Is every number compared with infinity virtually zero?

Question

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 ?.

Answer 1

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: $$\lim_{x \to +\infty} a/x = 0$$

Answer 2

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.

Answer 3

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.