# Is every number compared with infinity virtually zero?

When thinking about infinity people think that a great number like googolplex or a googolplexian is closer to infinity, but infinity never ends every huge number that I can think off is closer to zero than to infinity does that make every number virtually zero ?.

First we need to define what is "comparing" two numbers. Based on your statement, I assume comparing is evaluating how many times a number is bigger than another.

$$1000000/10 = 100000$$

So a million compared to ten is pretty large, as it's a hundred thousand times larger. So how large is infinity to a finite number?

As it turns out, infinity is not really a number, and division is not well defined for infinity. But we can use limits. Think of a really big finite number $a$. If you divide it by $x$, and you can make $x$ arbitrarily large, $a/x$ eventually gets really close to zero. Even if $a$ is really big (like a googolplex), $x$ can be bigger, and the bigger $x$ gets, the closer the quocient approaches zero: $$\lim_{x \to +\infty} a/x = 0$$

Yes. Every finite number is small compared to $\infty$.

It's kind of hard to say more without defining “virtually,” but if you give me any positive number $n$ which you might call large, and positive number $\epsilon$ which you might call small, I can find a number $N$ such that $0 < \frac{n}{N} < \epsilon$. In other words, $$\lim_{x\to\infty} \frac{n}{x} = 0$$ for any number $n$, no matter how large.

Sort of yes --- the only issue is often infinity isn't necessarily "number" (often it's introduced as a short hand for a particular kind of limiting behavior of a function). There are formal ways to consider it a number, and then it's bigger than any other number. So, if you have whatever massive number you want to compare to infinity (call it $N$), then you have $N<\infty$. You also have $2N<\infty, 3N<\infty,\dots, N^2<\infty,\dots, N^N<\infty,\dots$, etc.