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.

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