## What does “close to” mean?

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

$0.333333333\dots$

with an infinite number of 3′s after the decimal point. Now, you probably know that this represents $1/3$. 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 $0.333333333\dots$ as a shorthand for the limit of the sequence

$0.3, 0.33, 0.333, 0.3333, \dots$

That is, the sequence of rational numbers $0.3$, $0.33$, and so on, get infinitely close to some number, namely, $1/3$, 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 $0.3$, $0.33$, 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 $x$ and $y$ is defined to be $|x - y|$, where $|a|$ 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 $|a|_{10}$. Using this size function will give us a different meaning of “close to”: two numbers $x$ and $y$ will be “close to” each other when $|x - y|_{10}$ is small.