How do you prove it?

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If I say 1/0, how do I proved it's "undefined" and not some actual number? I can't use "undefined" in algebra to solve for a variable, why should it be a real solution? I could see how it makes sense orally, but it seems like there's weird unpredictable guidelines for where human assumption comes in and what the math carries out to be. Like if I say "x approaches infinity", that's used in all sorts of algebra but that's not an actual numeric thing that's just us basically saying "as x increases indefinitely", but in mathematics x never actually has a value of infinity.

But anyway, if I say 1/0=undefined, and I have x = 0 * undefined, shouldn't I logically be able to solve for it if "undefined" is actually the correct answer? It honestly seems like the answer is infinity, but there's no way to actually prove something is equal to infinity using math because it's not a real number, so it seems like some sort of dilemma I could literally put "nothing" into something an infinite amount of times, it never get's used up because it's nothing. I'm doing it right now as you're reading, I'm putting "nothing" into 1 bottle.

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What do you mean "there's no way to actually prove something is equal to infinity using math because it's not a real number"? Mathematics is a set of rules relating made-up mathematical objects; mathematicians define new rules and new objects and explore their properties. If one can construct a set of rules that describe when some expression evaluates to infinity, then one can prove that an expression is equal to infinity.

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Do you understand what the word "defined" means?

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What do you mean "there's no way to actually prove something is equal to infinity using math because it's not a real number"? Mathematics is a set of rules relating made-up mathematical objects; mathematicians define new rules and new objects and explore their properties. If one can construct a set of rules that describe when some expression evaluates to infinity, then one can prove that an expression is equal to infinity.

But if you just have the logic of math on its own without humans to make all sorts of inferences and explorations, how would you find infinity? Where does it "naturally" occur in the logic of mathematics and it's proof?

Do you understand what the word "defined" means?

Ok yeah so if you can't seemingly do operations on it or if it doesn't pass the vertical line test or w/e other test, but how do we know the answer couldn't be some thing or some number? Couldn't "infinity" be thought of as one thing in of itself? How do you prove that it isn't some value?

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It honestly seems like the answer is infinity

Look at a graph of f(x)=1/x. Now, look at the value of f(x) as x approaches 0 from the left side of the graph. Now, do the same from the right side. The left-handed and right-handed limits aren't equal, so it's not a continuous function. Given that the graph diverges so much there, where do you think we should put f(0)?
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Ok yeah so if you can't seemingly do operations on it or if it doesn't pass the vertical line test or w/e other test, but how do we know the answer couldn't be some thing or some number? Couldn't "infinity" be thought of as one thing in of itself? How do you prove that it isn't some value?

You seem to think math is some ethereal thing which we merely discover facts about. That's not the case. Math is a set of rules. We define what it means to divide two numbers, and if our definition doesn't handle the case of 1/0, then it has no value. You could propose a new way to define division, of course.

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You seem to think math is some ethereal thing which we merely discover facts about. That's not the case. Math is a set of rules. We define what it means to divide two numbers, and if our definition doesn't handle the case of 1/0, then it has no value. You could propose a new way to define division, of course.

Well being an administrator I trust you can find the two different topics that shows I do not think that. While I do not think math is "ethereal", I do have to respect that it is a logical system in many respects and can make predictable results.

You're right that it isn't continuous, it wouldn't be a normal type of function if there were literally infinite points going vertically up and down, but conceptually doesn't it make sense that you can put "nothing" into something an infinite amount of times? I mean there's all sorts of math that basically indirectly does stuff like that, like derivatives and integrals, they use infinitesimally small numbers, in reality it doesn't make sense to have an infinitely small box, yet somehow we find the exact area. What where do you find that exact area? In none other than exactly on a horizontal asymtote.

Edited by SamBridge
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This discussion comes up more often than it probably should.

Let's assume that $\frac{1}{0} = \infty$ is valid and true. This means $1 = 0 \infty$. But what happens if we multiply by a real number $n \neq 1$? Then we have

$(1)(n) = (0)(\infty)(n) \implies n = (0)(\infty) = 1$

But $n \neq 1$. Thus $\frac{1}{0}\neq\infty$.

If we just went with the concept of "putting nothing into something forever," it seems to me that would result in $\frac{1}{0} > \infty$, which contradicts the definition of infinity as being greater than any value.

Edited by John
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This discussion comes up more often than it probably should.

Let's assume that $\frac{1}{0} = \infty$ is valid and true. This means $1 = 0 \infty$. But what happens if we multiply by a real number $n \neq 1$? Then we have

$(1)(n) = (0)(\infty)(n) \implies n = (0)(\infty) = 1$

But $n \neq 1$. Thus $\frac{1}{0}\neq\infty$.

If we just went with the concept of "putting nothing into something forever," it seems to me that would result in $\frac{1}{0} > \infty$, which contradicts the definition of infinity as being greater than any value.

I know I've worked out that whole scenario before, but I just don't know what else it could be at this point. Why does it break everything if all of a sudden I rotate a stick to 90 degrees? Reality doesn't care, it seems that math freaks out more than anyone. Although the math your suggesting would merely mean that it isn't a function, there are systems which can yield more than one result per intersection of a line at a given value.

Although this situation kind of reminds me of something else, so perhaps there can be solutions in "higher" dimensions. If you have a plane, or I guess a square-like object, in order to inverse the plane, it would have to stretch literally infinitely outward to loop back around, but you can model the transformation of the plane using simple a sphere, something like this

and a sphere doesn't not stretch infinitely outward or anything its a finitely bound shape, but the errors for using that seem to occur at 0 and 180 degrees outside of a sphere, something to do with the tangent line being vertically up and down at those angles, but on a sphere it doesn't matter it seems. Edited by SamBridge
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I know I've worked out that whole scenario before, but I just don't know what else it could be at this point. Why does it break everything if all of a sudden I rotate a stick to 90 degrees? Reality doesn't care, it seems that math freaks out more than anyone. Although the math your suggesting would merely mean that it isn't a function, there are systems which can yield more than one result per intersection of a line at a given value.

No, his result shows that letting 1/0 be infinity creates inconsistencies in mathematics. A number n cannot simultaneously be one and not-one.

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No, his result shows that letting 1/0 be infinity creates inconsistencies in mathematics. A number n cannot simultaneously be one and not-one.

Well isn't there some triangle propertly?

Like

2*4 = 8

8/2 = 4

8/4 = 2

That's more what I'm thinking, but perhaps the inconsistencies come from the fact that infinity isn't really a number that we can mess around with as though it is a real number. I man you can't prove 1/infinity = 0, but wouldn't it make sense that if something was divided into infinite parts those parts would be of 0 size? Just like with an integral, the boxes get infinitely small as n approaches infinity.

Edited by SamBridge
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There is actually a flaw in my little proof, which is that it kind of assumes $\frac{0}{0} = 1$, which is its own related can of worms. This can be avoided by taking a few extra steps, though, and going from there. Regardless, you can manipulate the equation to get into all kinds of silliness, which reinforces the notion that the equation is wrong.

Your errors here are common, and I can't really fault you for them (though a quick Google search would help a lot). You're treating infinity as if it were just some very large number (even though you give lip service to the idea that it isn't), and you're misunderstanding the notion of the limit. When we take derivatives or integrals, we're assuming some value (let's call it $\Delta x$) is getting closer and closer to (but never actually reaching) zero. $\Delta x$ gets arbitrarily small, and we call it an infinitesimal in recognition of that fact, but it's never actually zero.

So while $\lim_{x \to \infty} \frac{1}{x} = 0$, the fact remains that $\frac{1}{\infty} \neq 0$.

Edited by John
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Yeah I know what limits mean but that's not how we orally interpret them from an outside view of math, it's literally when h=0 that we find the exact derivative the old fashioned way, it's literally when n goes to infinity and what value is exactly on the actual horizontal asymtote that we treat as the exact integral. I'm well aware that the limit of 1/x as x approaches infinity is 0 and that 1/infinity itself is not proven to equal 0. But as it says on my calculator, the answer is "undefined", but that doesn't mean there can't be one, we just have no idea what it is, or I guess we have no idea how to prove what it is, but conceptually the concept makes sense, the math doesn't. It makes sense that nothingness can go into a bottle an infinite or never-ending or indefinite amount of times, and it conceptually makes sense that something broken into infinite pieces would have them yield a quotient of 0, but infinity just isn't a number that uses the same systems of axioms as arithmetic and algebra, it's more of a concept in of itself, so shouldn't the answer be more conceptual? I mean we don't use actually proven math to say infinity-1 = infinity, it's just a concept we determine by looking at evidence like how I did. Unless maybe is there some way to use cardinal numbers that proves infinity-1=infinity?

Edited by SamBridge
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No, his result shows that letting 1/0 be infinity creates inconsistencies in mathematics. A number n cannot simultaneously be one and not-one.

It also cannot simultaneously be 0 and one and infinity.

0/0 could be regarded as one and not one if anything divided by itself = one, none or not none if anything divided by 0 = 0, or infinity or not infinity if anything divided by 0 = infinity.

Edited by LaurieAG
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• 2 months later...

Um, I'm not much of a mathematician, and repling to an old thread, hah, and tired, but two thoughts occur:

1) Zero was not originally a number, but a placeholder. At that point in history, it didn't make sense to divide anything by 0 because 0 was not a number, just the sign for the absence of a number--a mark that meant 'no quantity'. Nothing to divide by. You can't put emptiness into a bottle an infinity of times, or zero times, or one time. Because it occupies the center of our coordinate graphs, however, we came to think of it as like other numbers; but it if like Sambridge we want to think about it 'outside the view of math' we'll notice that it is a little paradoxical--a quantityless quantity. It's really still NOT a number, just a sign that says 'no number exists here'. So it makes sense that you can't divide anything by 0--you can only divide numbers by other numbers.

2) But, oh my, we have redefined zero as one of the real numbers, because we want to play up and down the number line, and all across the grid, and we can’t
do it safely if there’s just a hole at the center of everything, now can we? I strongly suspect that this idea (0 = some number) sits in the center of our graphs like a little liar’s paradox or Gödel sentence ("'This statement cannot be proved in theory T' cannot be proved in theory T") to remind us of the incompleteness of any mathematical logic we happen to define. Perhaps because 0 is both a ‘real number’ and a quantityless quantity, any statement n/0 = x is simply a self-contradiction, like ‘this sentence is false’—if it is false, then it’s true… oh my. So, just as the liar’s paradox has no truth value—it is neither true nor false—n/0 has no numerical value, not n, not infinity.

If you must have a ‘value’ for 1/0 it would have to be 0 itself: 1/0 = 0 i.e. ‘not a number’, i.e. ‘this is not math’. Lol, trying to figure out how to turn ‘this is not math’ into a Godel sentence…lol… .

‘In my bad dreams I fear the zero-point on the number line, and when I wake
I am thankful that mathematics does not map reality. In our equations,

how easily we handle zero, making nothingness our toy; and then I tremble anew:

how much more fearsome are the real abysses we face?’

Effron T. R. Ygäll ‘The Logic of Absurdity’.

PS 1/0 can't = infinity because 0 times infinity = 0 not 1 ;;; keep on ...no, please stop ...ahahaha ... it tickles ... Charlie

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Numberphile posted a very informative video on just this. After spending at least 3 seconds looking through the thread, I don't think it's been posted yet. At least not in this thread.

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Good video

00:00 'Zero is a perfectly good number' then why when we try to use it as a number does it produce so many contradictions?

2:15 Infinity is not a number, it's a concept...

I would still argue that all these weird things happen 'at zero' because it's not a number, but a concept...the concept of 'no number here'....

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2:15 Infinity is not a number, it's a concept...

Though generally considered a concept, it's not just any concept. I hate the notion that "infinity" is no more legitimate a number than "justice" or "love". Unlike the latter entities, $\infty$ actually has a defined, ordered relationship to the real numbers. Even still, there are number systems, besides the reals, where $\pm\infty$ are in fact numbers with some defined arithmetic relations. And they work completely well in their own respects.

00:00 'Zero is a perfectly good number' then why when we try to use it as a number does it produce so many contradictions?

What contradictions? Arithmetic on zero only causes contradictions by assuming truth in certain statements which we conventionally regard as fallacies. Letting $\frac{1}{0}$ be undefined is the keystone example in this thread.

Also, there is nothing wrong with zero itself (insofar as I see). It's only when you perform certain operations where things go erky. Do you also have a problem with the number 1? Because 1 and logarithms don't always go well together. Treating zero like infinity may essentially eliminate all trivial mathematical relations it has with the other numbers.

Anyway, if you'd like to abolish zero as a number, go ahead. But you will have to reformalize your own rigorous framework of mathematics with many new rules and stipulations. Then somehow reconstruct the set of real numbers without zero in a way that makes math work sensibly (and the complex numbers also since they are so useful). And then continue to rework all of arithmetic, all of analysis, all of algebra, geometry, calculus, topology, etc. since they will obviously not be the same without zero (yet you never hear about geometry being broken because we use zero as a number).

Edited by Amaton
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Also, there is nothing wrong with zero itself (insofar as I see).

Not only are there number systems that include cardinals (infinities) - albeit at the expense of loosing certain arithmetic operations - there are also number systems without a zero, eg the natural or counting numbers.

You might also like to consider whether zero should be counted as odd or even.

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Not only are there number systems that include cardinals (infinities) - albeit at the expense of loosing certain arithmetic operations - there are also number systems without a zero, eg the natural or counting numbers.

Fair. Yet you can only do so much with just counting numbers At least, without defining new kinds of numbers from them (i.e. their negatives for non-zero integers --> their quotients for non-zero rationals, etc.)

You might also like to consider whether zero should be counted as odd or even.

Interesting. I like to follow the convention that any $2k,\,\, k\in\mathbb{Z}$ is even. Therefore, zero is obviously even. More so, $2k+1,\,\, k\in\mathbb{Z}$ is odd, meaning zero is most definitely not odd. But I'm not sure if this is 'good enough' to rest the issue.

Edited by Amaton
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Rest assured, zero is even.

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OK that's parity, how about sign.

Is zero positive or negative?

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OK that's parity, how about sign.

Is zero positive or negative?

You usually hear zero included in "non-positive" and "non-negative" subsets of $\mathbb{R}$, so I would guess neither. The Wikipedia page agrees.

What the formal definition of 'sign' is, I don't know. I'd like to say that a number $n$ is either positive or negative if $n\ne -n$, where we consider the unary negation. So by this definition, zero is neither.

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So by this definition, zero is neither.

One formal definition of sign is to use strictly greater than zero for positive and strictly less than zero for negative.

Which makes it different from every other number in the set.

Of course, one question is

Which set?

This is interesting because we require a set without a zero to be able to use proof by induction, which involves creating a sequence of conditions in one to one correspondence with the natural numbers, N. N does not include zero or we would not be able to do this.

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One formal definition of sign is to use strictly greater than zero for positive and strictly less than zero for negative.

Or that -- which is much simpler and obvious, lol.

Which makes it different from every other number in the set.

Of course, one question is

Which set?

This is interesting because we require a set without a zero to be able to use proof by induction, which involves creating a sequence of conditions in one to one correspondence with the natural numbers, N. N does not include zero or we would not be able to do this.

But zero can be a natural number, as per $\mathbb{N}_0={0,1,2,3...}$. It just depends on your preference, since there is no universal standard as to whether or not $\mathbb{N}$ includes zero.

This is a bit above my level as a student, but why would zero make a difference in inductive reasoning? From Natural number - Peano axioms (Wikipedia):

There is a natural number 0.

...

If a property is possessed by 0 and also by the successor of every natural number which possesses it, then it is possessed by all natural numbers. (This postulate ensures that the proof technique of mathematical induction is valid.)

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