I Learned How to Divide by Zero (Don't Tell Your Teacher)
2024 ж. 7 Мам.
932 701 Рет қаралды
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They say you can't divide by zero. But "they" say a lot of things. It's time to see how to divide by 0.
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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?
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!
0:15 Wow, I didn't know Ant is such a strong word 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.
"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
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?
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!
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
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?
4:00 Problem solved, right?? Not quite. Me ragequitting the video
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.
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!
i had no idea this was released today a year ago and that just makes this better
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*
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
So you mean we can't create a black hole dividing by zero. Fine, I'll go back to the blackboard.
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.
"...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
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
@@SolstitiumNatum he's living the 80's American Dream lol
it would be funny to see my math teacher have a popular yt channel
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.
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.
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
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.
You could also map out quaternions, octonions, and so on to multidimensional donuts. Great video.
I like the approach of how everything equals everything else, its almost like it too the definition away and left everything undefined
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
I've just watched this video and I'm gonna subscribe straight away because that is mind blowing
Thanks a ton!
Sharing this to my teacher🏃♂️
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.
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 ∞ × ∞ = ∞?
Makes sense honestly. Infinity is a quantity not a number, and if 0 has no sign it makes sense that infinity doesn't too
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.
incredible video. super fascinating
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.
BIG OUTOUTS :)
😂 that’s what I get for trying to break rules
Wow, thanks! You really blew my brain this time! :D
Glad to hear it!
Never gonna give you up!
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.
"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
For such a simple question I did not expect such a complicated answer
2:21 square roots: Am I a joke to you
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.
We focused so much on whether we COULD do it that we never stopped to think whether we SHOULD do it.
you have a penetrating mind... what a sharp wit! I say this regarding to three video about zeroth root, zeroth power and dividing by zero.
We haven't been lied to. When we say 'undefined' we mean "not defined in for these purposes or in this context."
why do you look so displeased whenever you're drawing something 😄
"God I hate writing backwards, why do I do this to myself?"
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!
The zeroth root of 1: °√1 = a, with "a" being every number, because every number to the power of zero is equals one: a^0 = 1 5^0 = 1 23^0 = 1 (-2897,3401)^0 = 1 °√1 = 5 °√1 = 23 °√1 = -2897,3401 And so: 5 = 23 = -2897,3401
Lol
After some thinking, I found this to be an interesting idea: 1/0 = 0^-1 There's probably some flaws, but here's my thought process: If you multiply each side by 0, you get 1/0 * 0 = 0^-1 * 0 Division and multiplication of 0 cancel out, and you're left with 1 = 0^-1 * 0 Every time you multiply a number by itself, its exponent increases by 1, so 1 = 0^0 And 0^0 = 1, therefore 1 = 1
that's a cool way to think of it, but one problem here is that 0^0 isn't necessarily 1. In many contexts, it is considered undefined. Another is that, in algebra, 0*x=0, so you chose one rule over the other. If we used that rule instead, your math will look like this: 1/0*0=0^-1*0 0=0^-1*0 0=0^0=1
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
Really great video I'm French guy but I understood your video
Glad you liked it! Thanks for watching!
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.
1:48 And this isn't a problem as -0 ≠ 0, just as -0.000000001 ≠ 0.000000001. Therefore, 1 / 0 = ∞ & 1 / -0 = -∞.
Zero is neither positive nor negative. -0 is not a number
Very happy to give this video the 1000th and more than deserved like, This is a really interesting qubject
Thanks so much!!
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 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 always had the idea that there are two types if 0, true 0 and false 0. True 0 is a number that exactly equals 0. False 0 is a number so small that we round/write it as 0. Since they appear the same, that's why 0÷0 is undefined.
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
1:16 So this is probably why people think something divided by 0 is Infinity
In the largest number question I alwys threw them infinity raised to infinity power. BTW, Yes I took calculus. For my major of Meteorology I had to take one course past calc. I took differential Equations. BTW, I'm 76, worked for A.F. as a meteorologist for almost 36 years, retired in 2009.
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.
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 :)
Can you do this in math: yes, as long as you're being consistent. Should you: only if it's useful. Done.
The answer is always infinity, unless it is negative, in that case it is negative infinity. ( Edit ) Take a pie chart and divide by zero slices, you have 1. Take 0 pies, you have 0. The value between 0 and 1 is equal to infinity. A very illegitimate way of how I came to the conclusion that 1/0 = infinity
But there’s no way of knowing if it’s positive or negative, since it depends on if you take the limit from the positive or negative side.
i reasoned to myself that 0/0 = 0. though all other numbers satisfy the equation, there are times where those numbers don't satisfy a equation that has a 0/0, except for 0 itself.
Thank you... Very informative and generous .. And yes i will not tell the prof or teacher.. 👍👍👍👍👍
You bet! 🤫🤫
4:28 "Infinity + 1 is infinity!" Lol. At my school people would just keep going with "infinity + 2" (3, 4 wtc) followed by "2x infinity" (3x, 4x etc)) followed by "always 1 more than you" followed by "always 2x as much as you" (then 3, 4 etc.). The worst part is the incorrect grammar in those sentences. In German, they would say "Immer zweimal mehr wie du!", Which is like saying "always two times more as you"
Infinity to the power of infinity
Hyper dimensional toroidal tension field physics... He describes a toroid perfectly.
Another correlation to division by 0, vertical slope, etc... is the tangent function at each PI/2 + P*N intervals. It's the same thing!
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.
Great video!
Glad you enjoyed it!
He looks so sad when writing stuff down, but so satisfied when he's done
There are few other ways to handle divisions by zero, and these are practical. 1) use replacement symbol (like with "i") and do the math; sometimes these entities cancel each other out or produce tangible results 2) conditional statements in programming languages is also a form of math. Sort of. "If something = divby0 then print something else" 3) in DSP fhere is a problem called "denormals". One way of handling them is to add some small offset to the computation.
In the first case, this works in very specific cases with very specific rules. For example, the degree of the 0 polynomial is often denoted by -infinity, a formal symbol satisfying: -infinity + -infinity = -infinity For all natural numbers x (including 0), -infinity + x = -infinity etc We do not have general rules for multiplication involving -infinity.
I'm glad there is another Bri the Math Guy out there! Well, I'm not really a math guy as much as a science guy. So I guess you could call me Bri the Science Guy! That feels taken somehow...
Very interesting !
Glad you thought so. Thanks for watching!
2:13 Bro that face just sums up any time I try to do anything math related and I go like "nah, it's doesn't worth it"
The nullity sounds like NaN in floating-point.
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?
Switches: 2:40
i have always thought that if you multiply by 0 on any number, you get 0. and since multiplying is dividing backwards, dividing would be multiplying backwards. so that would mean x / 0 = 0. Thats just what I think though
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.
There is a poetry to infinity in the Riemann sphere in that infinity has "arbitrary direction" just as 0 does.
"Don't tell your teachers" Teachers that are watching this video: you have become the very thing you swore to destroy
1 / 0 is both negative and positive infinity, a.k.a alif null. Simple.
As a engineer, the question is not really should we devide by zero.. it just will happen some times, but rather, what we wish the answer to be.
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.
I think people said to not divide with zero in order to make the final exams in 12th-13th grade easier. There is always a question where you have something like x+1/x and if 0 would be a possibility for that, than there would be a huge number of answers (I think, Im not that trained in maths).
I like your faceexpression during writing on the whiteboard
you're under arrest for destroying the universe
Thank you for your brilliant question and intelligible video about it. I had been asking myself this question in mid-November 2021 and finally found an answer that led to a new paradigm. I call it HyperMath, "Math 2.0" Math 20-21 (and 22 :) ), a contemporary math that "goes beyond" and unifies different branches of math in a single uniform vision. I share the same idea, that we should consider a space of numbers as a sphere (not a 2D-plane), so we treat "infinity" as a single point, just opposite to 0. "Plus inf" and "minus inf" differ just as path/direction to a single point ("infinity" itself IS really a single point, however there are many types of "infinities" in different "operator worlds"). Well, that answer is in short... and is... you can't divide a number by a number "zero", because THERE'S NO "number zero" in the "Multiplication World". What MW is? Consider a "product" as bijective mapping of (x,y)->z, where x,y,z are "numbers" (real positive numbers), z is a product (smth new "made from"/"based on" x and y) of multiplication. We can consider an equivalent mapping in MW. In a simple way, let's consider pairs (x,y), where y is "inverse/opposite in terms of multiplication" (reciprocal) to x and vice versa. Let's stay simpler and limit "number x" to a natural number (1,2,3..). Consider x as a natural number, and y as an inverse to it, so x*y=1. Members of our "Multiplication World" are grouped in pairs: (1, 1), (2, 1/2), (3, 1/3), ... Not that "1" is a special element. Call it as "self-applied"/"central" (stays the same during transformation)/"idempotent", so 1*1=1. It is just as "number zero" in a world of addition: 0+0=0 (and there are another ones in another "worlds"), by the way. Note that there's no "number 0" between members of MW: (1, 1), (2, 1/2), (3, 1/3), ..., (N, 1/N). "1/N" is no way 0, while N is a natural number. We can legally divide X to "1/N" and get X*N for any X,N in natural nums (MUL and DIV and opposite operations, so div(x,y) means mul(x,1/y)). But we can DEFINE a special element "INFTY", which is not a natural number, which means a "limit" of N. Then we must DEFINE its inverse as "1/INFTY" and represent it as "0" to match the pair (INFTY, 0). Then MW is extended by two mutually inverse items: (1, 1), (2, 1/2), (3, 1/3), ..., (N, 1/N), (INFTY, 0). These items are not "numbers" (natural or harmonic, e.g 1/N, numbers), I call them "pole". So they have a special meaning and special rules in multiplication world should be applied to them. This way, we DEFINE INFTY as inverse to 0, and 0 as inverse to INFTY. Simply put, we define 0*INFTY=1 (note that we define it just here. In a different place, we could define it differently). But how can we apply the pole to a number? We can define mul(x, ZERO) = ZERO (pole ZERO is a "killing" element, a very bizzare beast, breaking a bijective mapping mul(x,y)=z, so pole ZERO is different from "number", because a legal number keeps this bijection), mul(x, ZERO) = mul(x, 1/INFTY) = div(x, INFTY). This rule introduces UNCERTAINTY (just as shown in the video), which is "smth (that can be anything at once) in a range". We can invent "U(X)" is "anything" from "set X" ("everything at once"). Sometimes, the UNCERTAINTY is in range 0..1. Let's consider a graph of x^(1/100). When it comes to extreme (e.g. x^(1/1000), x^(1/100000), ...), then the graph goes close to a vertical segment (0,0)-(0,1). This is a sample of UNCERTAINTY(0..1). Another example of UNCERTAINTY is zero-vector. It is said, it has NO certain direction (or all directions at once). So, div(x, ZERO) = div(x, 1/INFTY) = mul(x, INFTY). But how we treat "x*INFTY"? The first approach is to treat it as UNCERTAINTY(0..INFTY), e.g. any number: x*INFTY = U. You can depict it as a vertical line x=y. So U/INFTY="any X"=U. The funny thing is that our "defined" rule "0*INFTY=1" gets a broader sense: "0*INFTY=U" (and not only "1"). The second approach is to define a "hyper-natural" (in terms of multiplication), that "goes beyond" the natural number, so there's a bijection between "x" and product of (x, INFTY) = x*INFTY or single-term "x*INFTY", so div("x*INFTY", INFTY) = x. And we don't have to give up some basic algebra rules, but invent extended ones. If we are lied that's impossible, just remember imaginary i. MUL( 5, i) = 5i. It's smth that is not a number (in sense of naturals), but it is defined. And hence... exists!
"vertical line y=x" shoud read a line x=const. More accurately: (x, y), x=const, y=any
I aint reading allat
I have some (groups) of questions: 1) You invented the concept of uncertainity (U). Is U also part of the multiplication world (bacause some things are equal to it, i'd figure it should be, but i could be wrong), and if so, is U also one of the "poles"? What can you pair with U in the inverse pairings? 2) In the addition world, i suppose that the diffirent pairs of inverse numbers would always be a number and their negative. Zero is self pairing. Now - what can you pair infinity and U with? Or do they not exist in the addition world? They probably should exist there, if they exist in the multiplication world...
Thank you for the video! In my opinion, division by zero mostly is both theoreticały and practically meaningless and it's just fine for it to remain undefined then, but in some special contexts it can be useful. Cheers!
its an april fools joke
@@TULLIS-sl9tj it's not... He is speaking legit in the video
@@wojciechszmyt3360 wow.... I hope you're joking
@@notanoobx684 u trolling or what? Go read about it, he does speak legit absolutely.
@@wojciechszmyt3360 yes, but he defined it in a way that is practically useless. How old are you?
I was recently thinking about how its possible to divide by negative numbers? You have a pie and you divide it by negative 2 how many pieces do you have?
Many floating point implementations define 1 divided by 0 as +infinity, and 1 divided by -0 as -infinity.
Finally, someone understood the art of dividing by zero
lol I once tried to create math based on it by creating something like imaginary numbers and to define 1/0=r and created a few nice ideas like that dividing by 0 can connect dimensions and it was fun
Sounds interesting.
@@Sovic91 it is
@@shaharzamir88 I do have some, though. For instance, how do you define other numbers divided by 0? Or, in other words if 1/0=r, then what is 2/0? Is it 2r, or something else entirely?
2*1/0=2r
Btw you also need to make a patch for multiplying by 0
As positive infinity and negative infinity are not numbers, we are allowed to define both as the same limit, if it suits us. Just approached in opposite directions, like 0. This could allow us to define it as a special number. Then, 0 and infinity would be dual to each other in the relation between reals and their inverses.... which would again be the reals. Closed inversion.
We can define 1/0 as another imaginary number, say "j", forming another complex plane and a complex 3d space. Multiplying by i rotates numbers 90 degrees counterclockwise around the j axis, and multiplying by j rotates around the i axis. We can create extra dimensions for more undefined numbers.