I had a math professor who was careful to say, "For the purposes of THIS CLASS," ... such and so would not or could not be done. That left the door open for me to really appreciate this!
I feel like the rules remain, except the nullification factor, well... nullifies whatever it's a part of. You only "lose" rules in the sense that those rules do not apply to this special operator with a specific definition. The rules "lost" are the rules that exist being submitted to nullification. It's literally no different than saying 1 + 1 = 3 nul 1 instead of just 2. That's a logically factual statement with the additional statement without taking away from the rules. To me, it doesn't take away from anything, but rather adds a special case where the rules are bent only for that function while still applying anywhere else in the equation not attached to the nullification. To me it's no different than saying the square root of negative one equalling i breaks math. Yet after time it seems less and less of a strong argument against it.
Calling them "basic algebra rules" is misleading. Algebraic structures are defined by the axioms that we impose on them. On the real numbers, we impose the field axioms. With a wheel, we modify those field axioms slightly, making them more general, to accomodate for the intoduction of /0 and 0/0 as elements of the wheel. As such, the field axioms are special cases of the wheel axioms.
@@angelmendez-rivera351 Honestly, your comment gets to the point faster and in a way that's different given I am not familiar with wheel algebra. Very well said.
@@TheLethalDomain Well, you can also read the Wikipedia article on wheel theory. The Wikipedia article does a really decent job at explaining how does this all work, keeping it simple, but rigorous.
This answer for the 1/0 problem falls under the category of "University Gas". It's an answer that has no utility in the real world. We have NOT been lied to.. When your real-world problem solution boils down to something divided by zero, you know that you have departed reality, and something is wrong with your problem/solution formulation. The word "undefined" captures that pretty well. "Nullity" is an abstract way of saying that, but it's not an "answer" to the division problem.
Eh, I cannot think of a reason you would *need* linear algebra in order to understand abstract algebra. Rings, groups, and fields should all make just about as much (or as little) sense either way. Speaking of fields, the problem with defining 1/0 is that you are probably going to lose your nice field properties by doing that...
@@kennyb3325 Vector spaces and Vector Subspaces can be quite abstract Concepts that should be introduced in a course on linear algebra before one Endeavors into abstract algebra, at least in my experience
@@9WEAVER9 A first course in abstract algebra need not cover those things. Rings, fields, and groups are more familiar (since we can think of good examples like the integers, rational, or real numbers) and can serve as the entry point to abstract mathematical structures, perhaps better than vector spaces. Of course, one would want to be introduced to vector spaces before encountering modules.
A professor of mine said that it was mostly designed by mathematicians instead of electronics engineers. He complained that it could've been faster to compute had it used twos complement instead
The "nullity" reminds me of NaN ("not-a-number") in programming. According to standard floating point arithmetic, the result of any operation where NaN is one of the operands is always NaN. The difference there though is that 0 / 0 = NaN, but 1 / 0 = Infinity
That's kinda built into the code package you use. With quantum computing I suspect this to become way more complicated. Pretty sure with MATHLAB you will have different outcomes more robust than a simple Java math class.
@@reignellwalker9755as much as people who preach their religion annoy me, i must admit that someone with a roblox pfp praising someone for talking about coding for seemingly no reason gives off a powerful aura
0:21 no, they didn’t discover you could take the square root of negative 1, they invented a new number to allow us to, before that you couldn’t take the square root of negative 1, similar to how before they invented calculus you couldn’t do calculus
And it was treated initially as a mathematical trick. And mathematicians know that they are giing something up when they switch from real numbers to complex numbers: ordering. There is not good definition of < and > for complex numbers.
@@glassjester except that the idea of dividing by 0 doesn’t exist, we don’t actually know if 1/0 times two is still 1/0 (with 1/0 acting like 0 does in multiplication) or if it’s 2/0, with the square root of -1 we knew it was going to act like a constant, just like pi, but 1/0 could act like 0 or a non 0 constant, because we can’t agree on its behaviour as a concept
@@glassjester But what is 1/0, how does it behave, does it work like 0, a non zero constant, infinity, or something else entirely, and if you multiply this "z" by 0, do you get 1, if so how does that work? Since by multiplying 1 times z by 0 you can either do 1 times 0 and get 0 times z or you can do z time 0 and get 1, by mathmatical laws these would have to be the exact same, meaning 0z is 1, but with 2z times 0 you could get 2 or 0z, meaning that 1=0z=2 by mathmatical laws, which is a contradiction we don't encounter with i
There are algebraic structures where division by zero makes sense. A very straightforward example is the ring of remainders of division by ten or any other non-prime.
This reminds me of stuff I learned in engineering. One was the delta function which is defined as infinity at a single point and 0 everywhere else. If you integrate over it you get 1. I mentally imagine it as a rectangle with 0 width and infinite height and area of 1. And you could multiple delta by constants to get other areas. We used it for theoretically perfect spikes. Calculus classes hated this. I remember another where when a function went to infinity, it could “wrap around the plane” to negative infinity or even to positive infinity. I think it had to do with finding stable points by wrapping them or something. It’s been so long that I don’t remember clearly anymore. But it sounds similar to mapping the plane to a sphere to make all infinite points touch. (And thanks reminding people infinity is a ranging concept and not an actual number.)
The delta function does actually have a rigorous definition in terms of the concept known as distributions, or continuous linear functionals on the space of smooth functions with compact support.
that’s called abstraction. a*b=1, while a->0 and b->inf. but actually this is the essence of calculus/analysis: when we say that a continuous interval van be decomposed to infinitely many infinitesimal (0-like) intervals.
Here's another way to put it: If you want to define a new set of numbers, you need to show that it's possible to start with already-defined numbers, go into the undefined set, and come back out the other side into already-defined numbers. If I gain 5 apples and lose 3 apples, I make a net profit of 2 apples. This holds true even if I went into debt because I lost 3 apples *before* I gained 5. This shows we can go into negative numbers and come back out, which means we can define the set of negative numbers. We know that the area of a triangle is bh/2. Knowing this, we can easily prove that if we have two isosceles right triangles, and we put them together as halves of a new isosceles right triangle, the new triangle has an area equal to the side length of the original triangles. If our original triangles had side lengths of 1, this shows we can go into irrational numbers (since the hypotenuses have lengths of sqrt(2)) and come back out with the rational number 1, which means we can define the set of irrational numbers. And though I forget the exact formulas involved, imaginary numbers were proven valid the same way. There was some known formula to solve a certain kind of polynomial, but it was found that if instead of just using the formula outright you worked through the *proof* of the formula, you would end up having to evaluate negative numbers under radical signs at some point in the process, even though you might start and end with real numbers. Conversely, the video demonstrates that the idea of "nullity" swallows numbers like a black hole from which there is no escape, since you have to "give up some rules of algebra" in order to use it. In other words, this new system is demonstrably incomplete and likely has no practical use.
That's pretty much the best way to put it, and the reason why division by zero is impossible. Unlike other mathematical elements, you can't define it without breaking the laws that already exist. If assuming that giving up the rules that solidify 99.99% of Maths is worth to justify one insignificant operation, why even keep on playing with maths?
@Remix God In the real world you actually can divide a singular piece into more pieces. There's a whole scientific field that came out of that, known as Chemistry, but even if you want to go into something simpler, imagine a slice of bread. Now cut it to 4 pieces. You just divided 1 by 4 in the physical world. Just because the set of natural numbers doesn't allow that doesn't mean it doesn't exist. In that case, 1/1 is just 1. That also involves the concept that dividing anything by 1 gives you the same thing. If I have a cake and zero people on my birthday party, the only one left to eat it is me, and I will, that's a 1/1 in the physical world. A nullity, at least as described in the video, is an absorbing element. *That* doesn't exist in the physical world because, by physics laws, energy is not lost. It just becomes something different. Yet a nullity can absorb every other number it's given with any operation. 1/1 can't do that.
@@finnfinity9711 I mean, I guess you could. But aren’t you still breaking some rules? [Nullity]2 * 0 = 2 You’re multiplying something by 0 and getting something out that isn’t 0.
I asked why I can't divide by zero when I was younger. My teacher told me I could if I wanted to but until I find a reason to there will be no value in doing so.
Well,if we set up the "nullity"=b . Then b=1/0.If that's the case,Then b×0=1.Then multiply both sides by an algebra:a.It becomes b×0×a=1×a.On the left, first calculate 0×a=0.b×0=a.If b×0=a,then b×0 is also=1.Which means 1=a.That means every number is equal to one.
The issue with this proof is in multiplication by zero. You said that b=1/0, thus b*0=1, which is a really easy mistake to make. We always learn that (a/b)*b=a, but this is a shortcut for the truth that (a/b)*b= (a/b)*(b/1)= (a*b)/(b*1)= (a/1)*(b/b)= a*(b/b). In most cases, b/b=1. In your example however, b=0, thus you actually have b=(1/1)*(0/0) =1*nullity =nullity. It was a difficult mistake to catch and it took me several minutes to be able to find it myself
Notice: he never answered the question, the nullity is still not a valid solution, because 0 times the nullity would still be the nullity, so 1 divided by 0 is not the nullity, he’s just thrown a bunch of math Mumbo jumbo in our faces and hoped everyone who had a more comprehensive understanding of this wouldn’t watch the video since they already knew it was bs
@@OliverKraus-q6s Being angry in some situations can be the correct response. In other words getting angry is not always wrong. If you think he is wrong, refute his argument rather than a personal attack.
The proper name of the "unsigned" infinity is: complex infinity. No matter which direction you go in the plane, you tend towards infinity as you keep going.
I get your joke (don't whoosh me), but the square root is a function (which means only one output) defined to give only non-negative outputs for real inputs. It's when you try to solve x^2 = a that results in x=±√a where √a ≥0
This kind of explains the quadratic formula. (-b ± sqrt(b^2 - 4ac))/2 Square root takes the positive and multiplies it by + and - making two answers. So square root on it's own doesn't have 2 answers, but ± does
I remember doing these in college...However, it sounds like the problem becomes over complicated. What I learned from proofs is, if something is over complicated with too many rules, then usually there is an undiscovered proof that will disprove it. Occam's razor
Turning Ian Malcolm's quote on its heels toward his own profession: The mathematicians were so preoccupied with whether or not they could they didn't stop to think if they should.
I like that you come to the exactly same conclusions as I did when I first learned about the symbol i from complex numbers and had the idea to check what happens if we define a symbol standing for the division by zero.
1 divided by 1 is 1. 1 divided by 0.5 is 2. 1 divided by 0.25 is 4. Therefore we can say that the increase of the numerator is directly proportional to the decrease of the denominator. Therefore as the denominator approaches 0, the numerator approaches infinity. Therefore, 1 divided by 0 is infinity. 9 divided by 0 would also be infinity, but it’s a bigger infinity than the first one.
Yeah same here, since zero could go into any number forever without filling the gap. But it's more fun when you start to involve things middle schoolers wouldn't be able to figure out normally.
@@josephjoestar953 personally, I have always argued with my teachers that if we think of it algebraicly, that as long as we don't use imaginary numbers that division by zero is simply a conserved absolute value addition problem using an infinite series. If you were to graph a negative and positive infinite series with the same absolute value, they would be identical graphically except for which side of the graph they were on. If you think about this way, X + -1/0 is actually X - |1/0|. If we think about it this way, 1/0 is a smaller infinity than 2/0 and so on, but the negative counterparts conserve the value without being defined in the opposite direction. Similarly, an infinite series of zeros is still zeros so zero/zero would simply be zero. 0-D is just zero, 1-D is an infinite line, -1-D is also an infinite line, 2-D is an infinite flat grid, as is -2-D, so on so forth.
Be careful, dinosaurs destroyed their world when a dinosaur wrote 1/0 on its chalkboard. Then the asteroids crashed to the ground. According to a Far Side cartoon.
I've been puzzling over 1/0 for quite some time; it does feel like you should be able to treat it in a similar fashion to sqrt(-1) by creating a new axis of complex numbers, but I've struggled to imagine what such a function would graph. The idea of the "terminus" makes me think it should be treated more like the center point of a sphere. 1/X becomes the distance from the center, with 1/0 being the true center. 1/1 would then be the shell where "normal" numbers lie. I'm a philosopher, not a mathematician, so this might be a dumb way of looking at it. I don't know. Still, thanks for posting this; it was interesting.
Hello. I thought I'd like to comment that square root is just the inverse of a square. So X to the power of 2, is the square, the inverse is to the power of a half, or 1/2. The importance of odd and even numbers comes into play with a cube root, such as to the power of 1/3, and odd powers such as 1/5, 1/7 etcetera. This is because a negative squared is a negative multiplied by a negative which makes a positive. This is not the case for cubic functions (to the power of 1/3) or other odd root functions. ( Like to the power of 1/5, or 1/7 etc) The cube root of -2 is -1.259921. But the square root of -2 does not exist. This theoretical anomaly has perhaps been where the visualisation of things has led to the idea of black holes and negative particles, and string theory.
I'm not a philosopher or a mathematician, but it seems like pretty interesting idea. "j = 1/0" I can't think of any real world uses, but the same was said about negatives and square roots of negatives.
@danc.5509 Well is kinda depends First off if you limit yourself to the reals you can't solve sqrt(-4) but if you expand to allow complex numbers Then you get 2i i is defined as i =√(-1) It doesn't "exist" but using it you can solve for a lot of things and has some real world applications @whyme1698 While there are some ways to have x/0 not be undefined using a variable like "i" is because it can be used to make two different numbers equal each other which means that it can't exist (1/0 = j) Is because there are a lot of ways to mess with it So: (1/0) = j Assuming absolutely nothing about j: So then: 1 = 0j And because any number times 0 is 0 1 = 0 Which is a contradiction
You can not just define your way out of 1/0, because division is the undoing of multiplying. Since most any number n * 0 is 0, we just do not know what the original number could have been. Higher-dimensional numbers (complex -> quaternions -> octonions) become more problematic with division, because there is just too many ways to get the same product.
When I was in college I studied projective geometry and homogenous Cartesian coordinates. So, (x,y) would be expressed as (x,y,1) or (2x,2y,2) etc.. We determined that that there was a single point at infinity in each direction of x/y. Further, all the points at infinity formed the line at infinity. The notation would be (x,y,0) for any particular point at infinity. In addition, using the General Projective Transformation, we could transform a point at infinity to become local, but losing a point previously local to become inaccessible. This was done by matrix cross products. For example, a simple addition nomogram, with three parallel lines, could become three concurrently intersecting lines, with the point at infinity now appearing as the common intersection. As the three lines approached the central point, the associated scales grew greater from both the positive and negative directions. As far as I know, the GPT is how the math behind computer graphics is handled. It allows for a single technique to be used for scaling, rotation, magnification, etc.. And the transformations can be stacked and reversed. But I've never seen this used to handle the points at infinity.
Not so simple. The problem is that division is multiplication of a multiplicative inverse. To say we can divide by 0 is to say that 0 has a multiplicative inverse. Hence, if _z_ = 1/0 and _z_ = 2/0, we get that 1/0 = 2/0 (equality is transitive) and hence (1/0) * 0 = (2/0) * 0, implying that 1 = 2, a clear contradiction. That is, _z_ * 0 would not be well defined.
I’m so glad you brought light to this, because I’ve been thinking about this concept the exact way you mentioned it, and I’m really happy that this concept is out there, being explained so masterfully yet simply.
Funny, a few years ago, I pretty much had the same idea of defining 1/0 and I called it zeta. I just thought, well, we defined sqrt(-1) = i, what if we define 1/0=zeta. After playing around with it, I noticed 1/0=zeta -> 1/zeta=0 by algebra. I concluded I just made a complex sphere. Also x*zeta=zeta just like x*0=0. I came with the phrase "Zeta, the other zero on the other side" for a clickbait title if I ever gonna talk about this lol. Then I got stumped when I ask what about 0*zeta, which you also discussed. Interesting stuff. I didn't think of the nullity number though.
Finally someone makes a video on something related to the Riemann Sphere, which isn't a lecture. Can I also request a video on looking at complex functions and transformations on the Riemann Sphere, because they're really mind-blowing and eye-opening. What functions correspond to reflexions across the 3 main axes of the sphere, and stuff like that. Thanks for this video!
I never tell my students they can’t divide by zero I always remind them of the idea of new number sets. Aside from wheel algebra there are also the hyper real number sets. Good job
It is absolutely true that division by zero is undefined (impossible) on the field of real (and complex) numbers, which is the only field any high school or lower students will ever work with. In fact, tons of students get things confused because they don’t really understand that certain functions (especially trigonometric ones) have entirely different results based on what they’re defined in. I’ve seen a perfectly intelligent (probably too clever) kid disbelieve that 0.99…=1 because they heard about the hyperreals and said that 1>0.99…1>0.99… without really understanding how it actually works. I don’t even know if that statement is true in the hyperreals, but in the real numbers 0.99…=3/3=1. And indeed, anything else would cause problems.
@@maxthexpfarmer3957 In statistical thermodynamics, we work with the quantities temperature (T) and entropy (S). One thing you probably have heard a lot is that we cannot reach absolute 0 for temperature. This is true,... but despite that, we can actually reach negative temperatures in Kelvin. The idea is that some physical systems have a highest energy U they can attain. This energy U is a function of the entropy S of the system. Entropy, energy, and temperature are related by the equation T = dU/dS. Now, if that physical system attains its maximum energy possible, what happens if you increase S even more? Then U obviously cannot keep increasing. It can only decrease from there. If S is increasing while S is increasing, then dU/dS < 0. In other words, the temperature has to become negative. However, this makes the system unstable, so the temperature begins to decrease rapidly in the negative direction, and intuitively, this looks like "T is going to -♾, looping back around to +♾, and then continues decreasing until it reaches stability." With this picture in mind, it looks analogous to the idea that -♾ = +♾ = ♾. But while I can see why it seems superficially similar, it is far from the same thing. Why? 1. Because T = dU/dS is only an approximation. It is well-known today that at very high temperatures, statistical thermodynamics does not describe reality accurately. It is also likely that there exists a highest temperature attainable, the Planck temperature, and if that is accurate, then that means that there is no such a thing as infinite temperature, and that temperature could never loop around the way it is described here. Besides, in reality, entropy changes discretely anyway. Entropy is defined as S = k·ln(Ω), where Ω is the number of microstates corresponding to the macrostate of the system, and k is Boltzmann's constant. Ω is necessarily a positive integer, so it can only change from Ω to Ω + 1, there is no smaller possible change, making it discrete. So the smallest possible change in entropy is k·ln(1 + 1/Ω). However, we can approximately these discrete changes as continuous changes, because given how astronomically small k as a constant is, and given how even smaller 1/Ω is, these changes in entropy are so small, that we can approximate them with continuous changes, so using derivatives gives a remarkably accurate model for low temperatures. 2. Also, this idea of unsigned infinity does not correspond to physics because absolute zero is still unreachable, and thus the analogous of division by 0 is still not possible in it. So again, there is some very superficial similarity if you ignore the rigor, but otherwise, it is not really analogous.
I've always been told by my math teachers (since the 90s) dividing by 0 results in "null set" not 0 technically but functionally it's 0. Thanks for explaining why!
It is wrong. First of all, how do you devide by nothing? And second deviding by an infinitely small number != 0 will get you an infinitely big number (approaching infinity). So it cannot be 0.
Well, IEEE floating point numbers work a little bit like that. Except that they distinguish between +infinity and -infinity, but then there are also different representations for +0 and -0.
The different binary representations of +0 and -0 are really just an implementation detail. They are two different ways of describing the same number in the sense that +0 == -0 is required to evaluate to true. But you're right about how all the indeterminate forms (0/0, 0*Inf, Inf/Inf and Inf-Inf) all evaluate to NaN ("not a number") in IEEE 754. And I think NaN shares several other properties with the "nullity" in the video (like NaN-NaN = NaN).
@@weetabixharry +0 and -0 were there because you still want to retain a sign even when the truncation caused the number to be zero. It can be even argued that they really represent infinitesimals in some sense. The actual implementation detail is that they are kinda aliased to the real zero, which was considered an acceptable tradeoff.
Can't we see the nullity as abox where we put all numbers. The nullity = 1 and the nullity = 0. All numbers equal the nullity 0/0= the nullity The nullity × 0 = 0 The nullity is defined as every number at the same time. The whole numberline is the nullity. Isn't this the nullity Is therefore 1/0 (the 1 and the 0 taken from the nullity numberline) some other numberline Did I just revolutionize math?
I've always misunderstood why you can't divide by 0 when I was younger, my thought process was that if you divide by nothing then why wouldn't the answer just be whatever the numerator is
We had it in 10th grade (Germany). I don't really understand if it's this but it think so. Basically we are not dividing by 0 but with an infinite small number.
@@iranmaia91 i'd think it's not really used because having an ordered set extending the real line (i.e., separating -inf from +inf) is more useful than merely including an edge case for division.
Ph.D. mathematician / retired math professor here: I'm watching this and mostly nodding on your points, BUT it still remains incredibly irresponsbile and plain wrong to say, or even imply, that you are dividing by zero. You absolutely are not dividing by zero. Even the use of simple Calculus limits (which I knew you would do beforehand) does not get around that simple fact. And the creation of "the nullity" has no utility or value to more than 99% of humanity - and none to this mathematician, either.
The dividend remains the number into which zero is divided, because the number is divided exactly zero times. I learned this in second grade. My little toddler mind had a concept of enumeration when I was playing with building blocks, though I didnt have the vocabulary to express those ideas. It really isnt complicated. Matter is finite. Zero is not infinite, however. It is measurable in quantity, which is a quantity of none. The answer is NOT infinity.
I think maths needs a solution/ definition for 1/0. This one sounds quite interesting. It would be nice to see some long existing problems solved by that
@@rhubaruth IDK but I heard somethings in physics are unsolvable like singularities, which maybe solved if we can divide by 0, though I have absolutely no idea because I don't know anything about it
@@atharva2502 Although you said you have no idea, I do think there is a significant point in your statement. I think its obvious through the study of calculus and real analysis that the idea of 0 is very closely linked to the idea of infinity. In that respect I could see a solution regarding infinities in physics (such as center of black holes ie. singularities) being related in some way to the idea of dividing by 0.
There is a tiiny wiiny clumsy detail we're forgetting here: 1/0 = INF 2/0 = INF 1/0=2/0 WTF? And, by the rules of expanding fractions: x/0 = x*k/0*k = x*k/0 From which: x = x*k This contradicts basics of math. So, no, Infinity isn't that good of a solution. Not in common algebra at least. If it was, why wasn't it implemented yet?
One thing you lose in replacing positive and negative infinity with unsigned infinity is the differentiation between functions which blow up n the positive direction versus blow up in the negative direction. You’re basically replacing “becomes unboundedly positively large” with “becomes unbounded in some direction.” It’s useful to be able to, for example, have the notion that positive infinity is strictly greater than any finite number. Of course you can define singularities like in the video, but I suspect in most contexts it’s better to keep positive and negative infinity as separate concepts.
@@angelmendez-rivera351 Strange matter is a theoretical form of matter that converts any other type of matter it touches into itself. Imagine a gray goo scenario, only waay worse, since theoretically, if even one particle of strange matter touches something like a planet, it converts the entire thing into strange matter. Pretty freaky if you ask me.
Dividing by zero is absolutely easy. If I have a cake and divide it by zero... in other words, I don't cut it in two or more pieces, it simply remains the same... That's what I call common sense, and why I always struggled with maths...
That's dividing by 1. You are giving the cake not to several persons partially but to 1 person (maybe yourself) as a whole unit. A unit is 1 part of 100%.
the way i do division is that the answer is the number of times you add up the divisor until it makes the dividend so dividing by zero gives you infinity
"Should we divide by Zero?" I still say no, but I don't think the division operation even happens at all when trying to divide by zero. If I divide 12 by 2, I'm laying out, for example, a set of 12 empty boxes into 2 groups, with 6 per group: 🔲🔲🔲🔲🔲🔲 🔲🔲🔲🔲🔲🔲 Laying out that set of 12 empty boxes into 3 groups instead is 4 per group, and so on. If I divide 12 by 1, I'm laying out that set of 12 empty boxes into 1 group, with 12 per group: 🔲🔲🔲🔲🔲🔲🔲🔲🔲🔲🔲🔲 If I divide 12 by 1/2, I'm laying out that set of 12 empty boxes into 1/2 of a group, leaving room for 12 more in the whole group, resulting in 24 empty boxes per group, which corresponds with multiplying 12 by 2: 🔲🔲🔲🔲🔲🔲🔲🔲🔲🔲🔲🔲 🔲🔲🔲🔲🔲🔲🔲🔲🔲🔲🔲🔲 Laying out the original set of 12 empty boxes into 1/3 of a group instead is 36 per group, 12 into 1/6 of a group is 72 per group, and so on, with the denominator getter closer to Zero. What if I just make the denominator Zero? Here's the result: I just laid out the set of 12 empty boxes into 0 groups, which means I didn't lay them out at all. I'm still holding onto them. I didn't divide them into any groups. Division doesn't occur. NOW HOW ABOUT NEGATIVE TWO GROUPS No. Show me what -2 groups look like and then we can discuss. HOW ABOUT i AMOUNT OF GROUPS No. Show me what i groups look like and then we can discuss.
@@esajpsasipes2822 Actually, I've expanded on this elsewhere since my OP. Let's use total dollars divided into dollars per person to get the number of people I'm paying or getting paid by. $12/$2 per person = 6 people getting paid by me $12/$3 per person = 4 people getting paid by me $12/$1 per person = 12 people getting paid by me $12/$0.50 per person = 24 people getting paid by me $12/$0.25 per person = 48 people getting paid by me $12/$0 per person = I'm not paying anyone and no person is paying me 😂 $12/$-2 per person = 6 people owe me money $12/$i per person = Nope
@@pronounjow Thats because we use reals (R) to express money, and i is not in R. If you had something in complex numbers (C): 12/i = 12/sqrt(-1) = 12/sqrt(-1) * sqrt(-1)/sqrt(-1) = 12sqrt(-1)/-1 = -12i It would be -12i. Complex numbers are used (apart from pure math) in electrotechnics to calculate things around AC circuits with capacitors and coils, in 2D graphics to calculate rotations (as it's simpler than using vectors), and it is present in quantum theories.
I still see problems with this first since (like told in this video) you can sometimes make sense of terms like infinity - infinity specific to a function and can get normal numbers (but also +-infinity). That means the nullity can be equivelent to any number. second when you transform equations with variables you can sometimes get plain wrong results when not accounting for the case that the variable may be 0 when dividing through the variable
Actually we can divide anything by 0, but we won't get an end, so that means it's infinity, like this: 1 / 0 -0 1 8 = Infinity that never reaches nothing Another way is: 1 / 0 -Ind Inf Ind = Indeterminate Form So in both ways the answer is Undefined, because "we must agree that Answer × 0 is 1, and that cannot be true, because anything multiplied by 0 is 0."
Very good but there’s still a problem. If 1 = infinity * 0, and we say that infinity * 0 = the nulity, then 1 = the nulity. If you divide 2/0, you get 2 = nulity. So if you substitute for the nulity, you get 1=2. You can’t really just get rid of some of the rules of algebra. Throughout all the proofs out there, I think it’s best to just keep it undefined. Maybe it will be defined one day, but it’s true definition must keep math consistent.
I bet no-one else had a teacher who told the class that if they divided by zero they'd explode. I don't think any of us believed it, but I haven't taken the risk in the 50 years since :-)
my interpretation has always been that it's more convenient to assume that infinity is not possible unless we say it is, including infinitely small numbers. for example, 1 / 0 would equal infinity if you state that an infinite number of zeroes would fit in the numerator, without making it 0 (four 2s can be subtracted from 8 to make zero) but once we get our result of infinity, we're still left with 1 in the numerator. this implies that infinite zeroes manage to approach 1, as zero cannot truly be nothing when used infinitely. 0.00001 times infinity equals infinity, so any number approaching zero would be infinity, until zero itself. and that's good enough for my purposes.
Consider infinity to be like an "edge" instead. What happens when you cross it? (like in your Infinity +1 example). It wouldn't "absorb" it like you say - but rather the perspective moves beyond, into another "measure" of infinity. Using the stereographic projection point of view, it is like crossing into a different 2-dimensional plane, where if you were to stereographically project this new plane onto the initial plane, the sphere that it forms would have a point at zero which would map to *the same point* as the point at infinity/at 1 on the initial sphere. Imagine it stacked on top with the only intersection between the two spheres being the point which represents infinity on the first sphere and 0 on the second sphere. Then when we consider dividing by zero, we can understand it from a new point of view in that what we are actually doing in the process of dividing by zero is like "crossing the edge" and moving between different measures of infinity. IDK how useful this point of view is yet, but its another way of looking at it that I thought of. I've been mulling over an idea relating this to complex numbers/quaternions, where the additional measures of infinity are represented as a form of complex number.
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First (I am part of the problem)
second second days
uh... nooooooo....
Just leave it to undefined for god sake
@@brianlam4101 that’s not funny though
As James Tanton likes to say: We can do anything in math. We just have to live with the consequences.
I like it!
Pretty accurate, frankly
Member tanton?
Pros:
Innovation in engineering and sciences
Cons:
Harder exams
Conclusion:
Isn't worth it
But if we can do anything, doesn't that include avoiding the consequences?
dont divide by zero at home kids
*Adult supervision required*
Do it outside
Batteries not included
@@electronichaircut8801 And make sure to safely contain the resulting black hole
@@Sovic91 is that what Happens when I divide 0?
"One divided by 0 is undefined."
Me, a blissfully innocent middle schooler: "Why don't we just define it?"
(1:0)
We can define it but then it would make ZFC inconsistent and every statement is true
Eo
Oo
Ikr. I’m also a middle schooler
I had a math professor who was careful to say, "For the purposes of THIS CLASS," ... such and so would not or could not be done. That left the door open for me to really appreciate this!
Same
So you mean we can't create a black hole dividing by zero. Fine, I'll go back to the blackboard.
XD
Ngl I thought it would make a black hole in math💀💀💀
So basically, if you allow for division on zero, you have to give up some basic algebra rules
True!
I feel like the rules remain, except the nullification factor, well... nullifies whatever it's a part of. You only "lose" rules in the sense that those rules do not apply to this special operator with a specific definition. The rules "lost" are the rules that exist being submitted to nullification. It's literally no different than saying 1 + 1 = 3 nul 1 instead of just 2. That's a logically factual statement with the additional statement without taking away from the rules.
To me, it doesn't take away from anything, but rather adds a special case where the rules are bent only for that function while still applying anywhere else in the equation not attached to the nullification.
To me it's no different than saying the square root of negative one equalling i breaks math. Yet after time it seems less and less of a strong argument against it.
Calling them "basic algebra rules" is misleading. Algebraic structures are defined by the axioms that we impose on them. On the real numbers, we impose the field axioms. With a wheel, we modify those field axioms slightly, making them more general, to accomodate for the intoduction of /0 and 0/0 as elements of the wheel. As such, the field axioms are special cases of the wheel axioms.
@@angelmendez-rivera351 Honestly, your comment gets to the point faster and in a way that's different given I am not familiar with wheel algebra. Very well said.
@@TheLethalDomain Well, you can also read the Wikipedia article on wheel theory. The Wikipedia article does a really decent job at explaining how does this all work, keeping it simple, but rigorous.
Math is one of the few things that can make adults feel like children again
😀
We’re all such nerds.
key word: can
@TurboGamer 0/0 is indeterminate since n•0=0
@@Enderia2 key word: your mom
Me in Algebra One: I like your funny words magic man
And there's me in precalc thinking the same thing.
@@cerulean22b69 same
Me finishing my 3rd year as a math major: Interesting
i like your profile picture!!
@@thewatermelonkid1337 Thank you!
0:15 Wow, I didn't know Ant is such a strong word in math
😂😂😂😂😂
ua-cam.com/video/jOxRCJS3idc/v-deo.htmlsi=1W35wNcx1Nh7yVy5
Hannah Fry disagreea
This answer for the 1/0 problem falls under the category of "University Gas". It's an answer that has no utility in the real world. We have NOT been lied to.. When your real-world problem solution boils down to something divided by zero, you know that you have departed reality, and something is wrong with your problem/solution formulation. The word "undefined" captures that pretty well. "Nullity" is an abstract way of saying that, but it's not an "answer" to the division problem.
I can’t tell being this is April 1st if this is a joke or not😂👏🏻
Well yes but actually no
@@BriTheMathGuy LMAOOOO
@@randylejeune Conway's *
@@angel-ig I think that was a prank as well
@@BriTheMathGuy yesn't
I always wanted to learn abstract algebra. Maybe this is a good excuse to order an abstract algebra book with my nullity dollars in my wallet.
First you need to understand Linear Algebra and that’s complicate af.
You do realise that now you can use as much as money as you want and you'll still be left with what you have right noe
Eh, I cannot think of a reason you would *need* linear algebra in order to understand abstract algebra. Rings, groups, and fields should all make just about as much (or as little) sense either way. Speaking of fields, the problem with defining 1/0 is that you are probably going to lose your nice field properties by doing that...
@@kennyb3325 Vector spaces and Vector Subspaces can be quite abstract Concepts that should be introduced in a course on linear algebra before one Endeavors into abstract algebra, at least in my experience
@@9WEAVER9 A first course in abstract algebra need not cover those things. Rings, fields, and groups are more familiar (since we can think of good examples like the integers, rational, or real numbers) and can serve as the entry point to abstract mathematical structures, perhaps better than vector spaces.
Of course, one would want to be introduced to vector spaces before encountering modules.
I now realize just how mathematically accurate NaN actually is in the floating point standard. NaN for life!
True! Thanks for watching!
A professor of mine said that it was mostly designed by mathematicians instead of electronics engineers. He complained that it could've been faster to compute had it used twos complement instead
IEEE engineer 1: do you have an idea how to handle 0/0?
IEEE engineer 2: NaN to speak of
But NaN does not actually work anything like 1/0 and 0/0 do in wheel theory.
Angel Mendez-Rivera Floating point have two zero. +0 and -0 and they have a set of subnormals and NaN is also a set.
"Maybe the real question is 'SHOULD we divide by zero?'" is the best conclusion you could have tbh
7:25 But what is a "nullity"?
the opposite of an infinity. a finity, if you will
4:02
Problem solved. Right?
Vsauce2 (Kevin): WRONG!
Or is it...?
@@dominicstewart-guido7598 Look! Look! Look!
There's still 1 way to get around this. .
Idk how to do a Jake impression.
because every good punchlines has a qualifier in parentheses.
@@dominicstewart-guido7598 *vsauce music plays* Michal: I mean think about it...
*vsauce music plays*
"...and if you divide by zero, you go to hell." Cit.
I sure hope not!
You go to the "bottom" of it.
Hahalmao so funny
guess i go to hell
@@BriTheMathGuy see ya in hell i guess. I'll make sure to bring a 6 pack and some hotdogs for the tasty hellgrill
The "nullity" reminds me of NaN ("not-a-number") in programming. According to standard floating point arithmetic, the result of any operation where NaN is one of the operands is always NaN. The difference there though is that 0 / 0 = NaN, but 1 / 0 = Infinity
God bless you all and Jesus loves you so much, that is why he died for you. By putting your faith in him as lord and saviour you will be saved.
That's kinda built into the code package you use. With quantum computing I suspect this to become way more complicated. Pretty sure with MATHLAB you will have different outcomes more robust than a simple Java math class.
NA and ERR have a way of propagating through spreadsheets.
@@reignellwalker9755as much as people who preach their religion annoy me, i must admit that someone with a roblox pfp praising someone for talking about coding for seemingly no reason gives off a powerful aura
@@reignellwalker9755Saved from what?
0:21 no, they didn’t discover you could take the square root of negative 1, they invented a new number to allow us to, before that you couldn’t take the square root of negative 1, similar to how before they invented calculus you couldn’t do calculus
And it was treated initially as a mathematical trick. And mathematicians know that they are giing something up when they switch from real numbers to complex numbers: ordering. There is not good definition of < and > for complex numbers.
That's what I thought of, too. So we could just define "z" as 1/0, and use it like we use i. 2/0 = 2z, by defnition.
@@glassjester except that the idea of dividing by 0 doesn’t exist, we don’t actually know if 1/0 times two is still 1/0 (with 1/0 acting like 0 does in multiplication) or if it’s 2/0, with the square root of -1 we knew it was going to act like a constant, just like pi, but 1/0 could act like 0 or a non 0 constant, because we can’t agree on its behaviour as a concept
@@Speak22wastaken The idea of a square root of a negative doesn't exist either. We just define "i" to mean that. We could do the same with /0.
@@glassjester But what is 1/0, how does it behave, does it work like 0, a non zero constant, infinity, or something else entirely, and if you multiply this "z" by 0, do you get 1, if so how does that work? Since by multiplying 1 times z by 0 you can either do 1 times 0 and get 0 times z or you can do z time 0 and get 1, by mathmatical laws these would have to be the exact same, meaning 0z is 1, but with 2z times 0 you could get 2 or 0z, meaning that 1=0z=2 by mathmatical laws, which is a contradiction we don't encounter with i
There are algebraic structures where division by zero makes sense. A very straightforward example is the ring of remainders of division by ten or any other non-prime.
2:22, "You can't have 2 definitions for one thing".
English: *has 430 definitions for the word "set"*
Xd
Yea but numbers should never be contextual
This reminds me of stuff I learned in engineering. One was the delta function which is defined as infinity at a single point and 0 everywhere else. If you integrate over it you get 1. I mentally imagine it as a rectangle with 0 width and infinite height and area of 1. And you could multiple delta by constants to get other areas. We used it for theoretically perfect spikes. Calculus classes hated this.
I remember another where when a function went to infinity, it could “wrap around the plane” to negative infinity or even to positive infinity. I think it had to do with finding stable points by wrapping them or something. It’s been so long that I don’t remember clearly anymore. But it sounds similar to mapping the plane to a sphere to make all infinite points touch.
(And thanks reminding people infinity is a ranging concept and not an actual number.)
The delta function does actually have a rigorous definition in terms of the concept known as distributions, or continuous linear functionals on the space of smooth functions with compact support.
As a calculus student, I'm actually really intrigued
God bless you all and Jesus loves you so much, that is why he died for you. By putting your faith in him as lord and saviour you will be saved.
that’s called abstraction. a*b=1, while a->0 and b->inf.
but actually this is the essence of calculus/analysis: when we say that a continuous interval van be decomposed to infinitely many infinitesimal (0-like) intervals.
Isn't a rectangle with 0 width and infinite height a line?
Here's another way to put it:
If you want to define a new set of numbers, you need to show that it's possible to start with already-defined numbers, go into the undefined set, and come back out the other side into already-defined numbers.
If I gain 5 apples and lose 3 apples, I make a net profit of 2 apples. This holds true even if I went into debt because I lost 3 apples *before* I gained 5. This shows we can go into negative numbers and come back out, which means we can define the set of negative numbers.
We know that the area of a triangle is bh/2. Knowing this, we can easily prove that if we have two isosceles right triangles, and we put them together as halves of a new isosceles right triangle, the new triangle has an area equal to the side length of the original triangles. If our original triangles had side lengths of 1, this shows we can go into irrational numbers (since the hypotenuses have lengths of sqrt(2)) and come back out with the rational number 1, which means we can define the set of irrational numbers.
And though I forget the exact formulas involved, imaginary numbers were proven valid the same way. There was some known formula to solve a certain kind of polynomial, but it was found that if instead of just using the formula outright you worked through the *proof* of the formula, you would end up having to evaluate negative numbers under radical signs at some point in the process, even though you might start and end with real numbers.
Conversely, the video demonstrates that the idea of "nullity" swallows numbers like a black hole from which there is no escape, since you have to "give up some rules of algebra" in order to use it. In other words, this new system is demonstrably incomplete and likely has no practical use.
i wouldn't call it "incomplete" just because it includes an "error state"...
Why not invent a set of numbers then that become their "real" counterpart when multiplied by 0.
Eg. 2÷0 =[Nullity sign]2
[Nullity sign]2 x 0 = 2
That's pretty much the best way to put it, and the reason why division by zero is impossible. Unlike other mathematical elements, you can't define it without breaking the laws that already exist. If assuming that giving up the rules that solidify 99.99% of Maths is worth to justify one insignificant operation, why even keep on playing with maths?
@Remix God In the real world you actually can divide a singular piece into more pieces. There's a whole scientific field that came out of that, known as Chemistry, but even if you want to go into something simpler, imagine a slice of bread. Now cut it to 4 pieces. You just divided 1 by 4 in the physical world. Just because the set of natural numbers doesn't allow that doesn't mean it doesn't exist.
In that case, 1/1 is just 1. That also involves the concept that dividing anything by 1 gives you the same thing. If I have a cake and zero people on my birthday party, the only one left to eat it is me, and I will, that's a 1/1 in the physical world.
A nullity, at least as described in the video, is an absorbing element. *That* doesn't exist in the physical world because, by physics laws, energy is not lost. It just becomes something different. Yet a nullity can absorb every other number it's given with any operation. 1/1 can't do that.
@@finnfinity9711 I mean, I guess you could. But aren’t you still breaking some rules?
[Nullity]2 * 0 = 2
You’re multiplying something by 0 and getting something out that isn’t 0.
I asked why I can't divide by zero when I was younger. My teacher told me I could if I wanted to but until I find a reason to there will be no value in doing so.
Well,if we set up the "nullity"=b . Then b=1/0.If that's the case,Then b×0=1.Then multiply both sides by an algebra:a.It becomes b×0×a=1×a.On the left, first calculate 0×a=0.b×0=a.If b×0=a,then b×0 is also=1.Which means 1=a.That means every number is equal to one.
The issue with this proof is in multiplication by zero. You said that b=1/0, thus b*0=1, which is a really easy mistake to make. We always learn that (a/b)*b=a, but this is a shortcut for the truth that (a/b)*b= (a/b)*(b/1)= (a*b)/(b*1)= (a/1)*(b/b)= a*(b/b). In most cases, b/b=1. In your example however, b=0, thus you actually have b=(1/1)*(0/0) =1*nullity =nullity. It was a difficult mistake to catch and it took me several minutes to be able to find it myself
Oh my gosh! Brian! You were my math professor last semester! Hope you’re doing well!
Hey Reggie, I am! Hope you are too!!
Brian
Brain
He just solved ÷0 as a mathematician.
He's living the dream baby
it would be funny to see my math teacher have a popular yt channel
@@use2l wow, so enlightened
4:58 Literally my facial expression when solving math problems 😂
His face is when you think "wait, am I really solving this right or bullshitting myself?"
@@pandakekok7319 yes
1:30 i'm officially using the word "outouts" instead of "outputs" forever now.
i came here to say this, only to discover: i already had. 😮
@@jamieg2427 lmao
@@jamieg2427its been another year do it again
Notice: he never answered the question, the nullity is still not a valid solution, because 0 times the nullity would still be the nullity, so 1 divided by 0 is not the nullity, he’s just thrown a bunch of math Mumbo jumbo in our faces and hoped everyone who had a more comprehensive understanding of this wouldn’t watch the video since they already knew it was bs
Why are you so angry?
@@OliverKraus-q6s not angry. he's just spittin' facts
@@OliverKraus-q6s Being angry in some situations can be the correct response.
In other words getting angry is not always wrong.
If you think he is wrong, refute his argument rather than a personal attack.
As far as I can recall, the meaning of "division by zero is undefined" was that there are no real numbers or complex numbers satisfying 1:0=z.
Makes sense honestly. Infinity is a quantity not a number, and if 0 has no sign it makes sense that infinity doesn't too
I've just watched this video and I'm gonna subscribe straight away because that is mind blowing
Thanks a ton!
Actually on the playground I would say infinity times infinity, infinity to the infinite power, or if I was feeling really petty, infinity plus two
You're so right!! Wish I had put that in the video instead!
The aleph series
Anyone who says that is talking about transfinite numbers.
AKA, they're smart without knowing it.
Yeah, but isn't ∞ × ∞ = ∞?
The proper name of the "unsigned" infinity is: complex infinity. No matter which direction you go in the plane, you tend towards infinity as you keep going.
When u say “problem solved… right? Not quite.” My teacher: AHHHHHHH WTF AGAIN
We focused so much on whether we COULD do it that we never stopped to think whether we SHOULD do it.
"Can't have two definitions for one thing"
Square root of all numbers being both negative and positive:
I get your joke (don't whoosh me), but the square root is a function (which means only one output) defined to give only non-negative outputs for real inputs. It's when you try to solve x^2 = a that results in x=±√a where √a ≥0
No it is |x|
@@jamieee472 r/wooooshwith4osandnoh
@@shinjiikari4199 yeah, what changed?
This kind of explains the quadratic formula.
(-b ± sqrt(b^2 - 4ac))/2
Square root takes the positive and multiplies it by + and - making two answers.
So square root on it's own doesn't have 2 answers, but ± does
why do you look so displeased whenever you're drawing something 😄
"God I hate writing backwards, why do I do this to myself?"
I remember doing these in college...However, it sounds like the problem becomes over complicated. What I learned from proofs is, if something is over complicated with too many rules, then usually there is an undiscovered proof that will disprove it. Occam's razor
We haven't been lied to. When we say 'undefined' we mean "not defined in for these purposes or in this context."
1:16 So this is probably why people think something divided by 0 is Infinity
Turning Ian Malcolm's quote on its heels toward his own profession: The mathematicians were so preoccupied with whether or not they could they didn't stop to think if they should.
I like that you come to the exactly same conclusions as I did when I first learned about the symbol i from complex numbers and had the idea to check what happens if we define a symbol standing for the division by zero.
1 divided by 1 is 1.
1 divided by 0.5 is 2.
1 divided by 0.25 is 4.
Therefore we can say that the increase of the numerator is directly proportional to the decrease of the denominator. Therefore as the denominator approaches 0, the numerator approaches infinity.
Therefore, 1 divided by 0 is infinity. 9 divided by 0 would also be infinity, but it’s a bigger infinity than the first one.
At this point math reaches philosophy and has to be branched into different ways of thought and utilization.
This is a similar line of reasoning that I used back in middle school, the teachers weren't convinced but I thought it was pretty intuitive.
Yeah same here, since zero could go into any number forever without filling the gap.
But it's more fun when you start to involve things middle schoolers wouldn't be able to figure out normally.
@@josephjoestar953 personally, I have always argued with my teachers that if we think of it algebraicly, that as long as we don't use imaginary numbers that division by zero is simply a conserved absolute value addition problem using an infinite series. If you were to graph a negative and positive infinite series with the same absolute value, they would be identical graphically except for which side of the graph they were on. If you think about this way, X + -1/0 is actually X - |1/0|. If we think about it this way, 1/0 is a smaller infinity than 2/0 and so on, but the negative counterparts conserve the value without being defined in the opposite direction. Similarly, an infinite series of zeros is still zeros so zero/zero would simply be zero. 0-D is just zero, 1-D is an infinite line, -1-D is also an infinite line, 2-D is an infinite flat grid, as is -2-D, so on so forth.
Teachers probably didn't know this type of math...too busy teaching Common core math which makes far LESS sense than anything.
It introduces more problems than it solves, meaning it's useless.
Be careful, dinosaurs destroyed their world when a dinosaur wrote 1/0 on its chalkboard. Then the asteroids crashed to the ground. According to a Far Side cartoon.
I've been puzzling over 1/0 for quite some time; it does feel like you should be able to treat it in a similar fashion to sqrt(-1) by creating a new axis of complex numbers, but I've struggled to imagine what such a function would graph.
The idea of the "terminus" makes me think it should be treated more like the center point of a sphere. 1/X becomes the distance from the center, with 1/0 being the true center. 1/1 would then be the shell where "normal" numbers lie.
I'm a philosopher, not a mathematician, so this might be a dumb way of looking at it. I don't know. Still, thanks for posting this; it was interesting.
Hello. I thought I'd like to comment that square root is just the inverse of a square. So X to the power of 2, is the square, the inverse is to the power of a half, or 1/2.
The importance of odd and even numbers comes into play with a cube root, such as to the power of 1/3, and odd powers such as 1/5, 1/7 etcetera.
This is because a negative squared is a negative multiplied by a negative which makes a positive.
This is not the case for cubic functions (to the power of 1/3) or other odd root functions. ( Like to the power of 1/5, or 1/7 etc)
The cube root of -2 is -1.259921.
But the square root of -2 does not exist.
This theoretical anomaly has perhaps been where the visualisation of things has led to the idea of black holes and negative particles, and string theory.
@@danc.5509 the square root of -2 does exist, just not within the real numbers
I'm not a philosopher or a mathematician, but it seems like pretty interesting idea. "j = 1/0" I can't think of any real world uses, but the same was said about negatives and square roots of negatives.
@danc.5509
Well is kinda depends
First off if you limit yourself to the reals you can't solve sqrt(-4) but if you expand to allow complex numbers
Then you get 2i
i is defined as i =√(-1)
It doesn't "exist" but using it you can solve for a lot of things and has some real world applications
@whyme1698
While there are some ways to have x/0 not be undefined using a variable like "i" is because it can be used to make two different numbers equal each other which means that it can't exist
(1/0 = j)
Is because there are a lot of ways to mess with it
So:
(1/0) = j
Assuming absolutely nothing about j:
So then:
1 = 0j
And because any number times 0 is 0
1 = 0
Which is a contradiction
You can not just define your way out of 1/0, because division is the undoing of multiplying. Since most any number n * 0 is 0, we just do not know what the original number could have been. Higher-dimensional numbers (complex -> quaternions -> octonions) become more problematic with division, because there is just too many ways to get the same product.
BIG OUTOUTS :)
😂 that’s what I get for trying to break rules
When I was in college I studied projective geometry and homogenous Cartesian coordinates. So, (x,y) would be expressed as (x,y,1) or (2x,2y,2) etc.. We determined that that there was a single point at infinity in each direction of x/y. Further, all the points at infinity formed the line at infinity. The notation would be (x,y,0) for any particular point at infinity.
In addition, using the General Projective Transformation, we could transform a point at infinity to become local, but losing a point previously local to become inaccessible. This was done by matrix cross products.
For example, a simple addition nomogram, with three parallel lines, could become three concurrently intersecting lines, with the point at infinity now appearing as the common intersection. As the three lines approached the central point, the associated scales grew greater from both the positive and negative directions.
As far as I know, the GPT is how the math behind computer graphics is handled. It allows for a single technique to be used for scaling, rotation, magnification, etc.. And the transformations can be stacked and reversed. But I've never seen this used to handle the points at infinity.
I like the approach of how everything equals everything else, its almost like it too the definition away and left everything undefined
i had no idea this was released today a year ago and that just makes this better
Just like how we assigned a undefined number to the square root of -1, anything divided by zero could be _z_ for example.
Not so simple. The problem is that division is multiplication of a multiplicative inverse. To say we can divide by 0 is to say that 0 has a multiplicative inverse. Hence, if _z_ = 1/0 and _z_ = 2/0, we get that 1/0 = 2/0 (equality is transitive) and hence (1/0) * 0 = (2/0) * 0, implying that 1 = 2, a clear contradiction. That is, _z_ * 0 would not be well defined.
I’m so glad you brought light to this, because I’ve been thinking about this concept the exact way you mentioned it, and I’m really happy that this concept is out there, being explained so masterfully yet simply.
Funny, a few years ago, I pretty much had the same idea of defining 1/0 and I called it zeta. I just thought, well, we defined sqrt(-1) = i, what if we define 1/0=zeta. After playing around with it, I noticed 1/0=zeta -> 1/zeta=0 by algebra. I concluded I just made a complex sphere. Also x*zeta=zeta just like x*0=0. I came with the phrase "Zeta, the other zero on the other side" for a clickbait title if I ever gonna talk about this lol.
Then I got stumped when I ask what about 0*zeta, which you also discussed. Interesting stuff.
I didn't think of the nullity number though.
Would be interesting to learn of more properties of zeta!
Have you talked about zeta yet?
@@kovanovsky2233
(Zeta/zeta)*(0/i)
This takes you to Buzz Lightyear territory! 🚀♾➡️➡️
1/0 = zeta -> 1 = 0*zeta
1/zeta = 0 = 0/1 -> zeta/1 = 1/0 = zeta
Simple 1/0 is infinity and 1/-0 is negative infinity and split them fully
"Don't tell your teachers"
Teachers that are watching this video: you have become the very thing you swore to destroy
Can you do this in math: yes, as long as you're being consistent.
Should you: only if it's useful.
Done.
Finally someone makes a video on something related to the Riemann Sphere, which isn't a lecture. Can I also request a video on looking at complex functions and transformations on the Riemann Sphere, because they're really mind-blowing and eye-opening. What functions correspond to reflexions across the 3 main axes of the sphere, and stuff like that. Thanks for this video!
I'm pretty sure "Your Scientists Were So Preoccupied With Whether Or Not They Could, They Didn’t Stop To Think If They Should" applies to this one.
i wont tell my teacher, im graduating
I never tell my students they can’t divide by zero I always remind them of the idea of new number sets. Aside from wheel algebra there are also the hyper real number sets. Good job
Can't divide by zero in the hyperreal number system either, but still cool.
@@edomeindertsma6669
Technically no but very close to the real thing
It is absolutely true that division by zero is undefined (impossible) on the field of real (and complex) numbers, which is the only field any high school or lower students will ever work with. In fact, tons of students get things confused because they don’t really understand that certain functions (especially trigonometric ones) have entirely different results based on what they’re defined in. I’ve seen a perfectly intelligent (probably too clever) kid disbelieve that 0.99…=1 because they heard about the hyperreals and said that 1>0.99…1>0.99… without really understanding how it actually works. I don’t even know if that statement is true in the hyperreals, but in the real numbers 0.99…=3/3=1. And indeed, anything else would cause problems.
Because its immposible
The thing about -∞ = +∞ is that it actually has some physical significance. I'm referring to the absolute (Kelvin) temperature scale.
Well... yes, but actually, no. (I say that as a physicist)
@@angelmendez-rivera351 Wait! I need to know more about this!
@@maxthexpfarmer3957 In statistical thermodynamics, we work with the quantities temperature (T) and entropy (S). One thing you probably have heard a lot is that we cannot reach absolute 0 for temperature. This is true,... but despite that, we can actually reach negative temperatures in Kelvin. The idea is that some physical systems have a highest energy U they can attain. This energy U is a function of the entropy S of the system. Entropy, energy, and temperature are related by the equation T = dU/dS. Now, if that physical system attains its maximum energy possible, what happens if you increase S even more? Then U obviously cannot keep increasing. It can only decrease from there. If S is increasing while S is increasing, then dU/dS < 0. In other words, the temperature has to become negative. However, this makes the system unstable, so the temperature begins to decrease rapidly in the negative direction, and intuitively, this looks like "T is going to -♾, looping back around to +♾, and then continues decreasing until it reaches stability." With this picture in mind, it looks analogous to the idea that -♾ = +♾ = ♾. But while I can see why it seems superficially similar, it is far from the same thing. Why?
1. Because T = dU/dS is only an approximation. It is well-known today that at very high temperatures, statistical thermodynamics does not describe reality accurately. It is also likely that there exists a highest temperature attainable, the Planck temperature, and if that is accurate, then that means that there is no such a thing as infinite temperature, and that temperature could never loop around the way it is described here. Besides, in reality, entropy changes discretely anyway. Entropy is defined as S = k·ln(Ω), where Ω is the number of microstates corresponding to the macrostate of the system, and k is Boltzmann's constant. Ω is necessarily a positive integer, so it can only change from Ω to Ω + 1, there is no smaller possible change, making it discrete. So the smallest possible change in entropy is k·ln(1 + 1/Ω). However, we can approximately these discrete changes as continuous changes, because given how astronomically small k as a constant is, and given how even smaller 1/Ω is, these changes in entropy are so small, that we can approximate them with continuous changes, so using derivatives gives a remarkably accurate model for low temperatures.
2. Also, this idea of unsigned infinity does not correspond to physics because absolute zero is still unreachable, and thus the analogous of division by 0 is still not possible in it.
So again, there is some very superficial similarity if you ignore the rigor, but otherwise, it is not really analogous.
@@angelmendez-rivera351 I had no idea!!!!! Thank you for taking the time to let us know
@Angel Mendez-Rivera your comment motivates me to continue persuing physics :)
Very happy to give this video the 1000th and more than deserved like, This is a really interesting qubject
Thanks so much!!
I've always been told by my math teachers (since the 90s) dividing by 0 results in "null set" not 0 technically but functionally it's 0. Thanks for explaining why!
It is wrong. First of all, how do you devide by nothing? And second deviding by an infinitely small number != 0 will get you an infinitely big number (approaching infinity).
So it cannot be 0.
Instructions unclear, divided my home by 0 and now am missing a ceiling.
2:28. Me at this point: Well 0 is negative and positive. Math is already weird so x/0= [infinity] and [minus infinity] wouldn't shock me
Well, IEEE floating point numbers work a little bit like that. Except that they distinguish between +infinity and -infinity, but then there are also different representations for +0 and -0.
The different binary representations of +0 and -0 are really just an implementation detail. They are two different ways of describing the same number in the sense that +0 == -0 is required to evaluate to true. But you're right about how all the indeterminate forms (0/0, 0*Inf, Inf/Inf and Inf-Inf) all evaluate to NaN ("not a number") in IEEE 754. And I think NaN shares several other properties with the "nullity" in the video (like NaN-NaN = NaN).
@@weetabixharry +0 and -0 were there because you still want to retain a sign even when the truncation caused the number to be zero. It can be even argued that they really represent infinitesimals in some sense. The actual implementation detail is that they are kinda aliased to the real zero, which was considered an acceptable tradeoff.
Math is even more broken when you prove the sum of all the counting numbers equals -1/12
If you just ignore the absorbing properties of zero and infinity, these numbers become a lot more beautiful a lot quicker
1/0 has always been defined as Infinity; it is only recently that "undefined" became the standard answer because computers became involved.
Really great video I'm French guy but I understood your video
Glad you liked it! Thanks for watching!
you're under arrest for destroying the universe
Can't we see the nullity as abox where we put all numbers. The nullity = 1 and the nullity = 0. All numbers equal the nullity
0/0= the nullity
The nullity × 0 = 0
The nullity is defined as every number at the same time.
The whole numberline is the nullity.
Isn't this the nullity
Is therefore 1/0 (the 1 and the 0 taken from the nullity numberline) some other numberline
Did I just revolutionize math?
I've always misunderstood why you can't divide by 0 when I was younger, my thought process was that if you divide by nothing then why wouldn't the answer just be whatever the numerator is
You could also map out quaternions, octonions, and so on to multidimensional donuts. Great video.
My programming teacher presented this to us in 2009 at high school. I still don't know why we still don't see this normally.
We had it in 10th grade (Germany). I don't really understand if it's this but it think so. Basically we are not dividing by 0 but with an infinite small number.
@@zekiz774 yes, the idea is that divide by zero tends (I think it is the word) to infinity.
@@iranmaia91 i'd think it's not really used because having an ordered set extending the real line (i.e., separating -inf from +inf) is more useful than merely including an edge case for division.
For the reasons given in the second half of the video. It breaks a load of things.
Ph.D. mathematician / retired math professor here: I'm watching this and mostly nodding on your points, BUT it still remains incredibly irresponsbile and plain wrong to say, or even imply, that you are dividing by zero. You absolutely are not dividing by zero. Even the use of simple Calculus limits (which I knew you would do beforehand) does not get around that simple fact. And the creation of "the nullity" has no utility or value to more than 99% of humanity - and none to this mathematician, either.
0:44 bros face looks like hes questioning how that one child is screaming at his ipad on why u can divide by zero
The dividend remains the number into which zero is divided, because the number is divided exactly zero times. I learned this in second grade. My little toddler mind had a concept of enumeration when I was playing with building blocks, though I didnt have the vocabulary to express those ideas. It really isnt complicated. Matter is finite. Zero is not infinite, however. It is measurable in quantity, which is a quantity of none. The answer is NOT infinity.
I think maths needs a solution/ definition for 1/0. This one sounds quite interesting. It would be nice to see some long existing problems solved by that
What problems for example?
@@rhubaruth the amount of biscuits I have eaten in my life
@@rhubaruth IDK but I heard somethings in physics are unsolvable like singularities, which maybe solved if we can divide by 0, though I have absolutely no idea because I don't know anything about it
@@atharva2502 Although you said you have no idea, I do think there is a significant point in your statement. I think its obvious through the study of calculus and real analysis that the idea of 0 is very closely linked to the idea of infinity. In that respect I could see a solution regarding infinities in physics (such as center of black holes ie. singularities) being related in some way to the idea of dividing by 0.
There is a tiiny wiiny clumsy detail we're forgetting here:
1/0 = INF
2/0 = INF
1/0=2/0 WTF?
And, by the rules of expanding fractions:
x/0 = x*k/0*k = x*k/0
From which:
x = x*k
This contradicts basics of math.
So, no, Infinity isn't that good of a solution. Not in common algebra at least. If it was, why wasn't it implemented yet?
One thing you lose in replacing positive and negative infinity with unsigned infinity is the differentiation between functions which blow up n the positive direction versus blow up in the negative direction. You’re basically replacing “becomes unboundedly positively large” with “becomes unbounded in some direction.” It’s useful to be able to, for example, have the notion that positive infinity is strictly greater than any finite number. Of course you can define singularities like in the video, but I suspect in most contexts it’s better to keep positive and negative infinity as separate concepts.
Nullity is “strange matter” of numbers
Huh?
@@angelmendez-rivera351 Strange matter is a theoretical form of matter that converts any other type of matter it touches into itself.
Imagine a gray goo scenario, only waay worse, since theoretically, if even one particle of strange matter touches something like a planet, it converts the entire thing into strange matter. Pretty freaky if you ask me.
Dividing by zero is absolutely easy.
If I have a cake and divide it by zero... in other words, I don't cut it in two or more pieces, it simply remains the same...
That's what I call common sense, and why I always struggled with maths...
That's dividing by 1.
You are giving the cake not to several persons partially but to 1 person (maybe yourself) as a whole unit.
A unit is 1 part of 100%.
the way i do division is that the answer is the number of times you add up the divisor until it makes the dividend so dividing by zero gives you infinity
"Should we divide by Zero?" I still say no, but I don't think the division operation even happens at all when trying to divide by zero. If I divide 12 by 2, I'm laying out, for example, a set of 12 empty boxes into 2 groups, with 6 per group:
🔲🔲🔲🔲🔲🔲 🔲🔲🔲🔲🔲🔲
Laying out that set of 12 empty boxes into 3 groups instead is 4 per group, and so on. If I divide 12 by 1, I'm laying out that set of 12 empty boxes into 1 group, with 12 per group:
🔲🔲🔲🔲🔲🔲🔲🔲🔲🔲🔲🔲
If I divide 12 by 1/2, I'm laying out that set of 12 empty boxes into 1/2 of a group, leaving room for 12 more in the whole group, resulting in 24 empty boxes per group, which corresponds with multiplying 12 by 2:
🔲🔲🔲🔲🔲🔲🔲🔲🔲🔲🔲🔲
🔲🔲🔲🔲🔲🔲🔲🔲🔲🔲🔲🔲
Laying out the original set of 12 empty boxes into 1/3 of a group instead is 36 per group, 12 into 1/6 of a group is 72 per group, and so on, with the denominator getter closer to Zero. What if I just make the denominator Zero? Here's the result:
I just laid out the set of 12 empty boxes into 0 groups, which means I didn't lay them out at all. I'm still holding onto them. I didn't divide them into any groups. Division doesn't occur.
NOW HOW ABOUT NEGATIVE TWO GROUPS No. Show me what -2 groups look like and then we can discuss.
HOW ABOUT i AMOUNT OF GROUPS No. Show me what i groups look like and then we can discuss.
and yet, x/-2 and x/i makes sence but x/0 doesn't
@@esajpsasipes2822 Actually, I've expanded on this elsewhere since my OP. Let's use total dollars divided into dollars per person to get the number of people I'm paying or getting paid by.
$12/$2 per person = 6 people getting paid by me
$12/$3 per person = 4 people getting paid by me
$12/$1 per person = 12 people getting paid by me
$12/$0.50 per person = 24 people getting paid by me
$12/$0.25 per person = 48 people getting paid by me
$12/$0 per person = I'm not paying anyone and no person is paying me 😂
$12/$-2 per person = 6 people owe me money
$12/$i per person = Nope
@@pronounjow Thats because we use reals (R) to express money, and i is not in R. If you had something in complex numbers (C):
12/i = 12/sqrt(-1) = 12/sqrt(-1) * sqrt(-1)/sqrt(-1) = 12sqrt(-1)/-1 = -12i
It would be -12i.
Complex numbers are used (apart from pure math) in electrotechnics to calculate things around AC circuits with capacitors and coils, in 2D graphics to calculate rotations (as it's simpler than using vectors), and it is present in quantum theories.
3:36
JESUS CHRIST, THIS GUYS RIGHT THUMB IS BROKEN! SOMEONE FIX IT NOW, FIX IT NOW!!!
I was looking for this comment 😂😂😂
He probably has hitchhikers thumb
I still see problems with this
first since (like told in this video) you can sometimes make sense of terms like infinity - infinity specific to a function and can get normal numbers (but also +-infinity). That means the nullity can be equivelent to any number.
second when you transform equations with variables you can sometimes get plain wrong results when not accounting for the case that the variable may be 0 when dividing through the variable
Bro casually increasing calculus students syllabus!
Actually we can divide anything by 0, but we won't get an end, so that means it's infinity, like this:
1 / 0
-0
1 8
= Infinity that never reaches nothing
Another way is:
1 / 0
-Ind Inf
Ind
= Indeterminate Form
So in both ways the answer is Undefined, because "we must agree that Answer × 0 is 1, and that cannot be true, because anything multiplied by 0 is 0."
7:01 Why x-x=0x^2, and not x-x=0x?
to make the nullity in the positive domain
There is a poetry to infinity in the Riemann sphere in that infinity has "arbitrary direction" just as 0 does.
3:30 - Are you folding space? Without SPICE‽
Isn't this topology?
WHAT IS THAT SYMBOL ‽ ‽ ‽
Very good but there’s still a problem.
If 1 = infinity * 0, and we say that infinity * 0 = the nulity, then 1 = the nulity. If you divide 2/0, you get 2 = nulity. So if you substitute for the nulity, you get 1=2. You can’t really just get rid of some of the rules of algebra.
Throughout all the proofs out there, I think it’s best to just keep it undefined. Maybe it will be defined one day, but it’s true definition must keep math consistent.
I bet no-one else had a teacher who told the class that if they divided by zero they'd explode. I don't think any of us believed it, but I haven't taken the risk in the 50 years since :-)
Teachers *HATE* him for this 1 simple trick!
I hope not!
4:00 Problem solved, right?? Not quite.
Me ragequitting the video
Thank you... Very informative and generous .. And yes i will not tell the prof or teacher.. 👍👍👍👍👍
You bet! 🤫🤫
my interpretation has always been that it's more convenient to assume that infinity is not possible unless we say it is, including infinitely small numbers. for example, 1 / 0 would equal infinity if you state that an infinite number of zeroes would fit in the numerator, without making it 0 (four 2s can be subtracted from 8 to make zero) but once we get our result of infinity, we're still left with 1 in the numerator. this implies that infinite zeroes manage to approach 1, as zero cannot truly be nothing when used infinitely. 0.00001 times infinity equals infinity, so any number approaching zero would be infinity, until zero itself. and that's good enough for my purposes.
Consider infinity to be like an "edge" instead. What happens when you cross it? (like in your Infinity +1 example).
It wouldn't "absorb" it like you say - but rather the perspective moves beyond, into another "measure" of infinity. Using the stereographic projection point of view, it is like crossing into a different 2-dimensional plane, where if you were to stereographically project this new plane onto the initial plane, the sphere that it forms would have a point at zero which would map to *the same point* as the point at infinity/at 1 on the initial sphere. Imagine it stacked on top with the only intersection between the two spheres being the point which represents infinity on the first sphere and 0 on the second sphere. Then when we consider dividing by zero, we can understand it from a new point of view in that what we are actually doing in the process of dividing by zero is like "crossing the edge" and moving between different measures of infinity. IDK how useful this point of view is yet, but its another way of looking at it that I thought of. I've been mulling over an idea relating this to complex numbers/quaternions, where the additional measures of infinity are represented as a form of complex number.