Complex Numbers

Complex numbers are the numbers with the format a + b i, where a and b are real numbers and
i ² = - 1. If we denote the set of complex numbers by C, then

C = { a + b i , where a and b are real numbers, i² = -1 }

If in the number x = a + b i , b is set to zero, then x = a, where a is a real number. Thus, all real numbers are complex numbers, i.e., the set of complex numbers includes the set of real numbers.

Integers are the numbers that are in either (1) the set of whole numbers, or (2) the set of numbers that contain the negatives of the natural numbers. If the set of integers is denoted by I, then

I = {......, -3, -2, -1, 0, 1, 2, 3, ......}

Positive integers are the numbers in I greater than 0. Negative numbers are the numbers in I less than 0.
The number zero is neither positive nor negative, i.e., it is both nonpositive and nonnegative.

Given the above definitions, the following statements about integers can be made:

(1) N is the set of positive integers.
(2) W is the union of N and the number 0.
(3) The set of numbers that contain the negatives of the numbers in N is
the set of negative integers.
(4) I is the union of W and the set of negative integers.

Natural numbers, also known as counting numbers, are the numbers beginning with 1, with each successive number greater than its predecessor by 1. If the set of natural numbers is denoted by N , then

N = { 1, 2, 3, ......}

Rational numbers are the numbers that can be represented as the quotient of two integers p and q, where q is not equal to zero. If the set of rational numbers is denoted by Q , then

Q = { all x, where x = p / q , p and q are integers, q is not zero}

Rational numbers can be represented as:

(1) Integers: (4 / 2) = 2, (12 / 4) = 3
(2) Fractions: 3 / 4, 13 / 3
(3) Terminating Decimals: (3 / 4) = 0.75, (6 / 5) = 1.2
(4) Repeating Decimals: (13 / 3) = 4.333....., (4 / 11) = .363636......

Conversely, irrational numbers are the numbers that cannot be represented as the quotient of two integers, i.e., irrational numbers cannot be rational numbers and vice-versa. If the set of irrational numbers is denoted by H, then

H = { all x, where there exists no integers p and q such that x = p / q, q is
not zero }

Typical examples of irrational numbers are the numbers p and e, as well as the principal roots of rational numbers. They can be expressed as non-repeating decimals, i.e., the numbers after the decimal point do not repeat their pattern.

Real Numbers

Real numbers are the numbers that are either rational or irrational, i.e., the set of real numbers is the union of the sets Q and H. If the set of real numbers is denoted by R , then

R = Q È H

Since Q and H are mutually exclusive sets, any member of R is also a member of only one of the sets Q and H. Therefore, a real number is either rational or irrational (but not both). If a real number is rational, it can be expressed as an integer, as the quotient of two integers, and it can be represented by a terminating or repeating decimal; otherwise, it is irrational and cannot be represented in the above formats.

Whole numbers are the numbers beginning with 0, with each successive number greater than its predecessor by 1. It combines the set of natural numbers and the number 0. If the set of whole numbers is denoted by N, then

N= { 0, 1, 2, 3, .......}