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Section 1.7 Properties of Real Numbers 71 �� ������������������������ Properties of Real Numbers ▶ OBJECTIVES As a result of completing this section, you will be able to 1. Apply the identity and inverse properties. 2. Apply the commutative and associative properties. 3. Apply the distributive property. The set of real numbers contains some interesting properties. We use some of the properties without even thinking about them. These properties are very important to the study of algebra, so we will state them explicitly in this section. The Identity and Inverse Properties The first property that we will discuss is the identity property. An identity element is a number which leaves another number unchanged when an operation is performed on it. • When we add numbers, the only number that can be added to another number without changing its value is zero. • When we multiply numbers, the only number that can be multiplied to another number without changing its value is one. Identity Property Identity Element Example (a = 6) Addition For all real numbers a, a + 0 = 0 + a = a Zero is the additive identity. 6 + 0 = 0 + 6 = 6 Multiplication For all real numbers a, a · 1 = 1 · a = a One is the multiplicative identity. 6 · 1 = 1 · 6 = 6 A related property is the inverse property. An inverse is a number which produces the identity element when an operation is performed on it. • The additive inverse of a number is its opposite since adding a number and its opposite results in 0. • The multiplicative inverse of a number is its reciprocal since multiplying a number and its reciprocal results in 1. Inverse Property Inverse Element Example (a = 5) Addition For all real numbers a, a + (-a) = (-a) + a = 0 -a is the additive inverse (or opposite) of a. The opposite of 5 is -5. 5 + (-5) = (-5) + 5 = 0 Multiplication For all real numbers a ≠ 0, a · 1 a = 1 a · a = 1 1 a is the multiplicative inverse (or reciprocal) of a, a ≠ 0. The reciprocal of 5 is 1 5 . 5a1 5 b = a1 5 b(5) = 1 �� ������������������������ ������������������ Find both the additive and multiplicative inverses of each number. Assume any variables are nonzero. Problems Solutions Additive Inverse, -a Multiplicative Inverse, 1 a 1a. -6 -(-6) = 6 1 -6 =- 1 6 1b. 3 4 -a3 4 b =- 3 4 1 3 4 = 1 · 4 3 = 4 3 1c. 2x -(2x) = -2x 1 2x = 1 2x 1d. -3y -(-3y) = 3y 1 -3y =- 1 3y 1e. x 7 -a x 7 b =- x 7 1 x 7 = 1 · 7 x = 7 x Objective 1 ▶ Apply the identity and inverse properties.


hendricks_beginning_algebra_1e_ch1_3
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