Integral Exponentiation
As multiplication is related to the concept of "repeated addition", exponentiation involving integers is akin to "repeated multiplication". The use of positive integers as exponents is illustrated as follows:
Algebra
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Arithmetic Axioms
The arithmetic operations with real numbers are governed by the following axioms:
(1) Closure Axiom of Addition / Multiplication
For real numbers a and b,
Multiplication and Division
The multiplication of two real numbers is similar to the concept of "repeated addition". This is best illustrated with the multiplication of two positive integers x and y:
Properties of Real Numbers
All real numbers have the following properties:
(1) Reflexive Property For any real number a, a = a. Example: 3 = 3, y = y, x + z = x + z (x, y and z are real numbers)
(2) Symmetric Property For any real numbers a and b, if a = b, then b = a. Example: If 3 = 1 + 2, then 1 + 2 = 3
Real Number System
The real number system is comprised of the set of real numbers and the arithmetic operations of addition and multiplication (subtraction, division and other operations are derived from these two). The rules and relationships that govern the real number system are the basis for most algebraic manipulations.
Natural Numbers
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, ......}
Whole Numbers
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, .......}
Integers
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, ......}
Rational and Irrational Numbers
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}
Discrete Algebra
Discrete Algebra
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