What does “close to” mean?

Continuing from last time, consider the (normal, decimal) number

with an infinite number of 3′s after the decimal point. Now, you probably know that this represents . But why? How do we define what such an infinite sequence of digits means?

The standard answer is that we think of the infinite decimal number as a shorthand for the limit of the sequence

That is, the sequence of rational numbers , , and so on, get infinitely close to some number, namely, , which is taken as the meaning of the sequence. (I am waving my hands a bit here; this is usually made more precise through the notion of a Cauchy sequence. But the intuition is the same.)

Now, in the previous paragraph I said that the numbers , , get infinitely close to some number. What do we mean by “close to”? You may think this a silly, obvious question. But it turns out that interesting things happen if we give a different answer than usual.

First, let’s think about what “close to” means in the context of the usual real numbers. The distance between two numbers and is defined to be , where denotes the usual absolute value of a number. We can think of the absolute value function as assigning a size to each number: 42 and -42 both have the same size, namely, 42. So the distance between two numbers is the size of their difference.

The name of the game now will be to define a different size function, which we will write . Using this size function will give us a different meaning of “close to”: two numbers and will be “close to” each other when is small.