Isn’t dividing by 0 similar to multiplying by infinity?

Question

A commonly cited proof for being unable to divide by zero is as such:

0 = 0 * 1
0 = 0 * 2
0 * 1 = 0 * 2
(divide both sides by 0)
1 = 2

That’s obviously unacceptable, but consider the following (assume ∞ is infinitely large and 0 is infinitely small):

0 = 1 / ∞
0 = 2 / ∞
1 / ∞ = 2 / ∞
(multiply both sides by ∞)
1 = 2

I see it mentioned often that dividing by zero isn’t possible, but I never see this. Is there a reason?

Answer 1

It is similar in that neither one is allowed within the standard real numbers. I think you don't see the second version because people are not used to dividing by infinity while they are used to multiplying by zero. You can then present the division by zero as canceling, which people don't think of as division and it doesn't trigger the reaction that "division by zero is not allowed".

Answer 2

I think the issue in your example is when you multiply both sides by ∞ in your final step, you'll get ∞/∞ on both sides, which is undefined, and does not equal 1. So the issue isn't that you multiplied ∞, the issue is claiming ∞/∞ = 1. In this respect, it's different than dividing by 0.

Answer 3

Another issue with dividing by zero, is that the "result" would have to differ depending on whether you approach zero from the positive or negative direction. For instance:

• $\lim_{x \to^- 0} 1/x = -\infty$.
• But $\lim_{x\to^+ 0} 1/x = +\infty$.

So dividing by zero is really just not well defined.

Answer 4

Dividing by zero is never possible, you can come near zero so much using a limit, but you can never reach zero, a fraction with a zero denominator is undefined.

Similarly multiplying by infinity, because infinity is not a number (and it doesnt act as one since,for example, $\infty +\infty = \infty$ but $a+a=2a \ne a$) So you can actually multiply by $x$ and take the limit as $x$ goes to infinity, but you won't be precisely multiplying by infinity.

Finally, $\frac{1}{x} \ne \frac{2}{x}$ but ONLY if you take the limit as $x$ goes to infinity you get that $\frac{1}{\infty}= \frac{2}{\infty}$ So you can't deduce from this that $1=2$ and nearly the same thing applies for the case you divide by $0$

Answer 5

You can't divide by $0$ for a very simple reason.

Division does not exist by itself (and Division is NOT the inverse of Multiplication by the way).

Multiplication is defined on the other hand, and division is defined by: dividing by a number $a$ is multiplying by the inverse of this number.

But $0$ has no inverse...

For an inverse of $0$ (let's call it $z$) would then satisfy $0 \times z=z \times 0=1$...