Number Systems

Abstract

We start with some notation and symbols. We denote the naturals by N={1,2,3,…}, the integers by Z={…,−2,−1,0,1,2,…}, the rationals (represented by fractions of integers) by Q, and the reals (which may be described by their decimal representation, see Chap. 8) by R. The symbol ∈ means ‘belongs to’. For instance, 1∈N means ‘1 is a natural’. The notation

$$\bigl\{x\in \mathbf{R}:x^2=2\bigr\} $$

designates the set of reals whose square is 2. The set

$$\{x\in \mathbf{R}:-1<x\leq2\} $$

is the set of real numbers strictly larger than −1 and smaller than or equal to 2. A shorter notation for such a set is (−1,2]. The set

$$\bigl\{x^2:x\in[-1,1]\bigr\} $$

is the set of squares of numbers in [−1,1]. The notation

$$\{-1,2,4\} $$

designates a set with three elements: −1, 2, and 4. The empty set will be denoted by ∅.