1.1 Some notation for number Systems

Here’s some notation for the number systems we will use:

• \(\mathbb {N} = \{ 0, 1, 2, \ldots \} \) is the set of natural numbers.

• \(\mathbb {Z} = \{ \cdots , -3, -2, -1, 0, 1, 2, 3, \cdots \} \) is the set of integers.

• \(\mathbb {R}\) is the set of real numbers.

• \(\mathbb {Q} = \{ a/b\in \mathbb {R}: a,b\in \mathbb {Z}, b\neq 0\} \) is the set of rational numbers.

• \(\mathbb {C} = \{ a + bi: a, b\in \mathbb {R}\} \) is the set of complex numbers. I’ll say more about \(\mathbb {C}\) below.

We have \(\mathbb {N}\subset \mathbb {Z}\subset \mathbb {Q}\subset \mathbb {R}\subset \mathbb {C}\).

If \(a\in \mathbb {R}\), then I set \(\mathbb {Z}_{{\gt}a} := \{ n\in \mathbb {Z}: n {\gt} a\} \). So, for example, \(\mathbb {Z}_{{\gt}0} = \{ 1, 2, 3,\ldots \} \). Sometimes I also write \(\mathbb {P} = \mathbb {Z}_{{\gt}0}\). The letter “P" is for positive.