DefinitionA.1.1
We will take several familiar number classes as objects which we assume we already understand. These are:
- natural numbers, N
- the set of positive integers (also known as whole numbers or counting numbers), which we enumerate as 1,2,3, … We will not define these at any point during the course.
- integers, Z
- the set of integers, including those positive, negative and zero, which we enumerate as … −2,−1,0,1,2, … We will not formally define these at any point in the course, although we will discuss how they could be defined in terms of the set N.
- rational numbers, Q
- the set of rational numbers, i.e., those numbers which may be written as a fraction of two integers of which the denominator is nonzero. After covering equivalence relations, we will formally define Q in terms of Z.
- algebraic numbers, ¯Q
- the set of algebraic numbers, i.e., those numbers which are roots of a polynomial with integer coefficients. For example, √2 is algebraic because it is a root of the polynomial x2−2. Similarly, the imaginary unit i is algebraic because it is a root of x2+1. Although it is not easy to prove, π is not an algebraic number. While you may not be familiar with algebraic numbers, they are in many ways a more natural class to work with than the real numbers.
- real numbers, R
- the set of real numbers, i.e., those numbers which correspond to all possible places on a number line, or alternatively, all numbers represented by infinite decimal expansions. These include, for example, √2,π,e and others, but not any numbers involving the imaginary unit i. If time allows and there is sufficient interest, we will cover the construction of the real numbers at the end of the semester.
- complex numbers, C
- the set of complex numbers, i.e., all those numbers of the form a+bi where a,b∈R and i is the imaginary unit satisfying the relation i2=−1. These can also be defined as those numbers which are roots of a polynomial with real coefficients (in fact, we can even restrict the polynomial to have degree two or less). This set includes all previous classes mentioned.