Is infinity divided by negative infinity equal negative infinity?!?!!?

Question

Using a simplistic rule from pre-algebra, i came up with a question. If you divide a positive number by a negative number, or vice-versa, you geta negative number as the answer. If infinity (positive) divides itself by negative infinity, won't it egual negative infinity?

Answer 1

Numbers can't divide; we divide. We divide 12 by 3 to get 4; the number 3 can't divide anything; it isn't a sentient being, and neither can infinity.

It depends on what you mean by infinity. If you mean potential infinity, maybe you mean something like

(1/x) / (-1/x)

when x = or approaches 0. If you take a number close to 0, you can perform the divisions. Simplifying the expression gives you -1 regardless of x, so if x approaches 0, this expression approaches -1. I.e., infinity divided by negative infinity is -1, not negative infinity. Actually the quotient infinity / (-infinity) is indeterminate and could be any value, including infinity.

If you mean actual infinities, such as the transfinite ordinal number w (omega), your equation still does not hold, for in the surreal numbers, w / (-w) = -1, not -w. You can cancel the w's as if they were 2s or something. You need to take a higher ordinal infinity such as w^2 to get -w: w^2/(-w) = -w.

Answer 2

Lots of good answers. In general, it is best to call infinity/infinity (or infinity - infinity) "undefined". Here is some motivation for this. You probably know that 2*infinity = infinity. So then if you had: 2*infinity/infinity, depending on the order of operations, you could get 1 or 2. Similarly, you know that infinity*infinity = infinity, so if you had infinity*infinity/infinity you could get infinity or 1 (or any other answer) depending upon your order of operations. So that is why it is best to call these things "undefined" since there are many plausible answers.

If you want to get more rigorous about these things, you can use limits. L'hopital's rule will help you with dividing infinities. Also in set theory or some other fun class, you can learn fun stuff like the number of integers is the same as the number of rational numbers, but the number of real numbers if larger. Marvin Gardner has a great book, which a believe is called Aha! Gotcha!, but might be called Aha! Insight!, which explains infinities with some comically primative cartoons =)

Answer 3

The thing is, you can't casually talk about infinity as if it were a number. Infinity is a concept, covered in more detail in the topic of limits in Calculus.

To think of it in terms of limits, we approximate infinity to be "a really large number" and negative infinity to be "a really small number". There's a chance that the approximation will be some fixed value, but there's also a chance it will be negative infinity.

But since you're treating infinity as a number, there's no answer to your question.

Answer 4

Because "infinity" is not an actual number, just a concept, there is no number 1/infinity. However, it is possible to say that the LIMIT of 1/x, as x goes to infinity, is 0.

Answer 5

Negative sign will sit in the beginning of the expression, but infinity/infinity is an undefined number, NOT infinity.
So (- infinity)/(infinity) = undefined number
It doesn't matter whether it is positive or negative.
Here you can apply L'Hospital's Rule. During this process you can keep your negative sign in the beginning of the expression whatever.

Answer 6

It would seem so, if you mean by infinity "an arbitrarily large number". That's not a rigorous definition of infinity, but that's what most people are tihnking of when they ask questions like tihs.

Actually, infinity is not a number, it is a limit, or rather, the lack of a limit. If you're asking what is the limit as n and m go to infinity of ........m/(-n), with n and m both function outputs, this is an indeterminate form that you'd have to evaluate with L'hopital's Rule, a famous result in calculus.

Moreover, there are different orders of infinity. See the work of G. Cantor for more info on this.

Answer 7

no. Infinity is not a number, division is an undefined operation on infinity, if you define the number rules you get -1. 0 ,negative infinity, undefined,and "igiveup". depending on the additional definitions/rules you choose. this is a good question to think about, It sorts the Physicists from the Mathematicians.

Answer 8

Yeah you'll learn in calculus that infinity is not a number but an expression of something so big its beyond imagination. So infinity divided by inifinity or zero, infinity multiplied by infinity or zero, is all: undefined.

Answer 9

Infinity isn't a number, it's a concept. The same rules of mathematics don't apply to it. For example, infinity divided by infinity is infinity, not one.

Answer 10

No it is not definded.

however you can sometimes say that the limit of (inf/-inf) -> -inf
that depends how you got the inf's

for instance lim x->0 ( 1/x / -1/x^2 ) = -inf
but lim x->0 ( 1/x^2 / -1/x ) = 0