YOUR MONDAY MATH with KP Hart: How Far is Infinity?

Here’s Mathematician KP Hart’s Math Question and Answer for Monday, January 18th!

How far is infinity?

One would say: infinitely far. But how far is that? It must stop once must it not?

For mathematicians the answer is quite simple: ∞ lies beyond all real numbers of, equivalently, all natural numbers. We have a mental picture of the number line: it looks a bit like the open interval (-1,1) and the, imaginary, points that you would want to associate with -1 and 1 we usually denote -∞ and ∞.

This picture may put you on the wrong foot because it makes it tempting to treat ∞ as a number and substitute it into expressions, for example when determining limits of sequences. The official definition of “the limit of the sequence x_n, for n to infinity, is equal to L” is not “substitute ∞ for n and see what happens” but:

“for every ε>0 there is a natural number N such that for n>N we have |x_n-L|<ε”This definition was formulated after a few centuries of working with sequences, functions etc. and it is basically the only one that one can work with in a mathematically unambiguous way. I wrote this definition down for two reasons. The first reason is to show what such a defintion actually looks like; these are not formulated for easy understanding but before anything else so as not to be misunderstood.

The second reason is connected to the subject of the question: ∞ does not play a role in the definition and this shows that in Mathematical Analysis we do not really need ∞: behind every compact formulation with ∞ there is an unambiguous and workable formulation without ∞.

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Read all of KP Harts math questions here!

About Dutch Mathematician KP Hart: In the beginning of this year the Dutch government opened a website, The Dutch Science Agenda, where everyone could post questions that they thought were of scientific interest. This was an attempt to involve the whole country in determining what the Dutch science agenda should be in the coming years.

I looked through the questions and searched for terms like `mathematics’, `infinity’ … to see what mathematical questions there were and I noticed various questions that already have answers (and have had for a long time). On a whim I decided to post answers to those questions, in Dutch. For your edification I will translate these posts into English.

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