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 number as , , , , , , , and .

The basic algebraic properties of the real numbers can be expressed in terms of the two fundamental operations of addition and multiplication.

__Basic Algebraic Properties__:

Let and denotes real numbers.

**(1) The Commutative Properties**

(a) (b)

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

**(2) The Associative Properties**

(a) (b)

The associative properties tells 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**

(a) (b)

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**

(a) (b)

We call the

**additive identity**and the

**multiplicative identity**for the 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 .

__Example__:

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

(a)

(b)

(c)

(d)

(e)

(f)

(g) If , then

__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 the 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:**

(a)

(b) If , then or (or both)

**(8) Properties of Negation:**

(a)

(b)

(c)

(d)

__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