating around our personal 0 has always existed - even if we didn't think mathematically this way, or could put it into formulas. Humans didn't fight the notion of mathematics that described rotations like we fought accepting that negative 2 bananas exist - no one accepted this for a very long time. If 2 represents 2 steps forward, then it is very easy to accept that we can rotate the +2 180 degrees to represent -2 steps behind us - but that of course opens the mind that we could have first rotated +2 by 90 degrees and then rotated this halfway stop another 90 degrees to get to the thing we are calling -2. Thus I am therefore going to DEFINE -2 to be a 180-degree rotation of +2 (which also happens to equal two separate 90 degree rotations). Let's pick a symbol (i) to DEFINE a 90-degree rotation - thus Mathematically -2 = 2 * i * i (2 rotated 90 degrees twice). I would similarly Define -3 = 3 * i * i. And now we are ready to ask the question, what is (-2) * (-3)? The Answer is 2 * i * i * 3 * i * i = 2 * 3 * I^4 = 6 * i^4 - and since i^4 is a 360 degree rotation back onto itself, (-2) * (-3) = 6 because we have rotated entirely back towards the direction that we have DEFINED 2 & 3 to exist in. I've used this explanation to explain these concepts to young children and they easily 'get it' - whereas if I ask educated adults why (-2)*(-3) = 6 - they really don't know why.
This is not actual proof because there is no proof how this system correlates with the algebraic and other systems we use .In this logic - x - is plus , but what is difficult to prove is the correspondence to algebraic system with your rotation system.
Why? Because (-) * (-) = (+) is most useful, so we defined fundamental axioms that lead to the properties we want once we use the logical inference of deduction on those axioms
yep, best answer that summarizes my intention as to why even introduce rings in the first place!! Since this video may have not been too explicit about it, you get a pin Edit: oops, can only pin 1 comment at a time. the other person gave the answer first, so I would have to give it to them
@NikoR96 I am stopping reading after your "Six Apples" example to remind you that _Numbers themselves_ are also an invention of the human mind, and don't exist in reality, without someone to perceive the objects being counted.... Food for thought 🤔
@NikoR96 Also, I suggest you use another even number besides two for your example because two plus two equals four, but coincidentally two _times_ two equals four, as well.
@@nikor9640 You are introducing superfluous stuff that isn't needed. I honestly hate when people respond like this because it just shows you have no idea what you are talking about. You are literally saying 1 rock times 3 bugs doesn't make sense, therefore 3 * 1 doesn't actually equal 3 since you cannot multiple rocks and bugs. My original comment is entirely correct, and you won't find a single mathematician that would disagree, because I just spoke straight facts. You had to introduce extra concepts, such as 'debt,' then you pretended that that is equivalent to the negative sign. You are basically introducing units, which is dimensional analysis. You need to remove this from your thoughts and start at a lower axiomatic level. If you want to build upon rings and introduce a logical concept of debt, you could do that, but I honestly don't even understand why you even responded when it isn't relevant to the discussion whatsoever. It is incredibly ironic that you think your superfluous additions are 'more fundamental' than a literal first axiomatic approach
Love this video!! Proving little theorems just like this one, ones that seem trivial to pretty much everyone, is one of my favorites! Understanding these basic and "trivial" things is crucial for understanding more advanced mathematical concepts imo. It is quite unbelievable that this is one of your first videos on this channel, it's awesome! Looking forward for more!!
Perhaps you are talking about the dimension of "Imaginary Time", discovered by Stephen Hawking. I wonder if it is why ordinary time only runs in one direction, in our universe?
This is actually very eye-opening... many people think that math is naturally inscribed into reality, until they find out about Gödel's incompleteness theorems and other things, like the fact that the result of multiplying two negative numbers is actually agreed upon 🙂 Seems like another brilliant channel has been born just now 🙂
@@caoinismyname So you say if I multiply 2 apples by -3 I should get -6 apples? ;-) These nice "explanations" you posted are actually just analogies or metaphors, but many of such "math phenomena" have nothing to do with direct experience in reality as we know it. What's even more mindblowing is that through such abstract thinking you can actually end up with something real, something that happens in this world ;-)
@@caoinismyname Wow, dude, what an ego! Look, the truth is nobody cares who you are and what prizedid you receive in your petty math olympiad. There are no authorities in science. On top of your enormous ego, you've clearly misunderstood the video. The author indeed referred to the difficulties of even imagining negative numbers and not mentioning manipulating them in real life. Get off your high horse, kid.
@@caoinismyname So, for internal logic, it is useful to caution one against (1) being too full of himself (as you did) or (2) acting like a genius (as you did), while (3) posting one's awards won (as you did)?
I mean, this rule is the most intuitive. -5*3=-15 can be read as "I borrowed $5 three times, therefore I owe $15", it then stands to reason that a negative multiplier must have the inverse effect: -5*-3=15 "I lent $5 three times, therefore I'm owed $15". Other rules can be defined, but this one maps fairly nicely to real life and the constraints thereof.
@@davidbarroso1960 yeah, you're right there is a flaw in the argument. You can't lend or borrow for a negative number of times as they HAVE to be Whole numbers.
@@davidbarroso1960 Yes his logic is flawed but I have a refined version of it... Logically in words, without mathematical notations, (-2)×(-3) means.. For example You have a certain number of candies and You get a temptation of eating 2 candies, it happens 3 times but You control Yourself and Don't (1st negative) eat (eating = subtracting 2 candies each time which is the 2nd Negative) 2 candies for all the 3 times that You were tempted to.. So, when You're asked how many candies did You save, Your answer will be 6 candies
It is intuitively much simpler. Negative numbers are VECTORS. They have both magnitude AND direction. Direction is defined relative to an origin POINT and a reference direction (usually the positive axis). The multiplication OPERATOR for vectors MULTIPLIES MAGNITUDES and ADDS DIRECTIONS.
@@pjay3028 It's possible to consider any number a vector; a scalar (standard number on its own) can be considered a one-dimensional vector. If you think of standard multiplication as the same thing as a vector dot product (probably the closest equivalent), one way to represent it is |v1| * |v2| * cos(angle between). Two positive or two negative numbers go the same direction, so angle between is 0, and cos(0) is 1. Otherwise the angle between them is 180, and cos(180) is -1. It's one way to think about it, but probably not very useful.
I strongly believe people first realized negative times negative is positive before defining rings that were probably inspired by that fact. It's like saying "why are wheels round?" and then presenting how cars drive on round wheels and showing that it's much better than square wheels.
This presents the mathematical demonstration not the historical reasons lmao. The people didnt "realise" negative times negative gives negative, but they prooved it using maths that can be, and where later seen as a part of set theory.
@@butwhoasked1821 Rings can be modelled in set theory, but do not require set theory. (One needs the naive notion of a collection on which to define (say) closure, but this need not entail the theoretical framework of ZF, and can even be done absent talk of collections altogether by introducing types.)
@@butwhoasked1821 All knowledge builds up on previous knowledge, so historical progression matters. If there was no ground to "realize" negative times negative is positive, they'd never carry it all the way to the rings and set theories. I hope I was clear enough
@@butwhoasked1821 Brahmagupta is the earliest known person to give rules for multiplication involving negative integers. Brahmagupta viewed positive numbers as fortunes/gains and negative numbers as debt/losses. Using these meanings, it is entirely reasonable to say Brahmagupta "realized" a negative times a negative is a positive, based on what he was using his numbers to model. In the end, he did define the rules for multiplication involving negative numbers, but this definition was based on what would accurately represent the scenario he cared about. From our modern perspective on mathematics, we can delve deeper into abstract reasoning to show why Brahmagupta's original definition is so robust, but doing so does not negate the history.
I think of negative numbers as opposites. If you multiply 2 by 2, you get 4, but if you multiply 2 by -2 you get the opposite of that which is -4, if you multiply -2 by -2 you get the opposite of *that* which is 4 again.
Exactly. It simply follows from definitions of positive and negative numbers and definition of multiplication. Why is negative times positive a negative? Because 'negative times' means changing the sign (opposite). Changing the sign (opposite) of negative is positive.
While teaching high school math, I explained it with a vector example (without using the word vector .) Forward is positive, reverse the gear and it's negative, reverse the reverse and you're back to positive.
I'm partial to an approach that to some degree combines the two examples you mentioned with the car and the complex numbers rotiation -- one where we imagine treating arithmetic as placing or removing arrows on the number line, sort of as simple roadmap instructions on how to arrive at the answer by simply counting our way there (starting at 0, a rightwards arrow with size 2 plus a leftwards arrow with size 3, brings you to -1). Multiplication is handled by treating it as statements of how many sets do we have of some arrow. A crucial point to make it work is to also state that we can remove an arrow even if it isn't explicitly said to be there - we'll just assume that it was added previously. That is to say that there's no functional difference between "go 3 steps to the right" and "assume that you had earlier been told to go 3 steps left, now undo that" - if you were at 0 when you got either instruction, both of them would bring you to 3. How does it work for multiplication? (+2)(-3): Start at 0, assume that the vector in question has size 3 and faces leftwards - place down such a vector two times. We end up at -6. (-2)(-3): Start at 0, assume that the vector in question has size 3 and faces leftwards - remove such a vector two times. We end up at 6. In other words, we can treat one operator as whether we are adding or removing an arrow, and the other operator as the distinction for which way the arrow points. This is no formal proof of course, but I've explained this way of thinking about it to kids at the junior high/middle school level who struggled with grasping the intuition behind multiplication with negatives, and almost without fail it's like a disco ball gets electrified behind their eyes. I imagine it's possible to express this rudimentary vector-arrow-simplification idea in terms of the Peano formalisms, though I don't know how rigorously (S is equivalent to a rightwards arrow, the inverse of S is equivalent to a leftwards arrow, both have the size of the unitary, etc). But whether it's workable from my starting point or not, isn't it possible to construct the same proof as you've done in this video using Peano, and if so, wouldn't that on some level possibly be "even more" mathematical?
I didn’t understand the (-2)(-3) example where you said to “remove the vector two times.” What do you mean by “remove”? I followed everything you said up to that point. I like this idea since I am also a school teacher, and visuals are best for young learners.
Excellent video, and I love the way the proofs were done systematically to lead us to the whole point of the video. I wish I had this channel back in college to help me with my math major!
My assumption before watching the video: using distributivity and commutativity, 4=(-1+3)(4-2)=-(4-2)+3(4-2)=-4+(-1)^2*2+3*4-3*2=2+2*(-1)^2, if (-1)^2=1 then this is correct. Sorry for the bad example though.
Ah, I see you're letting it handle itself by defining that single given instance, and assuming one could do that, I like that. You could use the cyclicality of i to a power to justify that, at least, so I'd think. Given that i^5 is i again, it implies the negative canceled to produce a 1*i.
When I taught K-4 I used the debt analogy. If you have a debt you "have" a negative quantity of currency. Canceling debt is negation of a negative. If one clerk cancels your debt and a second clerk cancels it too that is cancellation 2 times. I owe $5 so I "have" -$5. The first cancellation gives me +$5 so I pay my debt and I'm back to zero. -$5 + {-1 x (-$5)} = $0. Cancelling (negating) debt two times is multiplication by -2. So -2 x -$5 = +$10. My balance went up by $10 from -$5 to +$5. (-$5) + {-2 x (-$5)} = +$5. No need to get into group, ring, field algebra.
I'm sorry. I did start thinking long time ago like you did. I first owed $5 then I got bogged down because with the first cancellation, I saw that I did not owe $5. I failed to see why I have $5.
@@jeffleung2594 The story is that two clerks saw the cancel-debt order and both "cancelled" the same debt without the other's knowledge. To cancel debt is to add. Negating a debt is subtracting a negative. This was done twice. -2 x -$5 = +$10. The original balance was -5. 10 (2 cancellations) was added. -5 + 10 = +5. Clearer?
My way of explaining to my kids is that to remove something is a negative. And a debt is also a negative. To negative times a negative is I’m removing a debt. Which is positive !
The complex number explanation is particularly good because there is evidence that brains actually understand negation using wetware for rotation. That is, the wetware that controls and understands turning around and walking in the opposite direction.
Nice presentation, except that background music distracts from learning. Please don't get sucked into those anti-education patterns of presentation that are so popular because people like to play with technology.
I imagine the base rule for this is simply (-1) * (-1) = 1. And I imagine it's because in any ring, the rule is that if you multiply something by -1, you're taking 0 (the additive identity) and subtracting the number. So it boils down to explaining that when you subtract a negative, you're adding a positive.
This video makes something so simple extremely complicated. It is about double logical negation. A negative number represents the opposite direction. Multiplying two negative numbers means the opposite from the the opposite. Like: I'm not going left = I'm going right (assuming I am going somewhere). Or: I don't have credit at the bank = I have saving account at the bank (assuming there is a financial obligation between the bank and me).
Engineering lecturer here: this a good introduction to rings but does not attempt to explain why we would want integers to follow the rules of rings. I find the most satisfying explanation to relate to a personal bank account, where someone can make deposits but others can take debt tokens to the bank as well. This way negative numbers are introduced. Then, the bank can offset the negative tokens with positive deposits, reducing the total number of tokens in the account. Lastly, we allow for interest, so that debts or deposits can grow through multiplication. In setting up this sort of system we can motivate for the axioms of rings and then later give them the appropriate name.
Computer Science lecturer here: Agreed. The way I see it is that this framework is suitable to a wide range of mathematics and other fields' purposes (which usually is as good as we can get for really fundamental stuff), but saying: because this is leads to a good framework for X, Y and Z would make for a really short video. Though honestly, what really bothers me is the clickbaity language.
They have shown us all those proof at first year of my College. I regret it was shown so late, because it's fascinating to understand why math works like it works. I understand why it's not shown in primary school, but I think it should be taught in middle school.
I think the main problem with math education is that it focuses too much on the applied side too much. so students are taught basic algebra just to be able to learn calculus so they can be useful in engineering and finance and such. to be honest, I believe that is probably the best solution for an average person though, and we who are actually curious about how things work are few
@@EpsilonDeltaMain You are right. Even tho I'm not super into math, my college had a lot of it (Computer Science). And finding out for the first time, why basic math works like that was really interesting.
@@EpsilonDeltaMain I think your perception is greatly skewed. The vast majority of humans are stupid and cannot even learn algebra, let alone calculus (even though the intuition of calculus is super easy). I was always in the higher math classes, and even then, a bunch of people didn't get anything They don't even teach negative numbers until 7th grade now; math education is in the dumps; same with science education. It is as if they want a dumber population The people that are actually interested in this stuff learn on their own; I haven't really learned anything from school and just taught myself everything. Why wait to college to start 'real calculus,' when I can teach myself calculus at age 13, then real analysis at 14? Then I got a differential geometry textbook at 15 and was already solving problems in math and physics at what would be considered real college level and not those stupid AP classes I think the entire bachelor's degree in college is actually just high school level, if they didn't dumb everything down so much. And for those that don't go into higher learning, they should do trade skills and stuff that is actually useful to society. not everyone needs to know how stuff actually works So high school should end at age 13 now, then what is considered college should be 14-17, or those that do trades can learn trade skills, then be capable of working at 15 (two years of trades would be more than sufficient). Then, by the time we go to university, we would essentially be pursing PhDs I even thought college was way too slow and a waste of time and I finally felt that undergraduate stuff should personally be at the level of what graduate work is now
@@EpsilonDeltaMain I agree. The argument often goes that math like this is too abstract for students if they already struggle with the concrete stuff. I think it is the other way around. I think some students struggle with math because the rules and methods that you learn seem arbitrary and there is little understanding of _why_ it works. As an example I was tutoring my cousin and she was struggling with derivatives and at first I did not understand why this seemed so hard for her... at some point I figured out it was because she had no conceptual understanding of what a function is. Once I've explained that and why the rules are what they are she had no trouble with even the more difficult exercises.
@@Kaepsele337, yeah you're right. If maths were taught better with the concepts clearly explained I'm certain a lot of people would see that it's not that hard.
I always had a passing interest in math but never really entertained that interest, and always wondered about quirks like these. I liked I was able to understand the explanation too despite my lack of knowledge in the subject.
I am not sure what I was hoping for from this video, in the end it helped me to clear out what I already knew: Mathematics are a set of rules chosen because they fulfil certain requirements better than others, the explanation lies on an axiom that can be seen either as true or false but that in itself is void of reason other than its suitability in a set of operations. I am not a mathematician and I've learned axioms and their consequent statements almost by heart as they are useful if not vital on my everyday life and work as an architect. Through my experience in life maths has been a language to express and eventually solve problems more than anything else and as such (a language) its basis are defined to better suit and define the expression of problems involving the physical world that surrounds me. In a way I speak fluently English, Italian and Spanish whilst I only dabble in Mathematics.
as an architect you are not confronted with math but merely with calculus as any engineering profession. Math is the study of "essential relationships" stripped from all else what can disguise that. But this is my personal description for it. And it is not meant to cover all aspects of Math
The only reason arithmetic and algebras such as ring theory are widely known (albeit in the math community) is that it has a practical engineering and scientific use.
Follows from the distributive law and 0x = x•0 = 0 (which also follows from the distributive law): Assume A + a = b + B = 0. Then AB = AB + aB - aB = (A + a)B - aB = 0 - aB = a(b + B) - aB = ab + aB - aB = ab Note that x - y is short for x + (-y) here. Lemma: 0x = 0x + 0x - 0x = (0 + 0)x - 0x = 0x - 0x = 0.
I like this a lot, though it may need some explanation. What cmilkau is noting is that one can proove that A * B = (-A) * (-B) using the distributive law (Where -X is the additive inverse of X). They need 0x to equal 0 for the proof, which they also proove in the Lemma using the distributive law.
When I was a kid we were told to think of multiplying as " lots of".... so 3 lots of 2 = 6. So if we say, 3 lots of -2 , that will = -6 because it is -2 + -2 + -2. So -3 lots of -2 must = 6 because it is 3 negative lots of -2. So the double negative being a positive analogy is very good.
The classical definition of multiplication that Descartes gives in the Geometry is based on simple proportion theory. It can be shown, using that definition, that a negative times a negative equals a positive. It is really fascinating that you dealt with the question using ring theory! Cheers!
I was inclined to agree with you. I like the video's explanation, but like you said (if I'm interpreting right) it's just about thinking about it in terms of area being a positive manifestation (I guess I'm at a loss for an official term, but with magnitude of size being only able to be described in absolute terms, I just took it as it being multiplying coordinate values in a point and accepting that area ought to be positive. Sorry if this comes off as sounding like some pedantic rephrasing.
Absolute values are one way of thinking about it. If you would be interested, I can email you a short explanation of the definition, and how it applies to negatives.
The thing this is missing is the justification that "negative" and "additive inverse" are the same or related. The concept of "negative" has to do with ordering, which isn't mentioned at all. If you have that the additive inverse of a positive number is negative, then you've got a complete proof, but otherwise you've just proven a fact about inverses and not negatives.
@@turtledruid464 No, the term negative for integers is defined by the order. A negative integer is an integer which is less than zero. The fact that the additive inverse of a positive integer (one which is greater than zero) is a negative integer is a theorem, not a definition.
@@zapazap Ordering is completely independent from any algebraic properties. You can have either one without the other, both together, or neither one at all.
I was ordering a submarine sandwich at a shop, with my girlfriend who was doing likewise. I was surprised to discover that ordering sandwiches is a non-commutative operation. She is fussy about the preparation of her sandwich. In particular, she does not want her sandwich "contaminated" with pork products. So if I go ahead of her in line, and order a ham sandwich, she will ask the server to change his plastic gloves before he prepares her double-meat turkey sub. But if she goes first, I can order my ham sandwich and there is no need to ask the server to change his gloves. I have also discovered that ordering a meatball sub DOES commute with ordering a turkey sandwich, since the server does not have to handle the meatballs with his gloved hands. Rather he can use a ladle to add the meat, so no change of gloves is necessary. Which just goes to remind us that though a group may not be commutative for all pairings in a multiplication, there can be some elements that do commute. I believe multiplication by the identity element is always commutative, since I think that every right-inverse is also a left inverse. (Am I right?)
I would, too. 👍 But, I will not subscribe, because while I would like _some_ of your videos, not all of this kind of material "Will fit easily into my brain", so to speak. 🤔➡️😲➡️💥
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Even though explaining the axioms is important, the proper way to think about this is because we created math as a logical language. Setting true and false statements all the time, hence a false-false statement is a true statement. Remember that math is a language to proof propositions based on logic, and that this logic leads to paradoxes as well. Math is a powerful tool but is not perfect.
This is a great video but it definitely doesn’t help me how I should explain it to my students 😅😅. 1-(-1)=1+1 is not so bad: imagine you have a glass of ice water. The ice has a negative temperature and the water has a positive temperature (at least in Celsius, luckily I don’t live in Fahrenheit land). If you remove the ice, the average temperature goes up because you removed something negative. If you add a negative you are adding ice to the water in the thought experiment. It’s not a perfect analogy as it breaks down for -1-1 for example. For addition of negative numbers, a thermometer is an easier tool. It’s -1 degrees and then gets one degree colder, how cold is it? -1-1=-2
1) negative * positive = negative Simply because, in the same way that 3 * 7 = 7 + 7 + 7, then 3 * (-2) = (-2) + (-2) + (-2) = -6 2) negative * negative = positive Just plot 3 * (-2), then 2 * (-2), then 1 * (-2), then 0 * (-2) on a line with a graduation and think of the most logical place where (-1) * (-2) should be.
The "prime the intuition" example I always use is with money, specifically debt. If you're one dollar in debt, then you have a net of -1 dollars. To "remove" that debt - getting back to net 0 dollars - is to subtract that debt. Thus, in dollars: -1 - (-1) = 0. But another way to remove a one dollar debt is to gain a dollar, so -1 + 1 = 0. Thus -(-1) = +1. You can either add one dollar to remove a one dollar debt, or equivalently, you can subtract the one debt to remove it. Subtracting a one dollar debt is the same as adding one dollar to a one dollar debt.
Negative times a negative equals a positive can be reasoned out in a much more simply way. Consider (2)(3) this is equal to 6 because 3 + 3 = 6, we can also do 2 + 2 + 2 = 6. Now consider (2)(-3) this is equal to -6 because ( -3 ) + ( -3 ) = - 6, since (2)(-3) = (-3)(2) then (-3)(2) does the same as - (2) - (2) - (2) = -6. We usually think of repeated addition for multiplication, but as you have seen when the factor is a negative it basically performs repeated subtraction. This being said, it makes sense that (-2)(-3)=6 because - (-3) - (-3) = 6 . "There is also a simple explanation of why subtracting negatives does the same as adding positives." By the way I like your introduction of Ring Theory (easy to follow along) and how it can be used to show that a negative times a negative is a positive, although this explanation would scare off most people.
Because that way has been accepted at peer reviews as convenient for general use across mathematics using common multiplication. Multiplication can be defined other ways as we know but for most people it is good to think about it and ponder why learned people decided on that option and why it has existed for so long. If that rule failed it would be a major revision across the whole of mathematics?
I recall in high school, in algebra class, that there was a longish discussion about this. One of the objections raised by students was that in any multiplication that consisted only of positive integers, the correct answer could be obtained by simply adding the right side as many times as indicated on the left, and this statement could be reversed (commutatively), so that for a addition of a+b the answer could be arrived at by adding b to itself a times, and vice versa. You could step this answer out on the number line with dividers if you wanted. But this ceased to work as soon as you tried using even one negative. The teacher actually became frustrated and finally simply said "these are the rules."
@Artem Down The point where things got confusing is when you consider -5 X -4, which = 20, rather than -20. Negative signs are always "relative" to other negative signs. The survival of the negative depends on the number of negative signs in a multiplication, but not in addition. One mathematics teacher said that rather than an actual mathematical necessity, the action of negatives was grammatical. In Greek, a double negative is a negation of the initial negation. But in Old and Middle English no such rule operated. But by the Reformation there was a concerted effort to adjust English grammar to more classical lines in agreement with Greek and Latin. So, when as a kid we said something like "ain't no" a teacher would jump all over us for uttering nonsense, but it was really an older grammatical pattern. It would be interesting to think about a mathematical system that developed under different rules.
@Artem Down You are applying that originally grammatical rule. Consider what the result would be if the direction of movement along the number line were fixed as it is when the operation is addition of positive numbers. The negative sign is always relative to the other negative signs. I'm not disputing your result, because that is indeed what our mathematics teachers would mark as correct. I'm pointing out that + and - have different behaviours relative to themselves. Addition has only one behaviour on the number line unless a negative is present, but negatives _always_ modify the behaviour of other negatives, depending on whether there is an odd or even number of negative signs. In some of the ancestral languages of English "not not" was equivalent to emphasizing the negativity, rather than negating it. That is "not not" would be more like "very not." It's why English possesses "nor." It agrees with a prior "neither." These days people often use "or" in conjunction with "neither," but that isn't or at least wasn't a grammatical construction. That's hard to wrap our heads around now, but that is because grammar fanatics trained in classical languages insisted on "rationalizing" English to make it grammatically more like classical logic and emerging mathematical logic. The relics of previous structures linger as "bad grammar," which you chide your children for using. What I wonder about is what a mathematics would look like where the negatives had excatly the same rules as the positives. In the normal world you can't have a negative amount of anything real. Money is imaginary, and you cannot have a negative amount of cash, so put that to rest. Economics would look very different.
@Artem Down You do not seem to catch the point. I'm not arguing your results are wrong. I am saying that the mathematics we employ were evolved under a set of grammatical rules which are not native to English. "What would a mathematics evolved according to different grammatical rules be like," is the question I asked. Answer that and forget the simple minded arithmetic. You need to remember that all those simple initial rules are postulates. That is, saying that a negative times a positive is always negative and that a negative times a negative is always a positive are unprovable postulates similar to the definition of parallel lines, which is "true" only in Euclidian geometry but cannot be proven using Euclidean geometry. We do arithmetic that way because that is the set of rules we adopted and found useful. They are no profound philosophical truths.
Logically, a positive number multiplying a negative number is negative: e.g. 3 X (-10) = 3 'minus tens' = (-10) + (-10) + (-10) = -30. So dividing both sides by -10, +3 = (-30)/(-10) = (-30) X (-0.1). Also, the product of two negative numbers is the product of their absolute values times (-1)^2: (-a)(-b) = (-1)^2 ab; a, b > 0. If we make the 'choice' that the product on the left is negative, then (-1)^2 = -1, which results in the contradiction that -1= +1.
B4 I watch the video, Ill say how I visualize it and usually how I try to teach my students with a good degree of sucess Id say Think of numbers in a 2d plane where the y axis r the imaginary numbers and X axis the real numbers. When you take the natura number 1 and you multiply by -1, this is a 180 degrees rotation, landing you at the real axis, but in the opposite side of the axis, in the negatives: -1. So if you are a negative number -5, and you multiply by -1, a 180 degrees rotation lands u at positive 5. This also works great to understand complex numbers. If multiplying by -1 is a 180 degrees rotation, and i is the sqrt of -1, multiplying by i has to be a 90 degrees rotation. Hence why all the normal numbers make a plane, not a line. Its obviously not the full story and visualization requires rigorous proof , but I remember the 1st time I began to understand this when studying trig representation of complex numbers, and something just snapped like finally trigonometry and complex numbers made sense to me. And even the idea of negative times a negative is a positive. And it made me fell in love with it, like damn math indeed is beautiful
I see -1 more as a reflection across the origin. In the Real numbers, -1 acts as a mirror across 0. (And the reflection of a reflection is the original object.) Where Complex numbers come in is when trying to find "half of a reflection." x² = -1 is asking what operation, when done twice, gives a reflection across 0. One idea would be to just travel half the distance, but then you land on 0, which halts your progress for the next step. Instead you have to go _around_ 0, and when you expand your view to a 2D plane so that you _can_ go "around" 0, what looked like a mirror reflection before now also looks like a 180° rotation, and half of 180° is 90°. i² = 90°² = 180° = -1 Of course, in 2D you can go around 0 in two different directions, which give 90° = i and 270° = -90° = -i = i³, both of which square to -1. Adding additional dimensions gives even more paths around 0, and by extension more roots of -1.
Lol "These examples look outside in without fundamentally justifying why it's the case" *proceeds to list unjustified properties of an unrelated mathematical construct called ring and explaining through the lens of its properties* Justification implies some sort of goal/reason, I didn't catch any except "I like rings and their properties and want to show them to you", but that is your reason not "the real reason" as the title says and it's just as good or bad as saying it's a complex rotation. If it's a real fundamental reason, I want to see it emerge as you build numbers and multiplication and explore the design space to show numbers can't exist without rings(which is of course false, there's many systems out there) so the question shifts to why we make the choices and how do they result in the concept called ring. Whereas this video is backwards, boring textbook style, here's a thing and its properties let's see how it solves our problem, the properties are even numbered before shown, completely disregarding that someone had to choose and formulate those properties. But does anything above matter? No, as you said (-2)*(-3) can be anything you want and just like your video. Just know that by saying "real reason", "fundamental", "define any way you want" instead of "rigorously applying a mathematical framework I like" you attract my demographic which finds the video comically bad, dislikes, unsubs and leaves.
The real world and mathematics are two distinct things. The reasoning here is that multiplying negatives is not easily justifiable with real world examples, however we can, using mathematics, identify the main properties of integer summation and multiplication, which people have no problem justifying as it is quite intuitive in the real world, and reason what multiplying two negatives would to make sense in order to keep those properties. It is very common in mathematics to try and identify the main structure of the concepts we are studying and focus on the structure itself rather than the original concept. You are right, however, in that multiplying negatives could amount to anything we define, if we so wish. Unfortunately most of the mathematical concepts we take for granted since we learn them as school children, like rational numbers, continuous functions, and especially real numbers, are entirely unnatural and have a lot of mathematical heavy baggage behind them. Even the way we represent any number in base 10 has some heavy baggage, like number theory, infinite series and convergence. Mathematics is entirely unrelated with the real world, and these "simple" concepts we take for granted need the "boring textbook style" to be explained properly and for us to have some insight on them. Believing that you can understand mathematics fully without actually doing mathematics, just with analogies to the real world and simplifications, is akin to believing in magic.
@@trewajg I agree and I think you misunderstood my comment. The boring textbook style I refer to is using outside knowledge in the form of already defined properies(rules) which magically fit whatever problem in the intro. I want to see the design space of those properties explored, how choices were made and coming up with formulations for the properties(whose number you don't even know). If you don't go through this process of discovery and exploration and instead start with the framework you know it works, whatever is described is confirmation bias, not fundamental or the real reason.
What bugs me about it is that he tries to build this complex proof to show how the negatives can be pulled out and canceled using these ring properties, yet the very first theorem he shows looks as if it presupposes the property: (-a) + -(-a) = 0.
@@xxportalxx. that one comes from the fact that every element in the ring has additive inverse. The inverse property of addition. Every element has its respective negative. No presupposition there.
@@HaramGuys yes it derives from that, however the additive inverse itself is just a + (-a) = 0, the idea that the inverse of an inverse is the original seems to me to be the entire core point being discussed here, so imo he basically jumped the most important bit. Proving the rearrangement is trivial imo, the core idea is that -(-a) is a. This is the whole reason the 180° rotation explanation works, the product being the multiplicand scaled by the multiplier is all well and good, it's how the negatives cancel that was being discussed.
Because going the opposite of going the opposite side of North is still going North. Because the opposite of me owing you 3 dollars is you owing me 3 dollars. It doesn’t have to be complicated, only what you make it to be.
It's worth keeping in mind that this video is not primarily designed to teach people why a negative times a negative is a positive; rather, it is designed to be an introduction to ring theory (a subdiscipline in abstract algebra). Thus, this video is looking at the question "Why is a negative times a negative a positive" through the lens of ring theory. I think the title is clickbait.
i think a simpler answer is that.. taking the inverse (negative) of an inverse.. reverts/restores it back to a natural number/positive real. its pretty good to think about these things though
Ok so my take on why negative times negative is positive. In number line, we can define 3×5 as 3 steps forward from 0 for 5 times then it lands on 15. Negative multiplication can be defined by backward step like 3×-1 equal 3 steps backward for 1 time equal -3. So now for -ve times -ve means negative step backward, like -4×-5 means (4 step backward) for 5 times in backward direction, so it just cancel both backward and give positive direction. Any suggestions are welcome.
I kind of expected a defining line why it is the way it is, but I guess thisworks too. I’ll stick minus times minus as ”Minus reducing itself” as a go to explanation.
I really appreciate this explanation because I didn't know about rings and should learn more about them. I thought I had an intuitive geometric explanation for -a*-b > 0, but I never really though to question it too deeply and thinking about it just now I realized it was, no pun intended, circular reasoning.
Number line, for visual purposes. Starting at 0. +2, that's 2 to the right. +2, that's 2 more to the right, totaling 4. +2, that's 2 more to the right, totaling 6. We see how multiplication is addition a number of times we decide, in this case it is 2x3=6. So what is moving to the right negative 2 times? Negative removing, so we're moving left 2 times. We can do that three times also, (-2)x3=-6 So what if we want to negative, or remove, the negatives? Well if you're at -2 and you want to remove the -2, you move right, to 0. If we're at 0, and we remove the -2 three times, we're going to the right in three sets of 2. (-2)x(-3)=6 Who just memorized this, it's so simple and logical. The rules we use should describe the world we're observing. That CAN be a fictional world, but in the case of the everyday math that most of us use, it's going to relate to something probably tangible.
The way it was explained to me was that a positive times a positive is positive, so therefore, a negative times a negative is positive since it’s the same integer.
Ultimately "because it works." Consider: 0 + a + a + a = 3*a 0 - a - a - a = -3*a a + (-a) = 0 -3*[a + (-a)] = 0 (-3)*(a) + (-3)*(-a) = 0 (-3)*(-a) = -(-3)*(a) Let a be any positive number. From line 2 we have that -3*a is a negative number. From the last line we see that -3 times -a must be a positive number. Unless this is the rule then the consistency of the whole structure of arithmetic breaks down.
If you borrowed a sum of money at a negative interest rate, wouldn't that mean you were multiplying your negative equity by a negative and ending up with a positive - you'd make money on the deal?
If you graph and make squares out of (2, 2), (2, -2), (-2, 2), and (-2,-2) and make the square, the line through the origin gives slope, such that when (+,+) and (-,-) are positive slopes, and (-,+) and (+,-) are negative slopes. I would thing that is the most simple, most visual, and most intuitive way to show it to a child.
I find a little bit disturbing that he said the explanation with complex numbers is not a correct mathematical explanation... The complex numbers just shed light on the reason (-1)(-1) is 1 with numbers that we use on an everyday basis. It gives some geometric intuition and, an explanation to the identity (-1)(-1) = 1. There is no such thing as 'The Real Reason' for any concepts. Mathematicians are searching for any insights on their work and all any bit of information is a valid explanation. We should talk about 'More profound reasons', not 'The Real Reason'. That would be a more accurate choice of wording.
Using complex numbers explains this so much more easily Every complex number has an angle in radians, in which 0 represents positive real numbers, π/2 represents positive pure imaginary numbers, and π represents negative real numbers. There is an important exception that every angle can have 2πk added to itself (where k is an integer) and not change the number, because 2π represents a full rotation. For numbers with an angle of 0 (positive real numbers) is identical to numbers having an angle of 2π, 4π, -6π etc When you multiply any 2 numbers together, you add their angles together, meaning: i*i -> θ = π/2 + π/2 = π, meaning multiplying 2 pure imaginary gives negative -1*-1 -> θ = π + π = 2π, but 2π is the same as 0, meaning multiplying 2 negative gives positive
Or I suppose an even quicker way to sum it up would be: "negatives represent a halfway rotation, two halfway rotations represent a full rotation, which is positive"
@@glitchy9613 you use properties of complex numbers to prove the property, with which there were defined property of complex numbers. You can't do this. This isn't how proves work
@@glitchy9613 like wtf you use properties of complex numbers .. .Do you know that those properties are not the definition of complex numbers? Those properties were proved using much simpler properties like this one
@@glitchy9613 The idea that I think F_A_F 123 is getting at here is the following: How do you know that complex multiplication represents rotation in the complex plane? Don't you have to use ideas like "a negative times a negative equals a positive" in order to show that to begin with? Don't we also need "a negative times a negative equals a positive" to even have a need to define i to begin with? Suppose we lived in an alternative world where the alternative "real number" multiplication was x⊙y = x|y| (using standard multiplication of x and |y|). In this case, the "alternative square" of the real number x would be x|x|, and every real number would have exactly one real number square root. (If x is nonnegative, then the standard √x would be the alternative square root of x, and if x is negative, then the standard −√x would be the alternative square root of x.) Now, it's true that this alternative multiplication wouldn't have the nice arithmetic properties we've grown accustomed to, but what if things developed like this? There would be no need to introduce a new dimension to represent square roots of negative numbers since they would already exist on the number line. So, from that perspective, the motivation behind developing complex numbers in the first place is because a negative times a negative is a positive. Does this make sense?
Math videos like this introduce things like how religious people introduce their stuff "no those explanations are wrong because of their assumptions, now let's take my assumptions"
This video is solid for a mathematician, but it is too abstract for basically the rest of the world. I was expecting a video explaining what multiplication is and was curious to see if he would call it iterative addition and if so, how he was going to apply it. Also, this was a great illustration of how logic is often found independent of meaning.
Here is the simplest way I know to explain this in my opinion using -2 x -3, with the cow analogy. The first negative, in this case -2, represents how many you "are in the hole, or have lent out" of something, cows for example. Easy enough, right. Now, the second negative, in this case -3, represents a "decrease" in the amount lent out, rather than an "increase" in the amount that needs to be given back, which would be the opposite, +3. So, the first negative means "in the hole" as in you lost 2 cows, and the second negative is the amount that loss is decreased, meaning you now have a "gain" of 6 cows. Thus, a "positive" amount of cows. You can also imagine the opposite, on why -2 x 3= -6. If you again take the first negative, here -2, meaning again that you lost two cows, but then, say that loss is "increased" by 3 fold, you then arrive at -6, meaning you now lost 6 cows. So think of the second " - " sign as a "decrease" or "increase" relative to the first negative number. My 2 yen on how to think of it more logically rather than mathematically.
Think of it this way: Multiplication is a process of adding a value to itself a set number of times, and what is addition if that a positive term. So when we multiply an negative value by a negative value, we creating a set number of negative terms, which will be applied on the negative value inside the term, thus, since subtracting a negative means adding its positive. We are adding the positive of the negative value to itself for that set number of times.
Logically in words, without mathematical notations, (-2)×(-3) means.. For example You have a certain number of candies and You get a temptation of eating 2 candies, it happens 3 times but You control Yourself and Don't (1st negative) eat (eating = subtracting 2 candies each time which is the 2nd Negative) 2 candies for all the 3 times that You were tempted to.. So, when You're asked how many candies did You save, Your answer will be 6 candies.
a conflicting use of symbols occurs since we use the “-“ sign both to assign a negative to a number, as well as use it as a symbol for subtraction. In a geometric perspective, we use it to indicate the location of a number (i.e. 180 degrees opposite of the line natural numbers allign themselves in a square; or even more so a 3d representation like a sphere) and we use it to indicate a direction (of how an indicator moves from one number to another). From a geometrical point of view, numbers are nothing but locations, and tools to describe movements of an indicator through these locations. What exactly is an indicator is merely a defined point (or collection of points) in reference to zero. The meaning of zero can be related to space, for example the observer being the center indicator referred to as zero. Timewise the present moment is always zero. Metaphysically zero can represent the concept of Nothingness, though the nature of nothingness is hard to observe or proof - even to a Buddhist :) Of these three, since we perceive time to have a direction, moving from the present moment away from the past, we have a conditioned perspective of what is zero timewise. Zero meaning now. And as such backwards into time feels like going exactly backwards the way we came - if time viewed as a line, 180 degrees the opposite direction. But since we cant go that way, we assign a negative. “This happened at -2 minutes, and counting”. At quantum levels time doesn’t neccesarily move in one direction, nor does it only add 180 degrees opposite of its direction, as it’s second option. Time can move away from the present moment in ANY direction. Or rather: the present moment can be a result from both the past, the future or ANY direction in a sphere of possible directions of time.
I like the explanation that there are other possible rules for multiplying negative numbers, but that mathematicians found this rule to be more useful, so that's what we most often use.
The rules are whatever we make them.
yep, best answer that summarizes my intention as to why even introduce rings in the first place!! Since this video may have not been too explicit about it, you get a pin
Yes. I think this video explains why they are useful very well. If we want to define addition and multiplication as we do in a Ring, a negative times a negative is positive. He even goes through a series of proofs to show why this must be the case.
I guess EpsilonDelta could have spelled it out a bit more explicitly at the end, though.
@@EpsilonDeltaMain Hey thanks. I think it was clear in your video. I was just commenting my favorite bit.
Interesting take. Makes me think of what are necessary and sufficient conditions for an argument to be considered explanatory as opposed to illustrative. This dovetails into Godel’s Inconcompleteness theorem does it not?
I liked this introduction to rings, but this doesn't explain "the real reason why", it just shows that it must follow from several other properties we choose. This begs the question - what is the reason we choose these properties (distributivity, etc)? The answer is that it's properties we like as they describe things we want to describe, but the same reason justifies just choosing negative*negative=positive as an axiom on its own (with some other stuff) and then proving distributivity (or some other ring axiom we didn't include in this axiom set). Therefore the "real reason" is the car example at the start - because it describes natural things that we want to describe.
I think the only way to explain why a negative times a negative is positive is from some set of axioms. This answers "the reason why", but you say you want to know "the *real* reason why". That just results in trying to ask "why are the ring axioms the way they are", and you could try and find some even more fundamental axioms to prove the ring axioms from, but then you are still left with the question of "why are *those* axioms the way they are?" Your line of questioning is a never-ending regression of "what axioms can I use to prove the axioms I already have?". All you care about in math is the consequences of a set of axioms, not the reason they were chosen, although you may care to choose good axioms.
Okay that makes more sense. I was confused why this would be a better "why". That said, I always get suspicious when someone claims that there is a "why" in math - "why" requires an alternative, another way it could have been, some other axiom that could have been used, and a coherent reason to believe that the axiom really is the root and that replacing it should be assigned logical causality. the only "why" that makes sense to me is the evaluation order of the person or computer working through the problem - since math is made of equivalences and nonequivalences you can always go from anywhere to anywhere, afaict?
That's what he says in the beginning. It follows from the axioms we choose.
This is how math works. You creat a set of axioms that satisfy your needs. Math doesn't really tell reasons, it just proves statements.
@@tunafllsh well if you choose the wrong axioms…..?
I asked this question to my maths teacher in 5th grade and she probably couldn't understand that I was trying to think abstract and she thought I was dumb
Grade school teachers despise math so much, I'm surprised we have anyone that can use math in spite of it.
One of my students said, "It tickles my mind that a negative times a negative equals a positive."
Aww so cute!
You can also prove it using Peano's arithmetic.
The natural numbers are defined as a set that contains an initial element that we call 0. And there's a function S (for "successor"), where for every natural number *n* , *S(n)* is also natural; and *S(a) = S(b)* if, and only if, *a = b* ; also, no natural number satisfies *S(n) = 0* .
Given that, the natural numbers are {0, S(0), S(S(0)), S(S(S(0))), ...}, also known as {0, 1, 2, 3, ...}. Note that every natural number is either zero or a successor of another natural number, so we can use that to define the possible operations on this set.
Addition can be defined as:
a + 0 = a
a + S(b) = S(a + b)
And multiplication can be defined as:
a * 0 = 0
a * S(b) = a + (a * b)
Examples:
a + 1 = a + S(0) = S(a + 0) = S(a)
a * 1 = a * S(0) = a + a * 0 = a + 0 = a
By those definitions you can prove commutativity, associativity and distributivity, which will be needed for this proof. But it would be pretty verbose so I'm going to let it out of the comment (you can search it, though).
Also, you can define subtraction as simply as: *a - b = c* , if and only if *c + b = a*
However, Peano's arithmetic defines only natural numbers, if we want to extend it for negative integers, we can create an "imaginary" unit *w* (spoiler: we usually call it "-1") that by definition holds the property:
S(w) = 0
With that, we can simply apply the operations definitions:
a + S(w) = a + 0
S(a + w) = a
(a + w) + 1 = a
a + w = a - 1
a * S(w) = a * 0
a + (a * w) = 0
a * w = 0 - a
S(w) = 0
w + 1 = 0
w = 0 - 1
What would happen, though, when multiplying *w* by *w* ?
w * S(w) = w * 0
w + (w * w) = 0
w + (w * w) = S(w)
w + (w * w) = w + 1
w * w = (w + 1) - w
w * w = 1
Well, that's interesting, we just found another important property of *w* . Now, we're finally ready to prove that negatives cancel out on multiplication:
(0 - a) * (0 - b)
= (a * w) * (b * w)
= (a * b) * (w * w)
= (a * b) * 1
= a * b
Remember the spoiler I gave you earlier? So, we can use a simpler notation for "0 - n": we can simply write *-n* .
So, we can write *w* , or "0 - 1" as simply *-1* . And concluding my proof, we discovered that:
-a * -b = a * b
Thank you if you've read this far, if possible tell me what you think about this proof.
Ahhh yes, it is definitely cool system to consider, as there is a structured natural total ordering on N, and properties of the semirings are now theorems in the peano axioms.
Only problem I have with using peano axioms to prove negative times negative is positive is, well you need a video this long just to prove that 1+1=2 first in peano axiom, then extend the ordering to allow subtraction for all elements which is another video, and lastly, its use is exclusive to a totally ordered sets
I like your proof, but I think using Ring Theory is both simpler to explain and useful for a broader range of commonly-used mathematics.
For complete proof, you may check out the book Analysis 1 by Terence Tao
bruh
Nice 🤌🤌🤌
I have a Mathematics Degree with a Major in Statistics - so I shouldn't be commenting on this subject. That said, I would offer that as soon as we accept that a number like 2 exists, and it represents some distance from 0, it should follow that there are other directions away from 0 that must also exist. Humans have always had an innate understanding of rotations - we look left & right and we look behind us to see what might be trying to eat us - so the idea of rotating around our personal 0 has always existed - even if we didn't think mathematically this way, or could put it into formulas. Humans didn't fight the notion of mathematics that described rotations like we fought accepting that negative 2 bananas exist - no one accepted this for a very long time. If 2 represents 2 steps forward, then it is very easy to accept that we can rotate the +2 180 degrees to represent -2 steps behind us - but that of course opens the mind that we could have first rotated +2 by 90 degrees and then rotated this halfway stop another 90 degrees to get to the thing we are calling -2. Thus I am therefore going to DEFINE -2 to be a 180-degree rotation of +2 (which also happens to equal two separate 90 degree rotations). Let's pick a symbol (i) to DEFINE a 90-degree rotation - thus Mathematically -2 = 2 * i * i (2 rotated 90 degrees twice). I would similarly Define -3 = 3 * i * i. And now we are ready to ask the question, what is (-2) * (-3)? The Answer is 2 * i * i * 3 * i * i = 2 * 3 * I^4 = 6 * i^4 - and since i^4 is a 360 degree rotation back onto itself, (-2) * (-3) = 6 because we have rotated entirely back towards the direction that we have DEFINED 2 & 3 to exist in. I've used this explanation to explain these concepts to young children and they easily 'get it' - whereas if I ask educated adults why (-2)*(-3) = 6 - they really don't know why.
This is not actual proof because there is no proof how this system correlates with the algebraic and other systems we use .In this logic - x - is plus , but what is difficult to prove is the correspondence to algebraic system with your rotation system.
Why? Because (-) * (-) = (+) is most useful, so we defined fundamental axioms that lead to the properties we want once we use the logical inference of deduction on those axioms
yep, best answer that summarizes my intention as to why even introduce rings in the first place!! Since this video may have not been too explicit about it, you get a pin
Edit: oops, can only pin 1 comment at a time. the other person gave the answer first, so I would have to give it to them
@NikoR96 I am stopping reading after your "Six Apples" example to remind you that _Numbers themselves_ are also an invention of the human mind, and don't exist in reality, without someone to perceive the objects being counted....
Food for thought 🤔
@NikoR96 Also, I suggest you use another even number besides two for your example because two plus two equals four, but coincidentally two _times_ two equals four, as well.
@@nikor9640 You are introducing superfluous stuff that isn't needed. I honestly hate when people respond like this because it just shows you have no idea what you are talking about. You are literally saying 1 rock times 3 bugs doesn't make sense, therefore 3 * 1 doesn't actually equal 3 since you cannot multiple rocks and bugs.
My original comment is entirely correct, and you won't find a single mathematician that would disagree, because I just spoke straight facts. You had to introduce extra concepts, such as 'debt,' then you pretended that that is equivalent to the negative sign. You are basically introducing units, which is dimensional analysis.
You need to remove this from your thoughts and start at a lower axiomatic level. If you want to build upon rings and introduce a logical concept of debt, you could do that, but I honestly don't even understand why you even responded when it isn't relevant to the discussion whatsoever. It is incredibly ironic that you think your superfluous additions are 'more fundamental' than a literal first axiomatic approach
Love this video!! Proving little theorems just like this one, ones that seem trivial to pretty much everyone, is one of my favorites! Understanding these basic and "trivial" things is crucial for understanding more advanced mathematical concepts imo. It is quite unbelievable that this is one of your first videos on this channel, it's awesome! Looking forward for more!!
completely agree, makes things in higher math so much more intuitive if you understand these
This brings back memories of first year linear algebra for me.
"You are just proving that 0 = 0 - wtf?"
Perhaps you are talking about the dimension of "Imaginary Time", discovered by Stephen Hawking.
I wonder if it is why ordinary time only runs in one direction, in our universe?
This is an awesome video cannot wait till you become popular.
Fantastic video! I can't wait to see what the rest of the series has in store!
This is actually very eye-opening... many people think that math is naturally inscribed into reality, until they find out about Gödel's incompleteness theorems and other things, like the fact that the result of multiplying two negative numbers is actually agreed upon 🙂 Seems like another brilliant channel has been born just now 🙂
@@caoinismyname So you say if I multiply 2 apples by -3 I should get -6 apples? ;-) These nice "explanations" you posted are actually just analogies or metaphors, but many of such "math phenomena" have nothing to do with direct experience in reality as we know it. What's even more mindblowing is that through such abstract thinking you can actually end up with something real, something that happens in this world ;-)
@@caoinismyname Wow, dude, what an ego! Look, the truth is nobody cares who you are and what prizedid you receive in your petty math olympiad. There are no authorities in science. On top of your enormous ego, you've clearly misunderstood the video. The author indeed referred to the difficulties of even imagining negative numbers and not mentioning manipulating them in real life. Get off your high horse, kid.
@@caoinismyname perhaps we should stick to a more civil tone that knows nothing of trying to score points by ad hominem attacks?
@@caoinismyname So, for internal logic, it is useful to caution one against (1) being too full of himself (as you did) or (2) acting like a genius (as you did), while (3) posting one's awards won (as you did)?
@@caoinismyname not some math's ,most maths you can't give this debt concept in multiplication of two negatives
I mean, this rule is the most intuitive.
-5*3=-15 can be read as "I borrowed $5 three times, therefore I owe $15", it then stands to reason that a negative multiplier must have the inverse effect: -5*-3=15 "I lent $5 three times, therefore I'm owed $15".
Other rules can be defined, but this one maps fairly nicely to real life and the constraints thereof.
why would the difference of lending or borrowing lead to a sign change on the 3 instead of the 5?
@@davidbarroso1960 yeah, you're right there is a flaw in the argument. You can't lend or borrow for a negative number of times as they HAVE to be Whole numbers.
@@davidbarroso1960 Yes his logic is flawed but I have a refined version of it... Logically in words, without mathematical notations, (-2)×(-3) means.. For example You have a certain number of candies and You get a temptation of eating 2 candies, it happens 3 times but You control Yourself and Don't (1st negative) eat (eating = subtracting 2 candies each time which is the 2nd Negative) 2 candies for all the 3 times that You were tempted to.. So, when You're asked how many candies did You save, Your answer will be 6 candies
Or that you're -$15 down
It is intuitively much simpler. Negative numbers are VECTORS. They have both magnitude AND direction. Direction is defined relative to an origin POINT and a reference direction (usually the positive axis). The multiplication OPERATOR for vectors MULTIPLIES MAGNITUDES and ADDS DIRECTIONS.
This sounds like a better explanation, but is there a visualization for this?
Little elaboration would be appreciated.
Basically. Also the ring theorem he explained is essentially a vectorial space
You said "negative numbers are vectors". Are positive numbers vectors too? Thanks
@@pjay3028 It's possible to consider any number a vector; a scalar (standard number on its own) can be considered a one-dimensional vector. If you think of standard multiplication as the same thing as a vector dot product (probably the closest equivalent), one way to represent it is |v1| * |v2| * cos(angle between). Two positive or two negative numbers go the same direction, so angle between is 0, and cos(0) is 1. Otherwise the angle between them is 180, and cos(180) is -1. It's one way to think about it, but probably not very useful.
This simple explanation works for me:
2 x 2 = 4
2 x 1 = 2
2 x 0 = 0
2 x -1 = -2
2 x -2 = -4
1 x -2 = -2
0 x -2 = 0
-1 x -2 = 2
-2 x -2 = 4
I strongly believe people first realized negative times negative is positive before defining rings that were probably inspired by that fact.
It's like saying "why are wheels round?" and then presenting how cars drive on round wheels and showing that it's much better than square wheels.
This presents the mathematical demonstration not the historical reasons lmao.
The people didnt "realise" negative times negative gives negative, but they prooved it using maths that can be, and where later seen as a part of set theory.
@@butwhoasked1821 Rings can be modelled in set theory, but do not require set theory. (One needs the naive notion of a collection on which to define (say) closure, but this need not entail the theoretical framework of ZF, and can even be done absent talk of collections altogether by introducing types.)
@@butwhoasked1821 All knowledge builds up on previous knowledge, so historical progression matters. If there was no ground to "realize" negative times negative is positive, they'd never carry it all the way to the rings and set theories.
I hope I was clear enough
@@butwhoasked1821 Brahmagupta is the earliest known person to give rules for multiplication involving negative integers. Brahmagupta viewed positive numbers as fortunes/gains and negative numbers as debt/losses. Using these meanings, it is entirely reasonable to say Brahmagupta "realized" a negative times a negative is a positive, based on what he was using his numbers to model. In the end, he did define the rules for multiplication involving negative numbers, but this definition was based on what would accurately represent the scenario he cared about.
From our modern perspective on mathematics, we can delve deeper into abstract reasoning to show why Brahmagupta's original definition is so robust, but doing so does not negate the history.
I think of negative numbers as opposites. If you multiply 2 by 2, you get 4, but if you multiply 2 by -2 you get the opposite of that which is -4, if you multiply -2 by -2 you get the opposite of *that* which is 4 again.
Exactly. It simply follows from definitions of positive and negative numbers and definition of multiplication. Why is negative times positive a negative? Because 'negative times' means changing the sign (opposite). Changing the sign (opposite) of negative is positive.
While teaching high school math, I explained it with a vector example (without using the word vector .) Forward is positive, reverse the gear and it's negative, reverse the reverse and you're back to positive.
I remember my teacher doing the same.
I'm partial to an approach that to some degree combines the two examples you mentioned with the car and the complex numbers rotiation -- one where we imagine treating arithmetic as placing or removing arrows on the number line, sort of as simple roadmap instructions on how to arrive at the answer by simply counting our way there (starting at 0, a rightwards arrow with size 2 plus a leftwards arrow with size 3, brings you to -1). Multiplication is handled by treating it as statements of how many sets do we have of some arrow.
A crucial point to make it work is to also state that we can remove an arrow even if it isn't explicitly said to be there - we'll just assume that it was added previously. That is to say that there's no functional difference between "go 3 steps to the right" and "assume that you had earlier been told to go 3 steps left, now undo that" - if you were at 0 when you got either instruction, both of them would bring you to 3.
How does it work for multiplication?
(+2)(-3): Start at 0, assume that the vector in question has size 3 and faces leftwards - place down such a vector two times. We end up at -6.
(-2)(-3): Start at 0, assume that the vector in question has size 3 and faces leftwards - remove such a vector two times. We end up at 6.
In other words, we can treat one operator as whether we are adding or removing an arrow, and the other operator as the distinction for which way the arrow points.
This is no formal proof of course, but I've explained this way of thinking about it to kids at the junior high/middle school level who struggled with grasping the intuition behind multiplication with negatives, and almost without fail it's like a disco ball gets electrified behind their eyes.
I imagine it's possible to express this rudimentary vector-arrow-simplification idea in terms of the Peano formalisms, though I don't know how rigorously (S is equivalent to a rightwards arrow, the inverse of S is equivalent to a leftwards arrow, both have the size of the unitary, etc). But whether it's workable from my starting point or not, isn't it possible to construct the same proof as you've done in this video using Peano, and if so, wouldn't that on some level possibly be "even more" mathematical?
Brilliant!
I didn’t understand the (-2)(-3) example where you said to “remove the vector two times.” What do you mean by “remove”? I followed everything you said up to that point. I like this idea since I am also a school teacher, and visuals are best for young learners.
Excellent video, and I love the way the proofs were done systematically to lead us to the whole point of the video. I wish I had this channel back in college to help me with my math major!
Abstract algebra is the pinnacle of math fun.
My assumption before watching the video:
using distributivity and commutativity,
4=(-1+3)(4-2)=-(4-2)+3(4-2)=-4+(-1)^2*2+3*4-3*2=2+2*(-1)^2, if (-1)^2=1 then this is correct.
Sorry for the bad example though.
Ah, I see you're letting it handle itself by defining that single given instance, and assuming one could do that, I like that. You could use the cyclicality of i to a power to justify that, at least, so I'd think. Given that i^5 is i again, it implies the negative canceled to produce a 1*i.
When I taught K-4 I used the debt analogy.
If you have a debt you "have" a negative quantity of currency. Canceling debt is negation of a negative.
If one clerk cancels your debt and a second clerk cancels it too that is cancellation 2 times.
I owe $5 so I "have" -$5. The first cancellation gives me +$5 so I pay my debt and I'm back to zero. -$5 + {-1 x (-$5)} = $0. Cancelling (negating) debt two times is multiplication by -2. So -2 x -$5 = +$10. My balance went up by $10 from -$5 to +$5. (-$5) + {-2 x (-$5)} = +$5.
No need to get into group, ring, field algebra.
I'm sorry. I did start thinking long time ago like you did. I first owed $5 then I got bogged down because with the first cancellation, I saw that I did not owe $5. I failed to see why I have $5.
@@jeffleung2594 The story is that two clerks saw the cancel-debt order and both "cancelled" the same debt without the other's knowledge. To cancel debt is to add. Negating a debt is subtracting a negative. This was done twice. -2 x -$5 = +$10.
The original balance was -5. 10 (2 cancellations) was added. -5 + 10 = +5.
Clearer?
@@George4943 yeah, thank you😄
This makes much more sense than this complicated video.
@@dubio77 Yah, advanced math theory not needed.
When I learned what multiplying by i really means, negative times a negative made perfect sense. This is a nice explanation but above my level lol
I was thinking about this the other day. Cool video
My way of explaining to my kids is that to remove something is a negative. And a debt is also a negative. To negative times a negative is I’m removing a debt. Which is positive !
Wow! How old are your kids?! 😲
The complex number explanation is particularly good because there is evidence that brains actually understand negation using wetware for rotation. That is, the wetware that controls and understands turning around and walking in the opposite direction.
I don't know complicated stuff but if you are down sad crying listening to a sad song that you can relate to makes you a lil better
Nice presentation, except that background music distracts from learning. Please don't get sucked into those anti-education patterns of presentation that are so popular because people like to play with technology.
Long story short it's popularly agreed arbitrary decision.
That if you stick to the rules along with other arbitrary rules, you can predict the future with physics.
I imagine the base rule for this is simply (-1) * (-1) = 1. And I imagine it's because in any ring, the rule is that if you multiply something by -1, you're taking 0 (the additive identity) and subtracting the number. So it boils down to explaining that when you subtract a negative, you're adding a positive.
This video makes something so simple extremely complicated.
It is about double logical negation. A negative number represents the opposite direction. Multiplying two negative numbers means the opposite from the the opposite. Like: I'm not going left = I'm going right (assuming I am going somewhere). Or: I don't have credit at the bank = I have saving account at the bank (assuming there is a financial obligation between the bank and me).
Engineering lecturer here: this a good introduction to rings but does not attempt to explain why we would want integers to follow the rules of rings. I find the most satisfying explanation to relate to a personal bank account, where someone can make deposits but others can take debt tokens to the bank as well. This way negative numbers are introduced. Then, the bank can offset the negative tokens with positive deposits, reducing the total number of tokens in the account. Lastly, we allow for interest, so that debts or deposits can grow through multiplication. In setting up this sort of system we can motivate for the axioms of rings and then later give them the appropriate name.
What rules of rings would you consider superfluous for our naive use of numbers that did not involve multiplying negatives?
Computer Science lecturer here: Agreed. The way I see it is that this framework is suitable to a wide range of mathematics and other fields' purposes (which usually is as good as we can get for really fundamental stuff), but saying: because this is leads to a good framework for X, Y and Z would make for a really short video.
Though honestly, what really bothers me is the clickbaity language.
I cannot hear the background music without thinkinv of trash taste damnit
They have shown us all those proof at first year of my College. I regret it was shown so late, because it's fascinating to understand why math works like it works.
I understand why it's not shown in primary school, but I think it should be taught in middle school.
I think the main problem with math education is that it focuses too much on the applied side too much. so students are taught basic algebra just to be able to learn calculus so they can be useful in engineering and finance and such. to be honest, I believe that is probably the best solution for an average person though, and we who are actually curious about how things work are few
@@EpsilonDeltaMain You are right. Even tho I'm not super into math, my college had a lot of it (Computer Science). And finding out for the first time, why basic math works like that was really interesting.
@@EpsilonDeltaMain I think your perception is greatly skewed. The vast majority of humans are stupid and cannot even learn algebra, let alone calculus (even though the intuition of calculus is super easy). I was always in the higher math classes, and even then, a bunch of people didn't get anything
They don't even teach negative numbers until 7th grade now; math education is in the dumps; same with science education. It is as if they want a dumber population
The people that are actually interested in this stuff learn on their own; I haven't really learned anything from school and just taught myself everything. Why wait to college to start 'real calculus,' when I can teach myself calculus at age 13, then real analysis at 14? Then I got a differential geometry textbook at 15 and was already solving problems in math and physics at what would be considered real college level and not those stupid AP classes
I think the entire bachelor's degree in college is actually just high school level, if they didn't dumb everything down so much. And for those that don't go into higher learning, they should do trade skills and stuff that is actually useful to society. not everyone needs to know how stuff actually works
So high school should end at age 13 now, then what is considered college should be 14-17, or those that do trades can learn trade skills, then be capable of working at 15 (two years of trades would be more than sufficient). Then, by the time we go to university, we would essentially be pursing PhDs
I even thought college was way too slow and a waste of time and I finally felt that undergraduate stuff should personally be at the level of what graduate work is now
@@EpsilonDeltaMain I agree. The argument often goes that math like this is too abstract for students if they already struggle with the concrete stuff. I think it is the other way around. I think some students struggle with math because the rules and methods that you learn seem arbitrary and there is little understanding of _why_ it works.
As an example I was tutoring my cousin and she was struggling with derivatives and at first I did not understand why this seemed so hard for her... at some point I figured out it was because she had no conceptual understanding of what a function is. Once I've explained that and why the rules are what they are she had no trouble with even the more difficult exercises.
@@Kaepsele337, yeah you're right. If maths were taught better with the concepts clearly explained I'm certain a lot of people would see that it's not that hard.
I always had a passing interest in math but never really entertained that interest, and always wondered about quirks like these. I liked I was able to understand the explanation too despite my lack of knowledge in the subject.
I am not sure what I was hoping for from this video, in the end it helped me to clear out what I already knew: Mathematics are a set of rules chosen because they fulfil certain requirements better than others, the explanation lies on an axiom that can be seen either as true or false but that in itself is void of reason other than its suitability in a set of operations. I am not a mathematician and I've learned axioms and their consequent statements almost by heart as they are useful if not vital on my everyday life and work as an architect. Through my experience in life maths has been a language to express and eventually solve problems more than anything else and as such (a language) its basis are defined to better suit and define the expression of problems involving the physical world that surrounds me. In a way I speak fluently English, Italian and Spanish whilst I only dabble in Mathematics.
as an architect you are not confronted with math but merely with calculus as any engineering profession. Math is the study of "essential relationships" stripped from all else what can disguise that. But this is my personal description for it. And it is not meant to cover all aspects of Math
The only reason arithmetic and algebras such as ring theory are widely known (albeit in the math community) is that it has a practical engineering and scientific use.
Follows from the distributive law and 0x = x•0 = 0 (which also follows from the distributive law):
Assume A + a = b + B = 0. Then
AB
= AB + aB - aB
= (A + a)B - aB
= 0 - aB
= a(b + B) - aB
= ab + aB - aB
= ab
Note that x - y is short for x + (-y) here.
Lemma:
0x
= 0x + 0x - 0x
= (0 + 0)x - 0x
= 0x - 0x
= 0.
I like this a lot, though it may need some explanation.
What cmilkau is noting is that one can proove that A * B = (-A) * (-B) using the distributive law (Where -X is the additive inverse of X). They need 0x to equal 0 for the proof, which they also proove in the Lemma using the distributive law.
I was watching the Rings of Power videos, and this came up as suggested. The Ring theyory, indeed, is the one Ring to rule them all.
'Group' of Hobbits have other plans.
When I was a kid we were told to think of multiplying as " lots of".... so 3 lots of 2 = 6. So if we say, 3 lots of -2 , that will = -6 because it is -2 + -2 + -2. So -3 lots of -2 must = 6 because it is 3 negative lots of -2. So the double negative being a positive analogy is very good.
But what is a ‘negative lot’?
If you can understand this explanation, you were never confused in the first place.
The classical definition of multiplication that Descartes gives in the Geometry is based on simple proportion theory. It can be shown, using that definition, that a negative times a negative equals a positive. It is really fascinating that you dealt with the question using ring theory! Cheers!
I was inclined to agree with you. I like the video's explanation, but like you said (if I'm interpreting right) it's just about thinking about it in terms of area being a positive manifestation (I guess I'm at a loss for an official term, but with magnitude of size being only able to be described in absolute terms, I just took it as it being multiplying coordinate values in a point and accepting that area ought to be positive. Sorry if this comes off as sounding like some pedantic rephrasing.
Absolute values are one way of thinking about it. If you would be interested, I can email you a short explanation of the definition, and how it applies to negatives.
@@drakesmith471 I'll write up a doc, and leave a publicly available link for people to read at their leisure 👍
@@thomasaldredge653 sorry about that effort you’re going through. That said, thank you.
@@drakesmith471 nah you're good. It's not a big deal, I'm working on this stuff for a thesis anyhow
I really like that you're starting at the axioms. nice video !
The thing this is missing is the justification that "negative" and "additive inverse" are the same or related. The concept of "negative" has to do with ordering, which isn't mentioned at all. If you have that the additive inverse of a positive number is negative, then you've got a complete proof, but otherwise you've just proven a fact about inverses and not negatives.
Ordering is more specific than inverse. The ring of integers modulo 5 has inverses but no order structure .
I'm pretty sure the term negative for integers is defined by the additive inverse property, so the proof is implicit in the definition.
@@turtledruid464 No, the term negative for integers is defined by the order. A negative integer is an integer which is less than zero. The fact that the additive inverse of a positive integer (one which is greater than zero) is a negative integer is a theorem, not a definition.
@@zapazap Ordering is completely independent from any algebraic properties. You can have either one without the other, both together, or neither one at all.
Used to really confuse me when I was a kid why positive x positive was positive but negative x negative was not a negative. Very cool video.
I was ordering a submarine sandwich at a shop, with my girlfriend who was doing likewise. I was surprised to discover that ordering sandwiches is a non-commutative operation.
She is fussy about the preparation of her sandwich. In particular, she does not want her sandwich "contaminated" with pork products. So if I go ahead of her in line, and order a ham sandwich, she will ask the server to change his plastic gloves before he prepares her double-meat turkey sub. But if she goes first, I can order my ham sandwich and there is no need to ask the server to change his gloves.
I have also discovered that ordering a meatball sub DOES commute with ordering a turkey sandwich, since the server does not have to handle the meatballs with his gloved hands. Rather he can use a ladle to add the meat, so no change of gloves is necessary.
Which just goes to remind us that though a group may not be commutative for all pairings in a multiplication, there can be some elements that do commute. I believe multiplication by the identity element is always commutative, since I think that every right-inverse is also a left inverse. (Am I right?)
Your content is very good, so I would appreciate if you keep uploading.
Subscribed
I would, too. 👍
But, I will not subscribe, because while I would like _some_ of your videos, not all of this kind of material "Will fit easily into my brain", so to speak.
🤔➡️😲➡️💥
Even though explaining the axioms is important, the proper way to think about this is because we created math as a logical language. Setting true and false statements all the time, hence a false-false statement is a true statement. Remember that math is a language to proof propositions based on logic, and that this logic leads to paradoxes as well. Math is a powerful tool but is not perfect.
Geometry provides an elegant solution. The multiplication of two negative numbers is shown as the positive area.
_”Lucy! You’ve got some splanations to do!”_
-Ricky
I learnt this through jujutsu kaisen
Wish my elementary teachers could have just explained it like this!
Superb!
This is a great video but it definitely doesn’t help me how I should explain it to my students 😅😅. 1-(-1)=1+1 is not so bad: imagine you have a glass of ice water. The ice has a negative temperature and the water has a positive temperature (at least in Celsius, luckily I don’t live in Fahrenheit land). If you remove the ice, the average temperature goes up because you removed something negative. If you add a negative you are adding ice to the water in the thought experiment. It’s not a perfect analogy as it breaks down for -1-1 for example. For addition of negative numbers, a thermometer is an easier tool. It’s -1 degrees and then gets one degree colder, how cold is it? -1-1=-2
Fantastic video, you are amazing at explaining math concepts :) thanks
Your videos are so relaxing. I miss school math 🥲
1) negative * positive = negative
Simply because, in the same way that 3 * 7 = 7 + 7 + 7, then 3 * (-2) = (-2) + (-2) + (-2) = -6
2) negative * negative = positive
Just plot 3 * (-2), then 2 * (-2), then 1 * (-2), then 0 * (-2) on a line with a graduation and think of the most logical place where (-1) * (-2) should be.
The "prime the intuition" example I always use is with money, specifically debt.
If you're one dollar in debt, then you have a net of -1 dollars.
To "remove" that debt - getting back to net 0 dollars - is to subtract that debt.
Thus, in dollars: -1 - (-1) = 0.
But another way to remove a one dollar debt is to gain a dollar, so -1 + 1 = 0.
Thus -(-1) = +1.
You can either add one dollar to remove a one dollar debt, or equivalently, you can subtract the one debt to remove it.
Subtracting a one dollar debt is the same as adding one dollar to a one dollar debt.
Negative times a negative equals a positive can be reasoned out in a much more simply way. Consider (2)(3) this is equal to 6 because 3 + 3 = 6, we can also do 2 + 2 + 2 = 6. Now consider (2)(-3) this is equal to -6 because ( -3 ) + ( -3 ) = - 6, since (2)(-3) = (-3)(2) then (-3)(2) does the same as - (2) - (2) - (2) = -6. We usually think of repeated addition for multiplication, but as you have seen when the factor is a negative it basically performs repeated subtraction. This being said, it makes sense that (-2)(-3)=6 because - (-3) - (-3) = 6 . "There is also a simple explanation of why subtracting negatives does the same as adding positives." By the way I like your introduction of Ring Theory (easy to follow along) and how it can be used to show that a negative times a negative is a positive, although this explanation would scare off most people.
For me the background music puts ring theory into a mindless elevator.
I Recognized the music from Trash Taste podcast
This was a very informative video , great explantion
Because that way has been accepted at peer reviews as convenient for general use across mathematics using common multiplication.
Multiplication can be defined other ways as we know but for most people it is good to think about it and ponder why learned people decided on that option and why it has existed for so long.
If that rule failed it would be a major revision across the whole of mathematics?
nashe
I recall in high school, in algebra class, that there was a longish discussion about this. One of the objections raised by students was that in any multiplication that consisted only of positive integers, the correct answer could be obtained by simply adding the right side as many times as indicated on the left, and this statement could be reversed (commutatively), so that for a addition of a+b the answer could be arrived at by adding b to itself a times, and vice versa. You could step this answer out on the number line with dividers if you wanted. But this ceased to work as soon as you tried using even one negative. The teacher actually became frustrated and finally simply said "these are the rules."
@Artem Down The point where things got confusing is when you consider -5 X -4, which = 20, rather than -20. Negative signs are always "relative" to other negative signs. The survival of the negative depends on the number of negative signs in a multiplication, but not in addition.
One mathematics teacher said that rather than an actual mathematical necessity, the action of negatives was grammatical. In Greek, a double negative is a negation of the initial negation. But in Old and Middle English no such rule operated. But by the Reformation there was a concerted effort to adjust English grammar to more classical lines in agreement with Greek and Latin. So, when as a kid we said something like "ain't no" a teacher would jump all over us for uttering nonsense, but it was really an older grammatical pattern.
It would be interesting to think about a mathematical system that developed under different rules.
@Artem Down You are applying that originally grammatical rule. Consider what the result would be if the direction of movement along the number line were fixed as it is when the operation is addition of positive numbers. The negative sign is always relative to the other negative signs.
I'm not disputing your result, because that is indeed what our mathematics teachers would mark as correct. I'm pointing out that + and - have different behaviours relative to themselves. Addition has only one behaviour on the number line unless a negative is present, but negatives _always_ modify the behaviour of other negatives, depending on whether there is an odd or even number of negative signs. In some of the ancestral languages of English "not not" was equivalent to emphasizing the negativity, rather than negating it. That is "not not" would be more like "very not." It's why English possesses "nor." It agrees with a prior "neither." These days people often use "or" in conjunction with "neither," but that isn't or at least wasn't a grammatical construction.
That's hard to wrap our heads around now, but that is because grammar fanatics trained in classical languages insisted on "rationalizing" English to make it grammatically more like classical logic and emerging mathematical logic. The relics of previous structures linger as "bad grammar," which you chide your children for using. What I wonder about is what a mathematics would look like where the negatives had excatly the same rules as the positives. In the normal world you can't have a negative amount of anything real. Money is imaginary, and you cannot have a negative amount of cash, so put that to rest. Economics would look very different.
@Artem Down You do not seem to catch the point. I'm not arguing your results are wrong. I am saying that the mathematics we employ were evolved under a set of grammatical rules which are not native to English. "What would a mathematics evolved according to different grammatical rules be like," is the question I asked. Answer that and forget the simple minded arithmetic.
You need to remember that all those simple initial rules are postulates. That is, saying that a negative times a positive is always negative and that a negative times a negative is always a positive are unprovable postulates similar to the definition of parallel lines, which is "true" only in Euclidian geometry but cannot be proven using Euclidean geometry. We do arithmetic that way because that is the set of rules we adopted and found useful. They are no profound philosophical truths.
Logically, a positive number multiplying a negative number is negative:
e.g. 3 X (-10) = 3 'minus tens' = (-10) + (-10) + (-10) = -30.
So dividing both sides by -10, +3 = (-30)/(-10) = (-30) X (-0.1).
Also, the product of two negative numbers is the product of their absolute values times (-1)^2:
(-a)(-b) = (-1)^2 ab; a, b > 0.
If we make the 'choice' that the product on the left is negative, then (-1)^2 = -1, which results in the contradiction that -1= +1.
Slightly distracted by the music from trash taste lol
B4 I watch the video, Ill say how I visualize it and usually how I try to teach my students with a good degree of sucess Id say
Think of numbers in a 2d plane where the y axis r the imaginary numbers and X axis the real numbers. When you take the natura number 1 and you multiply by -1, this is a 180 degrees rotation, landing you at the real axis, but in the opposite side of the axis, in the negatives: -1.
So if you are a negative number -5, and you multiply by -1, a 180 degrees rotation lands u at positive 5. This also works great to understand complex numbers. If multiplying by -1 is a 180 degrees rotation, and i is the sqrt of -1, multiplying by i has to be a 90 degrees rotation. Hence why all the normal numbers make a plane, not a line.
Its obviously not the full story and visualization requires rigorous proof , but I remember the 1st time I began to understand this when studying trig representation of complex numbers, and something just snapped like finally trigonometry and complex numbers made sense to me. And even the idea of negative times a negative is a positive. And it made me fell in love with it, like damn math indeed is beautiful
I see -1 more as a reflection across the origin. In the Real numbers, -1 acts as a mirror across 0. (And the reflection of a reflection is the original object.) Where Complex numbers come in is when trying to find "half of a reflection." x² = -1 is asking what operation, when done twice, gives a reflection across 0. One idea would be to just travel half the distance, but then you land on 0, which halts your progress for the next step. Instead you have to go _around_ 0, and when you expand your view to a 2D plane so that you _can_ go "around" 0, what looked like a mirror reflection before now also looks like a 180° rotation, and half of 180° is 90°.
i² = 90°² = 180° = -1
Of course, in 2D you can go around 0 in two different directions, which give 90° = i and 270° = -90° = -i = i³, both of which square to -1. Adding additional dimensions gives even more paths around 0, and by extension more roots of -1.
Lol "These examples look outside in without fundamentally justifying why it's the case" *proceeds to list unjustified properties of an unrelated mathematical construct called ring and explaining through the lens of its properties*
Justification implies some sort of goal/reason, I didn't catch any except "I like rings and their properties and want to show them to you", but that is your reason not "the real reason" as the title says and it's just as good or bad as saying it's a complex rotation.
If it's a real fundamental reason, I want to see it emerge as you build numbers and multiplication and explore the design space to show numbers can't exist without rings(which is of course false, there's many systems out there) so the question shifts to why we make the choices and how do they result in the concept called ring.
Whereas this video is backwards, boring textbook style, here's a thing and its properties let's see how it solves our problem, the properties are even numbered before shown, completely disregarding that someone had to choose and formulate those properties.
But does anything above matter? No, as you said (-2)*(-3) can be anything you want and just like your video. Just know that by saying "real reason", "fundamental", "define any way you want" instead of "rigorously applying a mathematical framework I like" you attract my demographic which finds the video comically bad, dislikes, unsubs and leaves.
The real world and mathematics are two distinct things. The reasoning here is that multiplying negatives is not easily justifiable with real world examples, however we can, using mathematics, identify the main properties of integer summation and multiplication, which people have no problem justifying as it is quite intuitive in the real world, and reason what multiplying two negatives would to make sense in order to keep those properties. It is very common in mathematics to try and identify the main structure of the concepts we are studying and focus on the structure itself rather than the original concept.
You are right, however, in that multiplying negatives could amount to anything we define, if we so wish. Unfortunately most of the mathematical concepts we take for granted since we learn them as school children, like rational numbers, continuous functions, and especially real numbers, are entirely unnatural and have a lot of mathematical heavy baggage behind them. Even the way we represent any number in base 10 has some heavy baggage, like number theory, infinite series and convergence. Mathematics is entirely unrelated with the real world, and these "simple" concepts we take for granted need the "boring textbook style" to be explained properly and for us to have some insight on them. Believing that you can understand mathematics fully without actually doing mathematics, just with analogies to the real world and simplifications, is akin to believing in magic.
@@trewajg I agree and I think you misunderstood my comment.
The boring textbook style I refer to is using outside knowledge in the form of already defined properies(rules) which magically fit whatever problem in the intro.
I want to see the design space of those properties explored, how choices were made and coming up with formulations for the properties(whose number you don't even know).
If you don't go through this process of discovery and exploration and instead start with the framework you know it works, whatever is described is confirmation bias, not fundamental or the real reason.
What bugs me about it is that he tries to build this complex proof to show how the negatives can be pulled out and canceled using these ring properties, yet the very first theorem he shows looks as if it presupposes the property: (-a) + -(-a) = 0.
@@xxportalxx. that one comes from the fact that every element in the ring has additive inverse. The inverse property of addition. Every element has its respective negative. No presupposition there.
@@HaramGuys yes it derives from that, however the additive inverse itself is just a + (-a) = 0, the idea that the inverse of an inverse is the original seems to me to be the entire core point being discussed here, so imo he basically jumped the most important bit. Proving the rearrangement is trivial imo, the core idea is that -(-a) is a. This is the whole reason the 180° rotation explanation works, the product being the multiplicand scaled by the multiplier is all well and good, it's how the negatives cancel that was being discussed.
Because going the opposite of going the opposite side of North is still going North. Because the opposite of me owing you 3 dollars is you owing me 3 dollars. It doesn’t have to be complicated, only what you make it to be.
Agree, NEEDLESSLY complicated video: Eg 2, let's use imaginary complex numbers...
It's worth keeping in mind that this video is not primarily designed to teach people why a negative times a negative is a positive; rather, it is designed to be an introduction to ring theory (a subdiscipline in abstract algebra). Thus, this video is looking at the question "Why is a negative times a negative a positive" through the lens of ring theory. I think the title is clickbait.
i think a simpler answer is that.. taking the inverse (negative) of an inverse.. reverts/restores it back to a natural number/positive real.
its pretty good to think about these things though
Edsheran album joke always makes me laugh 😂
This channel is awesome
Ok so my take on why negative times negative is positive.
In number line, we can define 3×5 as 3 steps forward from 0 for 5 times then it lands on 15.
Negative multiplication can be defined by backward step like 3×-1 equal 3 steps backward for 1 time equal -3.
So now for -ve times -ve means negative step backward, like -4×-5 means (4 step backward) for 5 times in backward direction, so it just cancel both backward and give positive direction.
Any suggestions are welcome.
Indian
No actually, that causes an explosion of intensified DEBT
I kind of expected a defining line why it is the way it is, but I guess thisworks too. I’ll stick minus times minus as ”Minus reducing itself” as a go to explanation.
I really appreciate this explanation because I didn't know about rings and should learn more about them. I thought I had an intuitive geometric explanation for -a*-b > 0, but I never really though to question it too deeply and thinking about it just now I realized it was, no pun intended, circular reasoning.
Number line, for visual purposes.
Starting at 0.
+2, that's 2 to the right.
+2, that's 2 more to the right, totaling 4.
+2, that's 2 more to the right, totaling 6.
We see how multiplication is addition a number of times we decide, in this case it is 2x3=6.
So what is moving to the right negative 2 times?
Negative removing, so we're moving left 2 times.
We can do that three times also, (-2)x3=-6
So what if we want to negative, or remove, the negatives?
Well if you're at -2 and you want to remove the -2, you move right, to 0.
If we're at 0, and we remove the -2 three times, we're going to the right in three sets of 2.
(-2)x(-3)=6
Who just memorized this, it's so simple and logical.
The rules we use should describe the world we're observing.
That CAN be a fictional world, but in the case of the everyday math that most of us use, it's going to relate to something probably tangible.
Saved it to watch later, I will create a python program feeding my computer with the functions of these theorems to know if it understands that.
i only heard the trash taste theme
nothing else
time to watch that again
The way it was explained to me was that a positive times a positive is positive, so therefore, a negative times a negative is positive since it’s the same integer.
Ultimately "because it works." Consider:
0 + a + a + a = 3*a
0 - a - a - a = -3*a
a + (-a) = 0
-3*[a + (-a)] = 0
(-3)*(a) + (-3)*(-a) = 0
(-3)*(-a) = -(-3)*(a)
Let a be any positive number. From line 2 we have that -3*a is a negative number. From the last line we see that -3 times -a must be a positive number.
Unless this is the rule then the consistency of the whole structure of arithmetic breaks down.
If you borrowed a sum of money at a negative interest rate, wouldn't that mean you were multiplying your negative equity by a negative and ending up with a positive - you'd make money on the deal?
Yoo it's the Trash Taste Music. Didn't expect that crossover
"You are NOT ugly!"...that's a POSITIVE statement from "two negatives"..."Not" and "ugly" :-)
If you graph and make squares out of (2, 2), (2, -2), (-2, 2), and (-2,-2) and make the square, the line through the origin gives slope, such that when (+,+) and (-,-) are positive slopes, and (-,+) and (+,-) are negative slopes. I would thing that is the most simple, most visual, and most intuitive way to show it to a child.
Clear as mud. Thanks.
Boy you deserve waay more subscribere than you got
It's appreciable how you have put this in front of us... Great effort...
But this phenomenon can be easily understood with mirroring effect ...
I find a little bit disturbing that he said the explanation with complex numbers is not a correct mathematical explanation... The complex numbers just shed light on the reason (-1)(-1) is 1 with numbers that we use on an everyday basis. It gives some geometric intuition and, an explanation to the identity (-1)(-1) = 1. There is no such thing as 'The Real Reason' for any concepts. Mathematicians are searching for any insights on their work and all any bit of information is a valid explanation. We should talk about 'More profound reasons', not 'The Real Reason'. That would be a more accurate choice of wording.
Using complex numbers explains this so much more easily
Every complex number has an angle in radians, in which 0 represents positive real numbers, π/2 represents positive pure imaginary numbers, and π represents negative real numbers.
There is an important exception that every angle can have 2πk added to itself (where k is an integer) and not change the number, because 2π represents a full rotation.
For numbers with an angle of 0 (positive real numbers) is identical to numbers having an angle of 2π, 4π, -6π etc
When you multiply any 2 numbers together, you add their angles together, meaning:
i*i -> θ = π/2 + π/2 = π, meaning multiplying 2 pure imaginary gives negative
-1*-1 -> θ = π + π = 2π, but 2π is the same as 0, meaning multiplying 2 negative gives positive
Or I suppose an even quicker way to sum it up would be:
"negatives represent a halfway rotation, two halfway rotations represent a full rotation, which is positive"
@@glitchy9613 you use properties of complex numbers to prove the property, with which there were defined property of complex numbers.
You can't do this. This isn't how proves work
@@F_A_F123 I'd say "I disagree" here, but... what are you even trying to say?
@@glitchy9613 like wtf you use properties of complex numbers .. .Do you know that those properties are not the definition of complex numbers? Those properties were proved using much simpler properties like this one
@@glitchy9613 The idea that I think F_A_F 123 is getting at here is the following:
How do you know that complex multiplication represents rotation in the complex plane? Don't you have to use ideas like "a negative times a negative equals a positive" in order to show that to begin with?
Don't we also need "a negative times a negative equals a positive" to even have a need to define i to begin with?
Suppose we lived in an alternative world where the alternative "real number" multiplication was x⊙y = x|y| (using standard multiplication of x and |y|). In this case, the "alternative square" of the real number x would be x|x|, and every real number would have exactly one real number square root. (If x is nonnegative, then the standard √x would be the alternative square root of x, and if x is negative, then the standard −√x would be the alternative square root of x.)
Now, it's true that this alternative multiplication wouldn't have the nice arithmetic properties we've grown accustomed to, but what if things developed like this? There would be no need to introduce a new dimension to represent square roots of negative numbers since they would already exist on the number line.
So, from that perspective, the motivation behind developing complex numbers in the first place is because a negative times a negative is a positive.
Does this make sense?
3:42 I feel guilty for taking associativity for granted for so long.
Math videos like this introduce things like how religious people introduce their stuff "no those explanations are wrong because of their assumptions, now let's take my assumptions"
Your great at explaining
This video is solid for a mathematician, but it is too abstract for basically the rest of the world. I was expecting a video explaining what multiplication is and was curious to see if he would call it iterative addition and if so, how he was going to apply it.
Also, this was a great illustration of how logic is often found independent of meaning.
Here is the simplest way I know to explain this in my opinion using -2 x -3, with the cow analogy. The first negative, in this case -2, represents how many you "are in the hole, or have lent out" of something, cows for example. Easy enough, right. Now, the second negative, in this case -3, represents a "decrease" in the amount lent out, rather than an "increase" in the amount that needs to be given back, which would be the opposite, +3. So, the first negative means "in the hole" as in you lost 2 cows, and the second negative is the amount that loss is decreased, meaning you now have a "gain" of 6 cows. Thus, a "positive" amount of cows.
You can also imagine the opposite, on why -2 x 3= -6. If you again take the first negative, here -2, meaning again that you lost two cows, but then, say that loss is "increased" by 3 fold, you then arrive at -6, meaning you now lost 6 cows. So think of the second " - " sign as a "decrease" or "increase" relative to the first negative number.
My 2 yen on how to think of it more logically rather than mathematically.
I think of the negative sign as a uno reverse card, used twice (or any even number of times) returns the direction of travel back to what it was.
Really great explanation, excellent presentation. Subscribed!
Think of it this way:
Multiplication is a process of adding a value to itself a set number of times, and what is addition if that a positive term.
So when we multiply an negative value by a negative value, we creating a set number of negative terms, which will be applied on the negative value inside the term, thus, since subtracting a negative means adding its positive. We are adding the positive of the negative value to itself for that set number of times.
I woulda guessed the Well Ordering Principle
"imagine a car moving backward in time."
remove "in time"
Logically in words, without mathematical notations, (-2)×(-3) means.. For example You have a certain number of candies and You get a temptation of eating 2 candies, it happens 3 times but You control Yourself and Don't (1st negative) eat (eating = subtracting 2 candies each time which is the 2nd Negative) 2 candies for all the 3 times that You were tempted to.. So, when You're asked how many candies did You save, Your answer will be 6 candies.
a conflicting use of symbols occurs since we use the “-“ sign both to assign a negative to a number, as well as use it as a symbol for subtraction. In a geometric perspective, we use it to indicate the location of a number (i.e. 180 degrees opposite of the line natural numbers allign themselves in a square; or even more so a 3d representation like a sphere) and we use it to indicate a direction (of how an indicator moves from one number to another). From a geometrical point of view, numbers are nothing but locations, and tools to describe movements of an indicator through these locations.
What exactly is an indicator is merely a defined point (or collection of points) in reference to zero.
The meaning of zero can be related to space, for example the observer being the center indicator referred to as zero.
Timewise the present moment is always zero.
Metaphysically zero can represent the concept of Nothingness, though the nature of nothingness is hard to observe or proof - even to a Buddhist :)
Of these three, since we perceive time to have a direction, moving from the present moment away from the past, we have a conditioned perspective of what is zero timewise. Zero meaning now. And as such backwards into time feels like going exactly backwards the way we came - if time viewed as a line, 180 degrees the opposite direction. But since we cant go that way, we assign a negative. “This happened at -2 minutes, and counting”.
At quantum levels time doesn’t neccesarily move in one direction, nor does it only add 180 degrees opposite of its direction, as it’s second option. Time can move away from the present moment in ANY direction. Or rather: the present moment can be a result from both the past, the future or ANY direction in a sphere of possible directions of time.