Figure 1-1. Real line
Natural numbers, N+, are the counting numbers that begin with 1 and with each successive number greater than its predecessor by 1.
N+ = {1, 2, 3, ...}
Negative numbers, N-, are the negatives of the natural numbers N+.
N- = {..., -3, -2, -1}
Integers, Z, are a combination of the natural numbers, the negative numbers, and the number 0.
Z = {N-, 0, N+}
Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}
Rational numbers, Q, are the numbers that can be represented as quotients of two integers a and b where b is not equal to zero.
Q = {a/b : a and b integers, and b ≠ 0}For example,
6/2, 3/4, 0.95, 1.2, 1.333..., 0.363636...The underscore indicates that these digits repeat. Integers can be defined as rational numbers; so, Z is a subset of Q.
Irrational numbers, H, are the numbers that are not definable as rational numbers. For example,
π = 3.14159..., √2 = 1.41421..., e = 2.71828...
From these definitions, the sets Q and H are mutually exclusive; that is, a number can be a member of one set but not the other.
Real numbers, R, are the numbers that are either rational or irrational; that is, R is the union of the sets Q and H. Real numbers can be represented graphically as points on a line. For each distinct real number, there is exactly one point on the line.
Complex numbers, C, are numbers with two components, a real number and an imaginary number. The set R of real numbers is a subset of C. Complex numbers are usually expressed as: a + bi, where a and b are real numbers and i2 = -1. The symbol, i, represents the square root of negative one which is imaginary. Complex numbers can be represented as ordered pairs. For example,
3 + 2i = {3, 2}Complex numbers may also be represented as points on an x-y plane called an Argand diagram (Jean Robert Argand, 1768 - 1822). An Argand diagram (also known as the Argand plane) is a plot of complex numbers as points in the complex plane using the x-axis as the real axis and y-axis as the imaginary axis.
-π + 2i = {-π, 2}
-2.2 - 3.3i = {-2.2, -3.3}
2 - (5/2)i = {2, -5/2}
