The numbers used to measure real-world quantities such as length, area, volume, speed, electrical charges, probability of rain, room temperature, gross national products, growth rates, and so forth are called real numbers. They include such numbers as , , , , , , , and .
The basic algebraic properties of real numbers can be expressed in terms of the two fundamental operations of addition and multiplication.
Basic Algebraic Properties
Let and denote real numbers.
(1) The Commutative Properties
The commutative properties say that the order in which we either add or multiply real number doesn’t matter.
(2) The Associative Properties
The associative properties tell us that the way real numbers are grouped when they are either added or multiplied doesn’t matter. Because of the associative properties, expressions such as and makes sense without parentheses.
(3) The Distributive Properties
The distributive properties can be used to expand a product into a sum, such as , or the other way around to rewrite a sum as product:
(4) The Identity Properties
We call the additive identity and the multiplicative identity for real numbers.
(5) The Inverse Properties
(a) For each real number , there is real number called the additive inverse of , such that
(b) For each real number , there is a real number called the multiplicative inverse of , such that
Although the additive inverse of , namely , is usually called the negative of , you must be careful because isn’t necessarily a negative number. For instance, if , then . Notice that the multiplicative inverse is assumed to exist if . The real number is also called the reciprocal of and is often written as .
State one basic algebraic property of real numbers to justify each statement:
(g) If , then
(a) Commutative Property for addition
(b) Associative Property for addition
(c) Commutative Property for multiplication
(d) Distributive Property
(e) Additive Inverse Property
(f) Multiplicative Identity Property
(g) Multiplicative Inverse Property
Many of the important properties of real numbers can be derived as results of the basic properties, although we shall not do so here. Among the more important derived properties are the following.
(6) The Cancellation Properties
(a) If then,
(b) If and , then
(7) The Zero-Factor Properties
(b) If , then or (or both)
(8) Properties of Negation
Subtraction and Division
Let and be real numbers,
(a) The difference is defined by
(b) The quotient or ratio or is defined only if . If , then by definition
It may be noted that division by zero is not allowed.
When is written in the form , it is called a fraction with numerator and denominator . Although the denominator can’t be zero, there’s nothing wrong with having a zero in the numerator. In fact, if ,
(9) The Negative of a Fraction
If , then