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Chapter 3 Numbers

In this chapter, we will work our way up to understanding the real line \(\RR\text{.}\)

To get there, we will start with our friend the natural numbers \(\NN_0\text{.}\) We notice that there are certain equations that we can't answer using only the natural numbers, such as \(x+1=0\text{.}\) In other words, we can't subtract in the world of the natural numbers. This inspires us to introduce the ring of integers \(\ZZ\text{.}\)

In \(\ZZ\text{,}\) you can add, subtract, and multiply, but you cannot divide. That means, again, that there are equations we can't solve, such as \(2x=1\text{.}\) This inspires us to expand our concept of number yet further, to include the field of rational numbers \(\QQ\text{.}\) There, we can add, subtract, multiply, and divide.

There's another structure we have on the rationals, which is an ordering — we can always say which of two different numbers is larger, and that idea of largeness interacts with the four operations in pleasant ways. But this raises certain problems: there seem to be numbers that ought to exist, but don't. These numbers can be arbitrarily well approximated by rational numbers, but they aren't themselves rational numbers.

When we include these numbers, we arrive at the unique complete ordered field — the real line \(\RR\text{.}\)