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What does Zero Times Infinity Equal?


At first, you may think that zero times infinity equals zero. After all, zero times any number is equal to zero, however infinity is not a number.

Logic dictates that zero multiplied by itself no matter how many times will always equal zero. However, I am going to proof that this answer is not quite correct when dealing with infinity.

First, I am going to define this axiom that any number divided by infinity is equal to zero:

0 = c




Where c is any real number.

Someone pointed out this is axiom is incorrect. Stating that any real number divided by infinity should be equal to 0.000..1. Well, 0.000..1 is equal to zero for the same reason that 1 equals 0.999... Take a look at these proofs for more information. However, any real number divided by infinity is equal to undefined, because you can never finish dividing something into infinite number of parts. Therefore, the axiom above is false.




So, lets prove what zero times infinity equals:

y = 0 * ∞


The first step is to substitute for zero with the axiom:

y = c


* ∞


Therefore, when the infinities cancel each other out, we get:

y = c


Two friends of mine just proved to me that infinity divided by infinity does NOT equal to one, therefore my proof does not work. If you are interested, here is the proof that infinity divided by infinity does not equal to one. In actuality, when any number (including zero) is multiplied with infinity, then the results are always undefined. Therefore, zero times infinity is undefined.

This can be rewritten as:

0 * ∞ = c


So, zero times infinity is an undefined real number. This is the definition of undefined. Therefore, zero times infinity is undefined.

Another way of looking at this is that no one can EVER finish multiplying zero times infinity, therefore the answer will always be undefined. Even though logic dictates that the answer will never not be zero, this answer will never be reached. Therefore, trying to multiple zero times infinity is undefined.

by Phil for Humanity
on 20061221

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