# Basic Algebraic Properties of Real Numbers

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 $10$, $- 17$, $\frac{{17}}{{14}}$, $0$, $2.71828$, $\sqrt 2$, $- \frac{{\sqrt 2 }}{2}$, $3 \times {10^8}$ and $\pi$.

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 $a,b$ and $c$ denote real numbers.

(1) The Commutative Properties

(a) $a + b = b + a$
(b) $a \cdot b = b \cdot a$

The commutative properties say that the order in which we either add or multiply real number doesn’t matter.

(2) The Associative Properties

(a) $(a + b) + c = a + (b + c)$

(b) $(a \cdot b) \cdot c = a \cdot (b \cdot c)$

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 $a + b + c$ and $a \cdot b \cdot c$ makes sense without parentheses.

(3) The Distributive Properties

(a) $a \cdot (b + c) = a \cdot b + a \cdot c$

(b) $(b + c) \cdot a = b \cdot a + c \cdot a$

The distributive properties can be used to expand a product into a sum, such as    $a(b + c + d) = ab + ac + ad$, or the other way around to rewrite a sum as product:  $ax + bx + cx + dx + ex = (a + b + c + d + e)x$

(4) The Identity Properties

(a) $a + 0 = 0 + a = a$

(b) $a \cdot 1 = 1 \cdot a = a$

We call $0$ the additive identity and $1$ the multiplicative identity for real numbers.

(5) The Inverse Properties

(a) For each real number $a$, there is real number $- a$ called the additive inverse of $a$, such that $a + ( - a) = ( - a) + a = 0$

(b) For each real number $a \ne 0$, there is a real number $\frac{1}{a}$ called the multiplicative inverse of $a$, such that $a \cdot \frac{1}{a} = \frac{1}{a} \cdot a = 1$

Although the additive inverse of $a$, namely $- a$, is usually called the negative of $a$, you must be careful because $- a$ isn’t necessarily a negative number. For instance, if $a = - 2$, then $- a = - ( - 2) = 2$. Notice that the multiplicative inverse $\frac{1}{a}$ is assumed to exist if $a \ne 0$. The real number $\frac{1}{a}$ is also called the reciprocal of $a$ and is often written as ${a^{ - 1}}$.

Example:
State one basic algebraic property of real numbers to justify each statement:

(a) $7 + ( - 2) = ( - 2) + 7$
(b) $x + (3 + y) = (x + 3) + y$
(c) $a + (b + c)d = a + d(b + c)$
(d) $x[y + (z + w)] = xy + x(z + w)$
(e) $(x + y) + [ - (x + y)] = 0$
(f) $(x + y) \cdot 1 = x + y$
(g) If $x + y \ne 0$, then $(x + y)[\frac{1}{{(x + y)}}] = 1$

Solution:
(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 $a + x = a + y$ then, $x = y$
(b) If $a \ne 0$ and $ax = ay$, then $x = y$

(7) The Zero-Factor Properties

(a) $a \cdot 0 = 0 \cdot a = 0$
(b) If $a \cdot b = 0$, then $a = 0$ or $b = 0$ (or both)

(8) Properties of Negation

(a) $- ( - a) = a$
(b) $( - a)b = a( - b) = - (ab)$
(c) $( - a)( - b) = ab$
(d) $- (a + b) = ( - a) + ( - b)$

Subtraction and Division

Let $a$ and $b$ be real numbers,

(a) The difference $a - b$ is defined by $a - b = a + ( - b)$

(b) The quotient or ratio $a \div b$ or $\frac{a}{b}$ is defined only if $b \ne 0$. If $b \ne 0$, then by definition $\frac{a}{b} = a \cdot \frac{1}{b}$
It may be noted that division by zero is not allowed.

When $a \div b$ is written in the form $a/b$, it is called a fraction with numerator $a$ and denominator $b$. Although the denominator can’t be zero, there’s nothing wrong with having a zero in the numerator. In fact, if $b \ne 0$, $\frac{0}{b} = 0 \cdot \frac{1}{b} = 0$

(9) The Negative of a Fraction
If $b \ne 0$, then $\frac{{ - a}}{b} = \frac{a}{{ - b}} = - \frac{a}{b}$