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What does Infinity Divided by Infinity Equal?At first, you may think that infinity divided by infinity equals one. After all, any number divided by itself is equal to one, however infinity is not a real or rational number. I am going to prove what infinity divided by infinity really equals, and you may not like the answer. First, I am going to define this axiom (assumption) that infinity divided by infinity is equal to one:
Since ∞ = ∞ + ∞, then we are going to substitute the first infinity in our axiom:
The next step is to split this fraction into two fractions:
Next, substitute the axiom twice into the equation, we get:
Finally, this can be rewritten as:
Therefore, infinity divided by infinity is NOT equal to one. Instead we can get any real number to equal to one when we assume infinity divided by infinity is equal to one, so infinity divided by infinity is undefined. If you still do not believe me, let's try proving this another way. Let's start with our axiom again.
Next, split this fraction into two parts.
Solve the fraction.
This is equal to:
At first, you may assume that infinity times zero is equal to zero and that would make zero equal to one too. However, infinity times any number is undefined. Therefore, infinity divided by infinity is also 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 B.
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