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.