# Infinity is NOT a number

When some people think of infinity, the first instinct is to naturally think of ‘a big number’ or ‘the largest thing that one can possibly be thought of’. This leads to a very common misconception commonly exhibited in the back and forth game played:

“I’m 100 times better than you”

“I’m a BILLION times better than you!”

“I’m a ZILLION times better than you!!”

“I’m INFINITY times better than you!!!!”

The possible upside to this which justifies its usage in this context is that your opponent cannot retort with an INFINITY+1 as by adding anything to infinity, you simply get back infinity again leading to an unrivalled victory.

This approach though, is problematic from the get go as infinity is not a ‘number’ in the sense that you are normally familiar with. A common trait that the normal numbers have, is that when you add 1 to them, you obtain a new, larger number than what you started with.

But we know that $\infty$ doesn’t satisfy this condition. In fact, from before we know that

$\infty+1=\infty$

Why is this problematic? Well, if $\infty$ was a ‘number’, simply deduct it from both sides to obtain

$1=0$

Uh oh. Something has gone terribly wrong here.

For this reason, infinity is not considered as a ‘number’ that is just really really big because numbers that are really really big are not the BIGGEST since you can always add 1 to that really large number to obtain something larger. Infinity is more representative of an IDEA. The horizon doesn’t appear to get closer when you walk towards it and the same can be said for something ‘infinitely far away’. Somewhere that cannot be reached no matter how far a distance you try to travel.

One such use of infinity similar to how we described it above, is in a ‘destination’ type format. When we want to describe a number $n$ that is growing larger and larger without end, we say ‘as $n$ tends to infinity’ (using notation $n\to \infty$)  and NEVER when $n = \infty$ for the reasons mentioned above.

Because of this play on words, many come to think of infinity as something which cannot be used or understood. The common belief is that infinity itself is not even a well defined concept in the first place. While this is subject to some discussion, I shall end this post on a neat little note.

If I guarantee to you that after every step I take I will surely take another step, and then I take a step, won’t I end up taking an infinite number of steps?

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