Saying _"you can't do XYZ"_ in maths is really just a shorthand for saying _"The systems of maths that arrises by expanding an existing one to include XYZ is not interesting / useful / non-trivial / connected to other branch of maths."_ This is probably obvious to anyone who has studied higher maths and is familiar with the idea of there being many different systems of maths (different number systems, different starting axioms, etc...) that we can choose between at will; but far more alien to those who haven't gone beyond high school maths and think of it as a single, rigid, god given, singular thing.
@@nel_tu_You very much can, infact you can with any real number. But it requires branching out to complex numbers. It's nicely explained in this video: ua-cam.com/video/3C_XD_cCeeI/v-deo.html
Exactly! Maths is just a game of how much stuff you can make up that isn't contradictory with itself. The only place where you can't really do something is when it creates a contradiction in itself. For example, just like in the above video, we know if the system included 1, it would have a contradiction as we would get 0=1, so we just say "nah screw that bch, I never even liked one" and kick it out of the system altogether. Can't have the contradiction if the system doesn't have the number 1!
After lots of hour I finally implemented a fully working calculator for the zero ring: def add_or_mul_or_div_or_sub(a, b): return 0 It was hard work but will be worth it for future calculations
This is a great example for why we "can't" divide by zero, as defining it using a zero ring serves little to no purpose. (What are you going to do with a number system where R is simply zero and only zero?)
I did it nodejs for you guys 😊 const zero = require('zero-int'); const fns = require('funcs'); function zeroFactory() { return zero.create(); } function addMulSubDiv(a, b) { return zeroFactory(); } fncs.assign(addMulSubDiv, zeroFactory); require('export').exportFncs(addMulSubDiv, zeroFactory);
Modern texts, that define fields as a special type of ring, include the axiom 0 ≠ 1 for fields (or its equivalent) so that the zero ring is excluded from being a field.
@@mortvald they literally made a black hole in a quantum simulation then sent information through it and it came out of the other end in another quantum simulation in a completely different computer. There's no reason to turn science into dogma when the math is actually making predictions.
I was not expecting pure math from this channel, but I probably should've given that I learned about semigroups from a one-off comment in one of your Python videos. This is awesome.
@@omegahaxors9-11 well no, it didn't fall apart. The system you get might not be useful for anything practical (that you could think of, it might well have abstract applications or implications), however saying things "fall apart" is disingenuous as it suggests the foundational theorems of mathematics are not sound, yet in this case they are, they gave you something that works as they dictate. It maybe doesn't work how you'd imagine or how you'd like, but it still completely works.
@@sofisofi-gx8te Incorrect, Edison. This is why Tesla was right. AC circuit voltages and currents cannot be calculated correctly without the use of i. The only exact solutions for many AC design and analysis problems include an imaginary component. As mind blowing as this may be, imaginary numbers are quite real and useful in electromagnetism, quantum mechanics, particle physics and light propagation. A better word for "imaginary" here is "complex." Euler's Identity e^iπ + 1= 0 and its proof demonstrate they are real. The fact that you exist does. Fundamental forces like gravity operate by mechanisms that can only be accurately described with the use of complex numbers. Complex numbers, like gravity, weren't invented. They were discovered.
In a commutative algebra lecture, the professor gave the important proposition that the localisation at a (multiplicative) set is 0 if and only if the set contains 0 a very fitting name: If you divide by zero, everyone dies (when something becomes zero, people often call that „killing the element“).
In a logical sense dividing by zero means doing nothing. Like multiplying by zero means doing nothing to the number you are multiplying it with. You are asking about performing a task zero times.
@@stirfrybry1 That's not at all what it means. If you're talking about the "normal" number system and not the weird "zero only" system from the video, then multiplying by zero is not doing nothing. You're turning the original number into zero. That's not nothing. Dividing by zero is also not nothing. The result is indeterminate, but if you would divide by something that goes very near zero, the answer goes to infinitity. So this also is not doing nothing.
The main difference here is that including a square root of -1 is a field extension of R. In fact, it is a very special field extension. It is the splitting field of R (in many ways, it is better than R). But the ignoring that, the important thing is that C has R embedded in it; the natural homomorphism from R to C is injective, or in other words, the kernel is trivial. This means R is isomorphic to a subring (subfield) of C, so this extension doesn't lose you any of R. On the other hand, if you include 1/0, the new ring no longer has R embedded in it - it is not an extension of R. The natural (and only) homomorphism from R into the zero ring is as far from injective as it could be - the kernel is the entire set R. So there is nothing that looks like R inside the zero ring. This should be pretty obvious given that R is uncountable, while the zero ring only has 1 element.
@@hach1kokoit's pretty straightforward (it's also just a UA-cam comment). The parts that stick out as not straight forward are things to explore. More fun ahead
Given the typical content of this channel, i was assuming the set of numbers we would arrive at would be blackboard F, for floating point as specified IEEE 754
Had a teacher once ask me if I take what's in your hand and take away half what do you have left? Once I answered he said and if I keep taking half what do you have? This was, of course his way of telling me about atoms. Then he said if I take away the atoms what do you have? I said nothing. I have nothing left. He said everyone keeps saying that, but the answer is you have everything else. Once its gone, you have the whole universe. I wondered where he got his drugs from. After watching this, apparently he was right.
I thought you were going to talk about the projective real number line, which has an inverse of 0, so division is defined on everything but now addition/subtraction isn't.
@@Metal_Master_YT There's way more than I can possibly explain in a comment but the tldr is that you add a single point at infinity to the real number line.
@@Tehom1 that's more like a tldr of a tldr. that was literally a single sentence. contrary to popular belief, I actually do have some patience to read. but hey, if you don't have time, then don't let me bother you.😅
@@Metal_Master_YT The projective line is the real number line bent into a circle and glued together at the ends. It adds a new number which you can think of as the point where the ends are glued. This number acts like infinity in a sense, but to make the algebra work nicely you need some more complicated stuff called homogeneous coordinates. Roughly these are like taking a diameter of the circle and taking the antipodal intersection points of the diameter with the circle as coordinates. You can then consistently define algebra with ∞ and 1/0. You can’t, however, do algebra with 0/0 still. In order to make a structure where that works, you need what is called a wheel. These are a bit like further extensions of the projective line, but they need more difficult algebraic rules than before to account for 0/0.
@@HPTopoG interesting, thanks for explaining it to me. although, are the points that you are generating being plotted on a standard coordinate plane? and which of the 2 antipodal values is the x or y?
Alternatively, 1/0 = infinity + 1. I think that checks out, but I'm not a mathematician. Also, it might cause an infinite improbability drive to power up somewhere.
@@homan-awa I understand that, but that is a deduction based on the claim (at least in this video) that everything times 0 is 0, which is not a trivial statement.
The hypothesis in the video is that we are working in a ring. en.wikipedia.org/wiki/Ring_(mathematics) What he proved is that a ring having an inverse of 0 is a ring with all numbers being equal. If you want 0 to have an inverse, you have to concede some ring properties. Properties that we are familiar with.
@@Ghost-Raccoon I would agree, this is video's process to be seems like using the rules of our maths system (which form a paradox) and deciding that we should let "1=0" be true instead of letting "some x multiplied by 0 could be non 0" be true
1:13, pausing here for a moment, 0² is 0 so the √0 = 0 but wait that's 0 / 0 which extrapolated to N / 0 means N/0 = 0 In simplest form this means division and multiplication can be represented as follows without adding any extra values: a/b = while ( a >= b && c < b ) { a -= b; c += 1; } a*b = while ( b >= 0 ) { c += a; b -= 1; } The destination (c) in both cases starts as 0, skipping c < b is what causes the infinity loop. Basically N / 0 is the edge case of faulty division definition/s. **Edit:** I've found that it's better to compare lengths of the remainder vs length of the divisor. The length of the divisor is always at least 1 while the length of the remainder is always decremented by at least 1, inevitably the length of the remainder is eventually declared as 0 (even if the is the digit 0) forcing the loop to break since anything with a length less than the divisor will obviously fail the >= check inside the loop
you can also tweak the rules slightly to create a useful system, like what was done with floats; 1 / 0 = infinity. 1/-0 = -infinity. 0 * infinity = NaN, NaN with most operators just produces NaN.
Little fun fact about NaNs is that they actually encode. Though due to most NaNs being the result of trying to do a mathematical operation on a NaN these almost universally just end up as a huge wall of "Tried to do math on a NaN" codes. Not always though. If you look at the binary of a NaN float you can use that as a sort of error code to determine what exactly caused it. Just don't be surprised if the value is something meaningless or random because NaNs have undefined behavior and are completely dependent on the implementation of the float itself. There is zero standardization or guidelines across the entire industry.
Eh.. it requires a lot more than slightly changing the rules though. If you do this, you'll have to give up some very basic properties of math that will make doing everything overwhelmingly more complicated and you'd need to reprove basically every formula (well, a lot of formulas won't be reproven because they won't be true anymore) because nearly every proof uses those basic properties. For instance, is x - x = 0? Normally you'd say that's obviously true.. but what if we have 1/0 - 2/0? 1/0 = infinity, 2/0 = infinity, so 1/0 - 2/0 = 0. 2/0 = 2(1/0) though - that means that 1/0 - 2/0 = 1/0 - 1/0 - 1/0, which evaluates to negative infinity.. which implies that negative infinity = 0, which is obviously nonsense. That means you can no longer say that x - x = 0 in that new numbering system (or maybe that 2x isn't equal to x + x which also causes a lot of problems).. which is going to be causing a whole lot of problems with a lot of proofs. In the end basically every formula will still not function with any of those new numbers, which makes it functionally the same thing as being undefined because it'll still be impossible to actually use it anywhere. It also has problems with "what is -0?" - after all, how do you know whether 1-1 is 0 or -0? 1 - 1 = -1 + 1 = - (1 - 1), therefore 1/(1-1) = 1/(-(1-1)) = -1/(1-1), which implies that infinity = -infinity. If you want to handle this, you'll have to say something like x+y =/= y+x., or maybe that x(y+z) =/= xy + xz (even with non-infinite numbers). This is going to cause a lot of problems. There are almost certainly a whole lot more problems with it - there are *very* good reasons that it's treated as undefined and that the numbering system you're describing isn't used. The only reason it "works" with floats is that floats aren't intended to be an accurate way of calculating things - they're by definition not exact values, so any time you're working with floats it's to be expected that sometimes you won't get correct answers and you just have to deal with it being incorrect sometimes. Floating point numbers already break most of the rules of math, so they don't really care that infinity also breaks them since they were already broken by regular numbers anyway.
When you got to the "1=0" part and said that's not a contradiction, I had to double-check the upload date to make sure I'm not watching an April Fool's prank video lol
Man, I really had hope that this video would actually explain how to divide by 0, but then starting from 2:13 everything goes wrong. Why can't people just understand how simple the extended complex plane, Riemann sphere, or wheel algebra is? The inverse of 0 is infinity, it's as simple as that.
eh, just make it a wheel. You get zero, you get infinity, you get any symbol [x, 0], and you get a special element [0,0] (where for any *regular* value [a, b], to translate it into the real numbers, is just a/b, though some values such as [x,0] can't be translated)
1:44 you say "if we also throw in[] inverses of every positive whole number" but that's somewhat redundant, right? Wouldn't it suffice to just use inverses of primes?
Excellent observation! A fortiori adding in reciprocals of primes is sufficient, but it's not necessary to make the construction of the rationals dependent on facts about primes. I didn't mention this in the video, but the first step in performing localization is to compute the multiplicative closure of the set you are adding inverses for, then to throw all those inverses in. So if you did start with the primes, you would quickly compute their closure to be all nonzero integers and arrive back to that point in the video ;)
Isn't x/0 like Schrödinger's cat because technically you're not taking anything from it so it could be x, but if you multiply with the reciprocal value (0/0 is a weird fraction, but we're talking about division by zero soooo...) it would be zero. As obvious, I am no mathematician ^^
Localization isn't the only option for extension. There is also the one point compactification, and the two point compactification of the real line, where you add one or two infinities respectively. They have the drawback of not being fields. In those spaces you still can't divide zero by zero. And IEEE floats are very similar in behavior to the two point compactification apart from floats only representing dyadic rationals and not even all of them.
How does division by 0 work in the two point compactification? I thought it couldn't work because 1/0 would be ambiguous as to whether it's positive or negative infinity?
@@SJGster You are using field axioms to manipulate that expression. It isn't a field. In particular neither addition nor multiplication are associative. What is true is that not every source considers 1/0 or -1/0 to be defined because they don't follow from the limit point construction of the space as robustly as other properties. And another reason why people sometimes choose to leave them out is because if you do you get a weak form of associativity and distributivity.
this pretty much sums it up. in a ring, if we allow the additive identity to be equal to the multiplicative identity, we get the trivial ring with a single element that technically has all the properties but is completely useless. It is in fact also a field, a vector space over itself, an algebra and so on but again not very useful... However, there is apparently other places where it is useful like in projective geometry where we treat unsigned infinity as a normal number and in riemman spheres.
@@greenwaldian Actually the rule "anything times zero is zero" applies in IR, it may not apply in the new set that we're creating, but we can still proof that 0*(1/0) = 0, by doing so : 0*1 = 0 so 0*(0*(1/0)) = 0 so (0*0)*(1/0) = 0 so 0*(1/0) = 0 so 1 = 0
Maybe a way of thinking about it is to understand how maths tends to do things. We've defined the solution to equation to (0*1/0) to be 1. Okay cool. But we have the other rule that says that anything multiplied by 0 is 0. And thus, we have shown that 1 = 0 the whole time. We started with assuming that 1 and 0 were not the same thing, but we followed our rules and it turns out they were the same all along. Imagine we were talking about something else. Let's say we are talking about the number 2/4. Is 2/4 the same as 1/2? Well no, just look at it, they have different numerators and denominators. They're not the same, right? Well, if we follow our rules about cancelling shared parts of the denominators and numerators, we reduce 2/4 down to 1/2 and voila, by our rules, it turns out that they actually ARE the same number after all. This is a common idea that pops up in higher maths. For a vague example, you define what a 'group' is (to simplify, just think of it as something which is like the integers), you get this thing called the 'identity'. This is the element you get when you take something with its inverse (say, for example, 2 + (-2) = 0, 0 here is the identity). Is the identity unique? Can there be multiple identities? Well, assume there are two identities, do some algebra using the rules you set out, and voila you show that actually they are the same after all. I hope that's helpful for you.
I think one of the factors of it being un-defined is that it doesn't explain or help with anything if it's localized. In comparision,complex numbers is quite useful in a variety of things from quantum physics to engineering. A different branch for a division of 0 quite literally and metaphorically gives us nothing.
@@Fluffy6555 The key expression being "approaching", ie. we're talking about limits in calculus. And with the way limits are defined, you actually never end up dividing by zero.
@@shockthetoast except, in calculus, every single time, we don't just get really close, we make sure to actually get on to the thing. Otherwise d/dx x² would be 2x+h, for 0≈h≠0.
Then what would happen if multiplying by zero doesn't *have* to result in zero? If you, for example, assume that infinity * 0 = 1 , would it work then? 1/0=infinity, infinity/0=1, infinity*0=1, you probably need a doubly lined 0 as done with those C and R and Z's.
The fact that 0*x = 0 is usually derived from other facts. So it becomes a question of which properties you're willing to drop. In the typical formulation there are three other properties used to prove this: We have 0+a = a. This is one of the defining properties of 0, so you probably want to keep that. The distributive property: (a+b)*x = a*x + b*x. Subtraction: a + b - b = a With these properties we can do as follows: 0 + 0 = 0 (0+0)*x = 0*x = 0 + 0*x 0*x + 0*x = 0 + 0*x 0*x + 0*x - 0*x = 0 + 0*x - 0*x 0*x = 0
It works, when 0 * ∞ = E, but not 1 (nor 0 or ∞). 1/0=∞ and 1/∞=0 make sense, but from that doesn't follow 0*∞=1, as you e.g. would need to multiply 1/0 with 0, yet you cannot reduce 0/0 to 1. From the first 2 rules you get E = 0/0 = ∞/∞ = 0*∞ = 1/E = E², which helps to solve all equations.
when you do this you have two choices (one-point vs two-point compactification) in the first you say 1/0 = inf and inf = 1/0, and you do away with the ordering relations, and inf is a number that can't be subtracted (like how 0 couldn't be divided by) in the second you define 0+ and 0- not just 0, and you say 1/0+ = inf and 1/0- = - inf, now we keep the ordering relations, and now we can't add or subtract at all since 0- and 0+ must be distinguished. In general we can also construct a system where values are sets of numbers rather than individual numbers, and consider operations as images over sets where we generate from singletons of another set and make all operations total and closed by imposing sets as their solutions, this gives us transfields from fields such as the transreals from the reals or the transcomplex numbers from the complex numbers
I disagree with the framing of this: When talking about the complex numbers or the dyadic rationals, you emphasized the "throw it into the existing set of numbers and follow the known rules of algebra" part. In these cases, you have a base structure (a ring for Z, a field for R) and extend it in a way, that still satisfies closure and the axioms (in your examples by adjoining solutions to x^2+1=0 or 2x-1=0). Thus it retains key properties and contains the original ring/field as a subring/subfield (or at least a ring/field that is isomorphic to them, depending on your precise definition of the extension). When taking 1/0 and adding it to the reals however, this process does not work any more, as it implies x=0 for any x in the new set (as you showed). Thus we cannot simply extend R to another field by adjoining 1/0, as you implied by your framing. Rather what you did, is define a set and operations on this set, such that this structure contains an additive identity that has a multiplicative inverse, and then proved that it must be the only element in this structure. This is not a field and has nothing to do with the real numbers or adjoining elements to existing structures, even though your framing would suggest otherwise.
3:36 It still doesn't really make sense to write "0/0". Most people would not refer to {0} as a division ring. Having 1=/=0 is a requirement to be an integral domain and have "cancelation" as well. Really this is to eliminate the degenerate case of {0} being a field. Seeing the title, I definitely thought you would be talking about floating point arithmetic! :)
I played around with the idea but from the other direction - tracking what was multiplied by zero to get the zero you are working with. I'm not a mathematician so I didn't get very far, but the idea was that if you had a 0 that used to be 0x6, you could divide that by regular/unknown 0 to get the 6 back OR divide by any factor of 6 and change what class of 0 you were working with. So, a 0sub6 divided by 3 would give a 0sub2. The visual I was mulling over was counting empty cups that made up the "zero". The question of what the difference was between 0sub0, 0sub1, and which one would count as "regular" zero was where I faltered and to me felt more like kicking the can down the line, but then I considered, just like you said with imaginary numbers, there could be some merit in tracking factors when a real number could pop out of it.
Thhe "zero ring" mentioned in the video is an algebraic structure with only one element. There is no "6" in this structure. There is only "0", which is neutral element for multiplication,neutral element for addition, inverse element of any element in this ring for addition,inverse element for any element in the ring for multiplication, ... This structure has one and only one element, and can not be expanded to something else without getting inconsistent.
@@juergenilse3259 The structure in the video cannot be expanded, but they're not talking about the structure in the video. They're talking about a different way of (potentially, IDK if it would work) making 1/0 valid. In the video at 2:25 an assumption is made that 0*(x/0) = 0. This is of course a reasonable assumption, but it is just that - an assumption. Or rather it is an axiom - part of the definition of the number system. What is being done here instead is changing this axiom to state that 0*(x/0) = x, with the zeroes cancelling. This creates a full number system with the inherent requirement for cause-tracking of 0s as they describe.
It seems that this number system merely isn't defined for all additions and subtractions. This is fine, the natural numbers do this to. In natural numbers subtraction isn't defined for 3-5 = ? The result should be negative, so the expression is undefined on the naturals. You just have a system where addition and subtraction are not universally defined but this still generally allows you to continue. As for whether it's helpful I have no idea, but it might work unless you can prove a contradiction in it.
@@alansmithee419 It is defined for all additions and subtractions.In the zero ring, we have: 0*0=0 0+0=0 0-0=0 0/0=0 All is defined in this structure. The onl rule from our "normal calculation rules" that is not fullfilled,is, that the neutral element for multiplication and for addition should be different ...
@@alansmithee419 If you accept 1 (which is the neutral element for multiplication) is the *same* as 0 (the neutral element for addition), 1/0 is 0 in this ring (and 1 is only another name for 0 in this ring). But this ring is really borng.
software engineering allows so you to think so abstractly. No other engineering is detached from our physical world as much as software engineering. It teaches you to be a good, wise God.
the reason why you can't divide by zero is because of the same reason why you cannot get one solutions of the equations with more than one roots. the one divide by zero and oids happen to have infinite roots
2:20 The statement "anything times zero is zero" is not true in this number system. The zeros in 0 * 1/0 = 1 should cancel, leaving 1 = 1, same as with 2 * 1/2 = 1 the twos cancel. You can also end up with things like (0/2) * (1/0) = 1/2. Normally we discard the denominator if the numerator is 0, but with division by 0 it can be returned to the real number world later, same as negative square roots.
What you essentially said is that you can divide by zero if you redefine every single number as being equal to zero. Yes of course. The problem with a number that is "undefined" is that it could be any number from 1 to infinite. If any number is 0, then that problem disappears. It also makes math completely pointless.
If it doesn’t satisfy your practical requirements, that doesn’t infer its wrongness. It is also a valid algebraic system, just to stay aware of. If there are black holes and dark energy in the entire Universe, why not this)
@@adammizaushev I don't think commenters have a problem with the fact this is possible, number fields and vector spaces do all sorts of seemingly goofy stuff like this and it does make sense in a way. The issue with this is more that this is a bit of a reddit comment-esque video, it's like a "uihm _actually_ you can do that you uneducated [insert colorful swearing]" about a problem that is generally only brought up by the average guy when talking about the standard number system we always use in day-to-day life. Admittedly, it's a smart one and not at all that pedantic, it's probably even attracting those that just thought numbers are the way they are just 'cause, and those people probably learned something new and perhaps even enlightening, defintiely something intersting if nothing else, but the video's essence is still 100% a reddit comment
Nice video. But I think a step is skipped in the proof of 0*(1/0)=0. Let's call 1/0=j. We want j to have some common properties of any other elements of R so we can work with it, like j-j=0, 1*j=j, and the distributive law: a(b+c)=ab+ac. But once we set these 3 axioms, then it goes 0=j-j=j(1-1)=j*0=1. Note that 0*a=0 is not an axiom in the system(a ring), it's a theorem.
There is confusion between the zero in the definition of a ring and the zero in the real numbers. If you add "infinity" as the inverse of zero, you lose the ring structure and the zero in the model would no longer have the properties that the zero in a ring would have. Topologically, you would have the Alexandroff compactification of the real numbers (basically a loop). The idea of extending a set is to create a superset, not reducing it to a set with one element. You are not extending the real numbers, you are showing that the the only ring where the zero has an inverse is the zero ring.
Very interesting video! At first I thought you were going to represent the real numbers with a circumference instead of a line, that way a new infinity exists and division by 0 also exists and is that new infinity, but I didn't think of inventing new math!
The hyper-reals are what you’re describing basically - they define two objects: H is greater than any real number, and L is the quotient of 1 and H (it’s smaller in magnitude than any real number; essentially, it’s like +0). That solves the traditional problem with treating division by zero as a blanket limit - namely that of signs (if you approach from positive you’d get positive infinity versus from negatives where you get negative infinity). In the hyper-reals, you sacrifice the normal multiplicative properties of zero - the one that says anything times zero is zero, and that it is neither positive nor negative- to allow for division by zero. Addition and subtraction work almost how you would expect, but the anticommutative property of subtraction applies to the additive inverse (that is, if a + b = L, then b + a = -L). You can convert a hyper-real expression to reals with limits, assuming the limit exists.
@@fahrenheit2101 that's the other one, but it works better if you're using complex numbers IIRC, since complex infinity somehow makes it neater. That one isn't something I remember all that well tbh
Yes I was thinking of something like the Riemann space where some singularities can be considered points embedded in a broader space, e.g. where parallel lines meet in non-Euclidean geometries. At least that kind of number space has some profoundly useful applications, particularly in relativity.
there's a difference, though. "i" has a use. it can be turned into a real number. 1/0 does not have a use. it can't be placed inside any formula without breaking it. that's why you aren't taught much about 1/0 in school, but you are taught about "i"
For further reading, there are still other approaches to division by zero. For example, hyperreal numbers where you can divide by an infinitesimal number (which is not actually a zero, but whose standard real part is)
@@cezarcatalin1406 you’re right. Though, it’s a little inconvenient to have the whole universe as a result of an operation since it makes everything trivial (still correct). For example: - How much money will I get? - 0^0 (maybe 0, maybe 1000000, maybe -300)
When you say at 2:25 than "anything times zero equals 0" that is true in R, but what if we consider it not true in the new set and we only consider that any Real number by zero is zero, but not any number in our new set (let's call j the inverse of 0). what then ?
The other rules, such as 1×t=t and 0+t=t, are the ones that implicitly create 0×t=0. If we want 0×j≠0, some of the other rules have to go as well. Which do you want to lose?
Just to remind you that _i_ being sqrt(-1) is not arbitrary at all, it is exactly what it needs to be a 90° right angle turn to define the complex plane. In classical physics _i_ was considered mostly a theoretical trick to make things work, but as our understanding of quantum physics expanded, we realized that *quantum physics requires imaginary numbers to explain reality.* It is still rather "new" concept/discovery, so there are still quite a bit of professional mathematicians/physicists that are not aware of this connection
Quantum physics is not new anymore lol. Defining i as the sqrt(-1) is in fact arbitrary, as you can define the complex plane using other choices of i. These other choices result in a field isomorphic to the normal complex plane, but may be a bit of a pain to work with.
I agree with the undefined definition. Of a number divided by zero because zero is a quantity of something that could be incredibly tiny. So tiny that it's virtually zero but not zero. But zero is also considered a placeholder. So it's a placeholder without complete definition and therefore undefined when another number is divided by it.
Really good addition to the channel. Very cool explanations, it brought me back to the days when I was studying commutative algebra from Atiyah-Mcdonald's book.
UA-cam's recommendations are wack. I found this video without having much background in math or coding, and I was confused throughout. But I still watched the video because the premise was interesting.
I like this because it kind of shows, broadly, what mathematicians do. They push boundaries. What are the limitations of a system or property? What happens if we do something different with it? How do things relate to each other? I suspect many people think mathematicians just make up random rules because they can.
Division is just iterated subtraction. Working division back into its component subtractions -- dividing 8 by 2, one 2 at a time -- gets you 8-2=6 (one 2 so far); then 6-2=4 (two 2s so far); then 4-2=2 (three 2s); and finally 2-2=0, for a total of four 2s in 8. When you try to divide 8 by 0 you get 8-0=8. Hmm. Ok, lets try again. 8-0=8. Third time's a charm, right? 8-0=8. Well this is going to go on forever without ever subtracting any amount from 8, let alone working down to x-0=0, where x>0. Dividing by 0 is no different from asking how many 0s must to be added to one number to arrive at a larger number? There is no answer to that question.
Since 1x0 = 0 and 2x0 = 0, we can say that 1x0 = 2x0. By then dividing both sides of the equation by zero, we find that 1=2. And in the context of dividing by zero, this is absolutely true. Because as you divide by smaller and smaller numbers, the result tends towards infinity. And relative to infinity, 1 really is the same things as 2, because no finite value can change an infinite value. Any finite value compared to an infinite value is worth nothing, so this 'version of maths where everything is equal to zero' is really just mathematics with infinite numbers.
2:24 well, not necessarily. Maybe you could also define i as the number where i*0 = 1. Then i would be 1/0, 2*i would be 2/0, 3*i would be 3/0 and so on. (And 0*i would be 0/0 and because of our definition it would be 1, which somehow makes sense, because any number divided by itself gives 1. Let's just say 0/x = 0 is only valid for numbers in R). Then 1/i would be 1/(1/0) = 0/1 = 0. So every real number r would be equal r + 1/i. I don't know if there's a way to imagine that in a 2D area like the "actual" i, or if my theory would somewhere be a contradiction, but maybe we just need to define the common rules we know for calculating are only true in the real numbers. (pls tell me if this could work)
I think a step is skipped in the proof of 0*(1/0)=0. Let's call 1/0=j. We have j-j=0, and the distributive law: a(b+c)=ab+ac. Then we have 0=j-j=j(1-1)=j*0=1. We need this because 0*a=0 is not an axiom in the system.
@@Leonex52 It's still a leap in logic to say that j-j=0. j-j=(1-1)/0=0/0. You must prove that 0/0=0, or explain why the denominator can be discarded if the numerator is zero.
Uh, I think I went too deep into youtube again, I barely understood any of this. It felt like watching someone explain the concept of objective mathematics instead of the subjective math that the human brain can comprehend. Or maybe I'm just dumb.
"We're gonna define 1/0" Oh so is this gonna be some black magic analytic continuation? I wonder- "Yeah this is just the set containing zero lol" Okay, that was correct... but kinda disappointing.
The issue I don't get is why do people start be saying "anything times zero is zero" but don't apply the same rules to 1/0? It seams to be that this would also require special cases, in the same way that multiplication by zero does
A step is skipped in the proof of 0*(1/0)=0. Let's call 1/0=j. We want j to have some common properties of any other elements of R so we can work with it, like j-j=0, 1*j=j, and the distributive law: a(b+c)=ab+ac. But once we set these 3 axioms, then it goes 0=j-j=j(1-1)=j*0=1. Note that 0*a=0 is not an axiom in the system(a ring), it's a theorem.
Great video. I rate it zero of zero. Way to go!
@TigranK115It’s not indeterminate. It‘s zero. Just as the video says.
who@TigranK115
is it full points or zero points?
Everything, yet nothing at the same time!
@@Yilmaz4full points is a 0
Saying _"you can't do XYZ"_ in maths is really just a shorthand for saying _"The systems of maths that arrises by expanding an existing one to include XYZ is not interesting / useful / non-trivial / connected to other branch of maths."_ This is probably obvious to anyone who has studied higher maths and is familiar with the idea of there being many different systems of maths (different number systems, different starting axioms, etc...) that we can choose between at will; but far more alien to those who haven't gone beyond high school maths and think of it as a single, rigid, god given, singular thing.
you can't calculate the sine inverse of pi
@@nel_tu_You very much can, infact you can with any real number. But it requires branching out to complex numbers. It's nicely explained in this video: ua-cam.com/video/3C_XD_cCeeI/v-deo.html
@@Adam-zt4cn you cannot calculate the determinant of rectangular matrix.
@@Adam-zt4cn nice video btw
Exactly! Maths is just a game of how much stuff you can make up that isn't contradictory with itself. The only place where you can't really do something is when it creates a contradiction in itself. For example, just like in the above video, we know if the system included 1, it would have a contradiction as we would get 0=1, so we just say "nah screw that bch, I never even liked one" and kick it out of the system altogether. Can't have the contradiction if the system doesn't have the number 1!
"If you divide by zero, all numbers are zero". That's a cruel punishment
Explains a DMT journey quite well. 😅”you get what you deserve”. 😮
After lots of hour I finally implemented a fully working calculator for the zero ring:
def add_or_mul_or_div_or_sub(a, b):
return 0
It was hard work but will be worth it for future calculations
This will revolutionise maths
Hard work in the zero ring I’m sure, but in the real world this takes zero effort 🤪
This is a great example for why we "can't" divide by zero, as defining it using a zero ring serves little to no purpose. (What are you going to do with a number system where R is simply zero and only zero?)
@@thatguynamedgeorge9218at some point it will be useful we just havent found the right situation yet
I did it nodejs for you guys 😊
const zero = require('zero-int');
const fns = require('funcs');
function zeroFactory() { return zero.create(); }
function addMulSubDiv(a, b) { return zeroFactory(); }
fncs.assign(addMulSubDiv, zeroFactory);
require('export').exportFncs(addMulSubDiv, zeroFactory);
Modern texts, that define fields as a special type of ring, include the axiom 0 ≠ 1 for fields (or its equivalent) so that the zero ring is excluded from being a field.
That’s boring.
That's interesting.
Just a fact. You may find it useful... or not. Depends on your needs and intentions.
Opps, sorry, I meant that whoever decided to exclude the zero ring from being a field was boring, not this fact itself.
Opposite of boring I would say, since the zero ring is very boring!
"It's not that you can't divide by zero, it just doesn't do anything useful to define" is what I gathered.
That we know of. For all we know there might be some really weird model that can't work without division by zero. Like say... a black hole.
@@omegahaxors9-11 and with that i know you are another one of those pseudo science bros
@@mortvald the fuck that come from?
@@omegahaxors9-11 the same place that black hole came from
@@mortvald they literally made a black hole in a quantum simulation then sent information through it and it came out of the other end in another quantum simulation in a completely different computer. There's no reason to turn science into dogma when the math is actually making predictions.
I was not expecting pure math from this channel, but I probably should've given that I learned about semigroups from a one-off comment in one of your Python videos. This is awesome.
this guy is a gd genius, hes an expert at python, c++ (which im literally scared of), and math
He also has a great video on proofs of 0.999... = 1, if you want to check out more mCoding math!
@@supermonkeyqwerty software engineering allows so you to think so abstractly.
@@prawnydagrategeometry dash genius? What
@@βΛΔΗΟΛΣ bro what 💀 gd = goddamn
You can't divide by 0 until you invent a rule that you can.
The difficulty is, to do the definition without getting inconsistencies ...
And the instant you do, everything else falls apart.
@@omegahaxors9-11 well no, it didn't fall apart. The system you get might not be useful for anything practical (that you could think of, it might well have abstract applications or implications), however saying things "fall apart" is disingenuous as it suggests the foundational theorems of mathematics are not sound, yet in this case they are, they gave you something that works as they dictate. It maybe doesn't work how you'd imagine or how you'd like, but it still completely works.
no number squared eguals a negative number until you invent a rule that it does
@@sofisofi-gx8te Incorrect, Edison. This is why Tesla was right. AC circuit voltages and currents cannot be calculated correctly without the use of i. The only exact solutions for many AC design and analysis problems include an imaginary component. As mind blowing as this may be, imaginary numbers are quite real and useful in electromagnetism, quantum mechanics, particle physics and light propagation. A better word for "imaginary" here is "complex." Euler's Identity e^iπ + 1= 0 and its proof demonstrate they are real. The fact that you exist does. Fundamental forces like gravity operate by mechanisms that can only be accurately described with the use of complex numbers. Complex numbers, like gravity, weren't invented. They were discovered.
In a commutative algebra lecture, the professor gave the important proposition that the localisation at a (multiplicative) set is 0 if and only if the set contains 0 a very fitting name: If you divide by zero, everyone dies (when something becomes zero, people often call that „killing the element“).
I’ve also heard “disappear” which makes it sound like Mafia.
so... zero...
@@DeRickz69420 nope, if for any number a: a / 0 = 0 then a = 0
In a logical sense dividing by zero means doing nothing. Like multiplying by zero means doing nothing to the number you are multiplying it with. You are asking about performing a task zero times.
@@stirfrybry1 That's not at all what it means. If you're talking about the "normal" number system and not the weird "zero only" system from the video, then multiplying by zero is not doing nothing. You're turning the original number into zero. That's not nothing. Dividing by zero is also not nothing. The result is indeterminate, but if you would divide by something that goes very near zero, the answer goes to infinitity. So this also is not doing nothing.
The main difference here is that including a square root of -1 is a field extension of R. In fact, it is a very special field extension. It is the splitting field of R (in many ways, it is better than R). But the ignoring that, the important thing is that C has R embedded in it; the natural homomorphism from R to C is injective, or in other words, the kernel is trivial. This means R is isomorphic to a subring (subfield) of C, so this extension doesn't lose you any of R.
On the other hand, if you include 1/0, the new ring no longer has R embedded in it - it is not an extension of R. The natural (and only) homomorphism from R into the zero ring is as far from injective as it could be - the kernel is the entire set R. So there is nothing that looks like R inside the zero ring. This should be pretty obvious given that R is uncountable, while the zero ring only has 1 element.
Beautifully worded.
Love this explanation
I agree, but you could have phrased that in a much more straightforward way without losing much meaning at all
@@hach1kokoit's pretty straightforward (it's also just a UA-cam comment). The parts that stick out as not straight forward are things to explore. More fun ahead
@@pedrov8868 What's the point of mentioning kernels for instance? I think this just ends up confusing people that don't know what he's referring to.
Given the typical content of this channel, i was assuming the set of numbers we would arrive at would be blackboard F, for floating point as specified IEEE 754
Same here!
blackboard F is usually reserved for fields, which IEEE 754 absolutely is not. It's a cursed imitation of a ring.
i mean, pick an open letter lol idc, call it 𝕀𝔼𝔼𝔼𝟟𝟝𝟜 for all i care
I thought so too haha. It's fun watching mechanical calculators try to divide by zero
Had a teacher once ask me if I take what's in your hand and take away half what do you have left? Once I answered he said and if I keep taking half what do you have? This was, of course his way of telling me about atoms. Then he said if I take away the atoms what do you have? I said nothing. I have nothing left. He said everyone keeps saying that, but the answer is you have everything else. Once its gone, you have the whole universe. I wondered where he got his drugs from. After watching this, apparently he was right.
I thought you were going to talk about the projective real number line, which has an inverse of 0, so division is defined on everything but now addition/subtraction isn't.
can you explain that to me? that actually sounds like something I stumbled across a while ago.
@@Metal_Master_YT There's way more than I can possibly explain in a comment but the tldr is that you add a single point at infinity to the real number line.
@@Tehom1 that's more like a tldr of a tldr. that was literally a single sentence. contrary to popular belief, I actually do have some patience to read. but hey, if you don't have time, then don't let me bother you.😅
@@Metal_Master_YT The projective line is the real number line bent into a circle and glued together at the ends. It adds a new number which you can think of as the point where the ends are glued. This number acts like infinity in a sense, but to make the algebra work nicely you need some more complicated stuff called homogeneous coordinates. Roughly these are like taking a diameter of the circle and taking the antipodal intersection points of the diameter with the circle as coordinates. You can then consistently define algebra with ∞ and 1/0. You can’t, however, do algebra with 0/0 still. In order to make a structure where that works, you need what is called a wheel. These are a bit like further extensions of the projective line, but they need more difficult algebraic rules than before to account for 0/0.
@@HPTopoG interesting, thanks for explaining it to me. although, are the points that you are generating being plotted on a standard coordinate plane? and which of the 2 antipodal values is the x or y?
Alternatively, 1/0 = infinity + 1.
I think that checks out, but I'm not a mathematician. Also, it might cause an infinite improbability drive to power up somewhere.
2:22 is this really a true deduction? We just defined that 0* 1/0 = 1 so clearly NOT everything multiplied by 0 is 0 anymore.
so what they're saying is, 1 is essentially another name for 0 in this number system
@@homan-awa I understand that, but that is a deduction based on the claim (at least in this video) that everything times 0 is 0, which is not a trivial statement.
The hypothesis in the video is that we are working in a ring. en.wikipedia.org/wiki/Ring_(mathematics)
What he proved is that a ring having an inverse of 0 is a ring with all numbers being equal.
If you want 0 to have an inverse, you have to concede some ring properties. Properties that we are familiar with.
@@Ghost-RaccoonThat's what I also put out in my comment.
@@Ghost-Raccoon I would agree, this is video's process to be seems like using the rules of our maths system (which form a paradox) and deciding that we should let "1=0" be true instead of letting "some x multiplied by 0 could be non 0" be true
1:13, pausing here for a moment, 0² is 0 so the √0 = 0 but wait that's 0 / 0 which extrapolated to N / 0 means N/0 = 0
In simplest form this means division and multiplication can be represented as follows without adding any extra values:
a/b = while ( a >= b && c < b ) { a -= b; c += 1; }
a*b = while ( b >= 0 ) { c += a; b -= 1; }
The destination (c) in both cases starts as 0, skipping c < b is what causes the infinity loop. Basically N / 0 is the edge case of faulty division definition/s.
**Edit:** I've found that it's better to compare lengths of the remainder vs length of the divisor. The length of the divisor is always at least 1 while the length of the remainder is always decremented by at least 1, inevitably the length of the remainder is eventually declared as 0 (even if the is the digit 0) forcing the loop to break since anything with a length less than the divisor will obviously fail the >= check inside the loop
you can also tweak the rules slightly to create a useful system, like what was done with floats; 1 / 0 = infinity. 1/-0 = -infinity. 0 * infinity = NaN, NaN with most operators just produces NaN.
Little fun fact about NaNs is that they actually encode. Though due to most NaNs being the result of trying to do a mathematical operation on a NaN these almost universally just end up as a huge wall of "Tried to do math on a NaN" codes. Not always though. If you look at the binary of a NaN float you can use that as a sort of error code to determine what exactly caused it. Just don't be surprised if the value is something meaningless or random because NaNs have undefined behavior and are completely dependent on the implementation of the float itself. There is zero standardization or guidelines across the entire industry.
interesting! I did know about float packing (how javascript stores booleans, nulls, and other things as floats), but I did not know about NaN codes.
Eh.. it requires a lot more than slightly changing the rules though. If you do this, you'll have to give up some very basic properties of math that will make doing everything overwhelmingly more complicated and you'd need to reprove basically every formula (well, a lot of formulas won't be reproven because they won't be true anymore) because nearly every proof uses those basic properties.
For instance, is x - x = 0? Normally you'd say that's obviously true.. but what if we have 1/0 - 2/0?
1/0 = infinity, 2/0 = infinity, so 1/0 - 2/0 = 0.
2/0 = 2(1/0) though - that means that 1/0 - 2/0 = 1/0 - 1/0 - 1/0, which evaluates to negative infinity.. which implies that negative infinity = 0, which is obviously nonsense.
That means you can no longer say that x - x = 0 in that new numbering system (or maybe that 2x isn't equal to x + x which also causes a lot of problems).. which is going to be causing a whole lot of problems with a lot of proofs. In the end basically every formula will still not function with any of those new numbers, which makes it functionally the same thing as being undefined because it'll still be impossible to actually use it anywhere.
It also has problems with "what is -0?" - after all, how do you know whether 1-1 is 0 or -0?
1 - 1 = -1 + 1 = - (1 - 1), therefore 1/(1-1) = 1/(-(1-1)) = -1/(1-1), which implies that infinity = -infinity.
If you want to handle this, you'll have to say something like x+y =/= y+x., or maybe that x(y+z) =/= xy + xz (even with non-infinite numbers). This is going to cause a lot of problems.
There are almost certainly a whole lot more problems with it - there are *very* good reasons that it's treated as undefined and that the numbering system you're describing isn't used. The only reason it "works" with floats is that floats aren't intended to be an accurate way of calculating things - they're by definition not exact values, so any time you're working with floats it's to be expected that sometimes you won't get correct answers and you just have to deal with it being incorrect sometimes. Floating point numbers already break most of the rules of math, so they don't really care that infinity also breaks them since they were already broken by regular numbers anyway.
Ah so it’s COMPLETELY and UTTERLY *pointless*
… but you CAN do it
Now if that doesn’t describe math, I don’t know what does!
I was expecting this to be a video on IEEE floating points, but this is interesting in its own right.
When you got to the "1=0" part and said that's not a contradiction, I had to double-check the upload date to make sure I'm not watching an April Fool's prank video lol
i guess the (number)universe does collapse if you try to divide by zero
Exactly that happened in the video ...
😇
Man, I really had hope that this video would actually explain how to divide by 0, but then starting from 2:13 everything goes wrong. Why can't people just understand how simple the extended complex plane, Riemann sphere, or wheel algebra is? The inverse of 0 is infinity, it's as simple as that.
eh, just make it a wheel. You get zero, you get infinity, you get any symbol [x, 0], and you get a special element [0,0] (where for any *regular* value [a, b], to translate it into the real numbers, is just a/b, though some values such as [x,0] can't be translated)
Me: “oh, he’s building up to Wheels”
[wheels never come up]
This feels like the start of a 0 cult.
"All is 0. Everything is a mere label for what is truly 0."
1:44 you say "if we also throw in[] inverses of every positive whole number" but that's somewhat redundant, right? Wouldn't it suffice to just use inverses of primes?
Excellent observation! A fortiori adding in reciprocals of primes is sufficient, but it's not necessary to make the construction of the rationals dependent on facts about primes. I didn't mention this in the video, but the first step in performing localization is to compute the multiplicative closure of the set you are adding inverses for, then to throw all those inverses in. So if you did start with the primes, you would quickly compute their closure to be all nonzero integers and arrive back to that point in the video ;)
2:52 how do you know the neutral element in the new number system is still 1?
Great video, congratulations! Making these theoretical details of math visible to the regular user/student is a valuable way to promote math studying.
“You get what you deserve “. Get answer. I like it.
Here I was expecting an overview of IEEE 754.
Isn't x/0 like Schrödinger's cat because technically you're not taking anything from it so it could be x, but if you multiply with the reciprocal value (0/0 is a weird fraction, but we're talking about division by zero soooo...) it would be zero.
As obvious, I am no mathematician ^^
Localization isn't the only option for extension. There is also the one point compactification, and the two point compactification of the real line, where you add one or two infinities respectively. They have the drawback of not being fields. In those spaces you still can't divide zero by zero. And IEEE floats are very similar in behavior to the two point compactification apart from floats only representing dyadic rationals and not even all of them.
beautiful comment, I was going to point out similar issues.
How does division by 0 work in the two point compactification? I thought it couldn't work because 1/0 would be ambiguous as to whether it's positive or negative infinity?
@@SJGster 1/0 = +infinity and -1/0 = -infinity
0/0 is still undefined.
@@timseguine2 why wouldn't this seeming contradiction pose a problem? (1/0)*(-1/-1) = -1/0 therefore 1/0=-1/0 therefore infinity=-infinity?
@@SJGster You are using field axioms to manipulate that expression. It isn't a field. In particular neither addition nor multiplication are associative.
What is true is that not every source considers 1/0 or -1/0 to be defined because they don't follow from the limit point construction of the space as robustly as other properties. And another reason why people sometimes choose to leave them out is because if you do you get a weak form of associativity and distributivity.
cool video, thanks. nice to see it wasnt clickbait
Great one! You've just zero-rolled me!
this pretty much sums it up. in a ring, if we allow the additive identity to be equal to the multiplicative identity, we get the trivial ring with a single element that technically has all the properties but is completely useless. It is in fact also a field, a vector space over itself, an algebra and so on but again not very useful...
However, there is apparently other places where it is useful like in projective geometry where we treat unsigned infinity as a normal number and in riemman spheres.
I'm sad that wheel theory wouldn't have earned at least a honorable mention in this video.
but can't we introduce a new number system such that 0 time x is not 0
2:23 how can 0*1/0 be 0 if we defined it to be 1?
Because anything times 0 is 0
@@greenwaldian Actually the rule "anything times zero is zero" applies in IR, it may not apply in the new set that we're creating, but we can still proof that 0*(1/0) = 0, by doing so :
0*1 = 0
so 0*(0*(1/0)) = 0
so (0*0)*(1/0) = 0
so 0*(1/0) = 0
so 1 = 0
Maybe a way of thinking about it is to understand how maths tends to do things. We've defined the solution to equation to (0*1/0) to be 1. Okay cool. But we have the other rule that says that anything multiplied by 0 is 0. And thus, we have shown that 1 = 0 the whole time. We started with assuming that 1 and 0 were not the same thing, but we followed our rules and it turns out they were the same all along.
Imagine we were talking about something else. Let's say we are talking about the number 2/4. Is 2/4 the same as 1/2? Well no, just look at it, they have different numerators and denominators. They're not the same, right? Well, if we follow our rules about cancelling shared parts of the denominators and numerators, we reduce 2/4 down to 1/2 and voila, by our rules, it turns out that they actually ARE the same number after all.
This is a common idea that pops up in higher maths. For a vague example, you define what a 'group' is (to simplify, just think of it as something which is like the integers), you get this thing called the 'identity'. This is the element you get when you take something with its inverse (say, for example, 2 + (-2) = 0, 0 here is the identity). Is the identity unique? Can there be multiple identities? Well, assume there are two identities, do some algebra using the rules you set out, and voila you show that actually they are the same after all.
I hope that's helpful for you.
@@greenwaldian And what if there's an exception to that rule? Remember we aren't working with ordinary numbers here
@@Amechaniaa check my answer above
... And you get what you deserve.
Man, that actually hits...
I think one of the factors of it being un-defined is that it doesn't explain or help with anything if it's localized. In comparision,complex numbers is quite useful in a variety of things from quantum physics to engineering. A different branch for a division of 0 quite literally and metaphorically gives us nothing.
Division by 0 is the foundation of calculus. Calculating the derivative of a function is finding what 0/0 is approaching.
@@Fluffy6555 The key expression being "approaching", ie. we're talking about limits in calculus. And with the way limits are defined, you actually never end up dividing by zero.
@@joeltimonen8268Exactly, the whole point in calculus is "we can't calculate this, but can we figure out something really really close".
@@shockthetoast except, in calculus, every single time, we don't just get really close, we make sure to actually get on to the thing. Otherwise d/dx x² would be 2x+h, for 0≈h≠0.
It's not that you *can't* divide by zero, it's that you can *only* divide by zero.
Then what would happen if multiplying by zero doesn't *have* to result in zero?
If you, for example, assume that infinity * 0 = 1 , would it work then?
1/0=infinity, infinity/0=1, infinity*0=1, you probably need a doubly lined 0 as done with those C and R and Z's.
The fact that 0*x = 0 is usually derived from other facts. So it becomes a question of which properties you're willing to drop. In the typical formulation there are three other properties used to prove this:
We have 0+a = a. This is one of the defining properties of 0, so you probably want to keep that.
The distributive property: (a+b)*x = a*x + b*x.
Subtraction: a + b - b = a
With these properties we can do as follows:
0 + 0 = 0
(0+0)*x = 0*x = 0 + 0*x
0*x + 0*x = 0 + 0*x
0*x + 0*x - 0*x = 0 + 0*x - 0*x
0*x = 0
It works, when 0 * ∞ = E, but not 1 (nor 0 or ∞).
1/0=∞ and 1/∞=0 make sense, but from that doesn't follow 0*∞=1, as you e.g. would need to multiply 1/0 with 0, yet you cannot reduce 0/0 to 1.
From the first 2 rules you get E = 0/0 = ∞/∞ = 0*∞ = 1/E = E², which helps to solve all equations.
@@anon8510 It is every number.
@@johngalmann9579 (b-b) is indeterminant when we have infinity in the mix so the subtraction property doesn't apply
when you do this you have two choices (one-point vs two-point compactification)
in the first you say 1/0 = inf and inf = 1/0, and you do away with the ordering relations, and inf is a number that can't be subtracted (like how 0 couldn't be divided by)
in the second you define 0+ and 0- not just 0, and you say 1/0+ = inf and 1/0- = - inf, now we keep the ordering relations, and now we can't add or subtract at all since 0- and 0+ must be distinguished.
In general we can also construct a system where values are sets of numbers rather than individual numbers, and consider operations as images over sets where we generate from singletons of another set and make all operations total and closed by imposing sets as their solutions, this gives us transfields from fields such as the transreals from the reals or the transcomplex numbers from the complex numbers
I disagree with the framing of this:
When talking about the complex numbers or the dyadic rationals, you emphasized the "throw it into the existing set of numbers and follow the known rules of algebra" part. In these cases, you have a base structure (a ring for Z, a field for R) and extend it in a way, that still satisfies closure and the axioms (in your examples by adjoining solutions to x^2+1=0 or 2x-1=0). Thus it retains key properties and contains the original ring/field as a subring/subfield (or at least a ring/field that is isomorphic to them, depending on your precise definition of the extension).
When taking 1/0 and adding it to the reals however, this process does not work any more, as it implies x=0 for any x in the new set (as you showed). Thus we cannot simply extend R to another field by adjoining 1/0, as you implied by your framing.
Rather what you did, is define a set and operations on this set, such that this structure contains an additive identity that has a multiplicative inverse, and then proved that it must be the only element in this structure. This is not a field and has nothing to do with the real numbers or adjoining elements to existing structures, even though your framing would suggest otherwise.
3:36 It still doesn't really make sense to write "0/0". Most people would not refer to {0} as a division ring. Having 1=/=0 is a requirement to be an integral domain and have "cancelation" as well. Really this is to eliminate the degenerate case of {0} being a field.
Seeing the title, I definitely thought you would be talking about floating point arithmetic! :)
In the zero ring, 0/0 absolutely makes sense, since 0 is a unit, so 0^-1 is perfectly well-defined.
Sure you can, just as long as you are willing to accept that you create a random universe each time that you do it.
2:26 you can't treat 1/0 as number
So you are only able to multiple 0 to numerator which is 1 and you get 0/0
Love the video! More math videos please!
Felt like I am watching 3blue1brown but shorts version :P
while this is true, if i ever used the zero ring to prove anything in a math test in school, i think it would get marked wrong
I played around with the idea but from the other direction - tracking what was multiplied by zero to get the zero you are working with. I'm not a mathematician so I didn't get very far, but the idea was that if you had a 0 that used to be 0x6, you could divide that by regular/unknown 0 to get the 6 back OR divide by any factor of 6 and change what class of 0 you were working with. So, a 0sub6 divided by 3 would give a 0sub2. The visual I was mulling over was counting empty cups that made up the "zero".
The question of what the difference was between 0sub0, 0sub1, and which one would count as "regular" zero was where I faltered and to me felt more like kicking the can down the line, but then I considered, just like you said with imaginary numbers, there could be some merit in tracking factors when a real number could pop out of it.
Thhe "zero ring" mentioned in the video is an algebraic structure with only one element. There is no "6" in this structure. There is only "0", which is neutral element for multiplication,neutral element for addition, inverse element of any element in this ring for addition,inverse element for any element in the ring for multiplication, ... This structure has one and only one element, and can not be expanded to something else without getting inconsistent.
@@juergenilse3259
The structure in the video cannot be expanded, but they're not talking about the structure in the video. They're talking about a different way of (potentially, IDK if it would work) making 1/0 valid.
In the video at 2:25 an assumption is made that 0*(x/0) = 0. This is of course a reasonable assumption, but it is just that - an assumption. Or rather it is an axiom - part of the definition of the number system. What is being done here instead is changing this axiom to state that 0*(x/0) = x, with the zeroes cancelling. This creates a full number system with the inherent requirement for cause-tracking of 0s as they describe.
It seems that this number system merely isn't defined for all additions and subtractions. This is fine, the natural numbers do this to. In natural numbers subtraction isn't defined for 3-5 = ? The result should be negative, so the expression is undefined on the naturals.
You just have a system where addition and subtraction are not universally defined but this still generally allows you to continue. As for whether it's helpful I have no idea, but it might work unless you can prove a contradiction in it.
@@alansmithee419 It is defined for all additions and subtractions.In the zero ring, we have:
0*0=0
0+0=0
0-0=0
0/0=0
All is defined in this structure. The onl rule from our "normal calculation rules" that is not fullfilled,is, that the neutral element for multiplication and for addition should be different ...
@@alansmithee419 If you accept 1 (which is the neutral element for multiplication) is the *same* as 0 (the neutral element for addition), 1/0 is 0 in this ring (and 1 is only another name for 0 in this ring). But this ring is really borng.
software engineering allows so you to think so abstractly. No other engineering is detached from our physical world as much as software engineering. It teaches you to be a good, wise God.
the reason why you can't divide by zero is because of the same reason why you cannot get one solutions of the equations with more than one roots.
the one divide by zero and oids happen to have infinite roots
2:20 The statement "anything times zero is zero" is not true in this number system. The zeros in 0 * 1/0 = 1 should cancel, leaving 1 = 1, same as with 2 * 1/2 = 1 the twos cancel. You can also end up with things like (0/2) * (1/0) = 1/2. Normally we discard the denominator if the numerator is 0, but with division by 0 it can be returned to the real number world later, same as negative square roots.
What you essentially said is that you can divide by zero if you redefine every single number as being equal to zero.
Yes of course. The problem with a number that is "undefined" is that it could be any number from 1 to infinite. If any number is 0, then that problem disappears.
It also makes math completely pointless.
If it doesn’t satisfy your practical requirements, that doesn’t infer its wrongness. It is also a valid algebraic system, just to stay aware of.
If there are black holes and dark energy in the entire Universe, why not this)
@@adammizaushev I don't think commenters have a problem with the fact this is possible, number fields and vector spaces do all sorts of seemingly goofy stuff like this and it does make sense in a way. The issue with this is more that this is a bit of a reddit comment-esque video, it's like a "uihm _actually_ you can do that you uneducated [insert colorful swearing]" about a problem that is generally only brought up by the average guy when talking about the standard number system we always use in day-to-day life. Admittedly, it's a smart one and not at all that pedantic, it's probably even attracting those that just thought numbers are the way they are just 'cause, and those people probably learned something new and perhaps even enlightening, defintiely something intersting if nothing else, but the video's essence is still 100% a reddit comment
When you said "Let's do it!", I held in my chair feeling like we were about to break the universe
Nice video. But I think a step is skipped in the proof of 0*(1/0)=0.
Let's call 1/0=j. We want j to have some common properties of any other elements of R so we can work with it, like j-j=0, 1*j=j, and the distributive law: a(b+c)=ab+ac. But once we set these 3 axioms, then it goes 0=j-j=j(1-1)=j*0=1.
Note that 0*a=0 is not an axiom in the system(a ring), it's a theorem.
What about the Riemann sphere?
There is confusion between the zero in the definition of a ring and the zero in the real numbers. If you add "infinity" as the inverse of zero, you lose the ring structure and the zero in the model would no longer have the properties that the zero in a ring would have. Topologically, you would have the Alexandroff compactification of the real numbers (basically a loop). The idea of extending a set is to create a superset, not reducing it to a set with one element. You are not extending the real numbers, you are showing that the the only ring where the zero has an inverse is the zero ring.
I was honestly expecting something like Wheel theory to come up.
Very interesting video!
At first I thought you were going to represent the real numbers with a circumference instead of a line, that way a new infinity exists and division by 0 also exists and is that new infinity, but I didn't think of inventing new math!
The hyper-reals are what you’re describing basically - they define two objects: H is greater than any real number, and L is the quotient of 1 and H (it’s smaller in magnitude than any real number; essentially, it’s like +0). That solves the traditional problem with treating division by zero as a blanket limit - namely that of signs (if you approach from positive you’d get positive infinity versus from negatives where you get negative infinity). In the hyper-reals, you sacrifice the normal multiplicative properties of zero - the one that says anything times zero is zero, and that it is neither positive nor negative- to allow for division by zero. Addition and subtraction work almost how you would expect, but the anticommutative property of subtraction applies to the additive inverse (that is, if a + b = L, then b + a = -L). You can convert a hyper-real expression to reals with limits, assuming the limit exists.
@@NathanSimonGottemerreally? Seemed almost certain to me that they meant the projective real line.
@@fahrenheit2101 that's the other one, but it works better if you're using complex numbers IIRC, since complex infinity somehow makes it neater. That one isn't something I remember all that well tbh
Yes I was thinking of something like the Riemann space where some singularities can be considered points embedded in a broader space, e.g. where parallel lines meet in non-Euclidean geometries. At least that kind of number space has some profoundly useful applications, particularly in relativity.
my calculator doesn’t like “5/0” but it does like “0/5”
Dude! I loved this. Can you tell us what you used to create these animations and share the source code for these as well?
there's a difference, though. "i" has a use. it can be turned into a real number. 1/0 does not have a use. it can't be placed inside any formula without breaking it. that's why you aren't taught much about 1/0 in school, but you are taught about "i"
For further reading, there are still other approaches to division by zero. For example, hyperreal numbers where you can divide by an infinitesimal number (which is not actually a zero, but whose standard real part is)
Also, who says the result of an operation has to necessarily be a scalar number?
any/0 = {+inf,-inf}
0/0 = {+inf,-inf} U *R*
@@cezarcatalin1406 you’re right. Though, it’s a little inconvenient to have the whole universe as a result of an operation since it makes everything trivial (still correct).
For example:
- How much money will I get?
- 0^0 (maybe 0, maybe 1000000, maybe -300)
When you say at 2:25 than "anything times zero equals 0" that is true in R, but what if we consider it not true in the new set and we only consider that any Real number by zero is zero, but not any number in our new set (let's call j the inverse of 0). what then ?
The other rules, such as 1×t=t and 0+t=t, are the ones that implicitly create 0×t=0. If we want 0×j≠0, some of the other rules have to go as well. Which do you want to lose?
Just to remind you that _i_ being sqrt(-1) is not arbitrary at all, it is exactly what it needs to be a 90° right angle turn to define the complex plane.
In classical physics _i_ was considered mostly a theoretical trick to make things work, but as our understanding of quantum physics expanded, we realized that *quantum physics requires imaginary numbers to explain reality.*
It is still rather "new" concept/discovery, so there are still quite a bit of professional mathematicians/physicists that are not aware of this connection
Quantum physics is not new anymore lol.
Defining i as the sqrt(-1) is in fact arbitrary, as you can define the complex plane using other choices of i. These other choices result in a field isomorphic to the normal complex plane, but may be a bit of a pain to work with.
You can define i^2=0 or i^2=+1
@@pierrecurie yes, admittedly the paper I was talking about came out in 2021, so not that new
@@kazedcat And you can define that moon = cheese as well
@@Songfugel You are clueless on how mathematics work.
this video is zero out of zero in zero ring number system. Great job!
Obrigado pela explicação! É fundamental sabermos disso, nós, professores de matemática.
I agree with the undefined definition. Of a number divided by zero because zero is a quantity of something that could be incredibly tiny. So tiny that it's virtually zero but not zero. But zero is also considered a placeholder. So it's a placeholder without complete definition and therefore undefined when another number is divided by it.
I’ve seen a bunch of videos saying “you can divide by zero”. I was not expecting anything different here. I was wrong and liked it!
What I've gathered from this video is that you can do anything in math, you just have to keep making stuff up until it works
Really good addition to the channel. Very cool explanations, it brought me back to the days when I was studying commutative algebra from Atiyah-Mcdonald's book.
That last sentence caught me off guard
Very clever and very well done - this vid is going to blow up
It already has 0 views!
0:19 "Checking all of the details might be a bit complex." 10/10 joke lol
Can you please stop using white background?
UA-cam's recommendations are wack.
I found this video without having much background in math or coding, and I was confused throughout.
But I still watched the video because the premise was interesting.
I like this because it kind of shows, broadly, what mathematicians do. They push boundaries. What are the limitations of a system or property? What happens if we do something different with it? How do things relate to each other? I suspect many people think mathematicians just make up random rules because they can.
Division is just iterated subtraction.
Working division back into its component subtractions -- dividing 8 by 2, one 2 at a time -- gets you
8-2=6 (one 2 so far); then
6-2=4 (two 2s so far); then
4-2=2 (three 2s); and finally
2-2=0, for a total of four 2s in 8.
When you try to divide 8 by 0 you get
8-0=8. Hmm. Ok, lets try again.
8-0=8. Third time's a charm, right?
8-0=8. Well this is going to go on forever without ever subtracting any amount from 8, let alone working down to
x-0=0, where x>0.
Dividing by 0 is no different from asking how many 0s must to be added to one number to arrive at a larger number? There is no answer to that question.
Me: So, it's all zero?
Man behind me with a gun: Always has been.
It seems somehow appropriate that the numeral zero looks like a tiny ring.
Since 1x0 = 0 and 2x0 = 0, we can say that 1x0 = 2x0. By then dividing both sides of the equation by zero, we find that 1=2. And in the context of dividing by zero, this is absolutely true. Because as you divide by smaller and smaller numbers, the result tends towards infinity. And relative to infinity, 1 really is the same things as 2, because no finite value can change an infinite value. Any finite value compared to an infinite value is worth nothing, so this 'version of maths where everything is equal to zero' is really just mathematics with infinite numbers.
they keep telling me "You can't divide by zero" i ask them "why not?" and they just go silent
Thanks for the clarification. The zero ring looks like some kind of poison.
You can't divide by 0, simply because it's nothing.
Gotta love the base zero number system. I think we should all switch to using base zero.
0 divided by 0 is defined by geometry.
2:24 well, not necessarily.
Maybe you could also define i as the number where i*0 = 1.
Then i would be 1/0, 2*i would be 2/0, 3*i would be 3/0 and so on.
(And 0*i would be 0/0 and because of our definition it would be 1, which somehow makes sense, because any number divided by itself gives 1. Let's just say 0/x = 0 is only valid for numbers in R).
Then 1/i would be 1/(1/0) = 0/1 = 0. So every real number r would be equal r + 1/i.
I don't know if there's a way to imagine that in a 2D area like the "actual" i, or if my theory would somewhere be a contradiction, but maybe we just need to define the common rules we know for calculating are only true in the real numbers.
(pls tell me if this could work)
I think a step is skipped in the proof of 0*(1/0)=0.
Let's call 1/0=j. We have j-j=0, and the distributive law: a(b+c)=ab+ac. Then we have 0=j-j=j(1-1)=j*0=1.
We need this because 0*a=0 is not an axiom in the system.
@@Leonex52 It's still a leap in logic to say that j-j=0. j-j=(1-1)/0=0/0. You must prove that 0/0=0, or explain why the denominator can be discarded if the numerator is zero.
What if you attach the numerator to the answer so that x/0 is infinity with x attached and when you multiply this by zero you get back to x?
Uh, I think I went too deep into youtube again, I barely understood any of this. It felt like watching someone explain the concept of objective mathematics instead of the subjective math that the human brain can comprehend. Or maybe I'm just dumb.
"We're gonna define 1/0"
Oh so is this gonna be some black magic analytic continuation? I wonder-
"Yeah this is just the set containing zero lol"
Okay, that was correct... but kinda disappointing.
That number system makes it a lot easier to learn the times table(s).
The issue I don't get is why do people start be saying "anything times zero is zero" but don't apply the same rules to 1/0? It seams to be that this would also require special cases, in the same way that multiplication by zero does
Looks like a trivial demostration, but definitely an interesting proposition.
Happily surprised the video title wasn't clickbait!
This was like saying, if I make up my own math, I can do anything!
For a more serious answer, one could look into "Wheel theory"!
2:28 Wouldn't Bodmas be applicable here? Because we need to solve 1/0 first to multiply with zero next.
A step is skipped in the proof of 0*(1/0)=0.
Let's call 1/0=j. We want j to have some common properties of any other elements of R so we can work with it, like j-j=0, 1*j=j, and the distributive law: a(b+c)=ab+ac. But once we set these 3 axioms, then it goes 0=j-j=j(1-1)=j*0=1.
Note that 0*a=0 is not an axiom in the system(a ring), it's a theorem.
Now I have a superpower! I can divide by 0. Finally!
What happens if we remove the rule that anything multiplied by 0 is 0? That is make an exception for 1/0?
My parents raised me JavaScript, so I was always able to divide by zero: x/0 = inf
As an electrical engineer, dividing by 0 = infinity unless that's inconvenient then it's just a really big number