Section 1.3 Properties of Real Numbers and Simplifying Algebraic Expressions 33 SECTION 1.3 Properties of Real Numbers and Simplifying Algebraic Expressions According to the U.S. Census Bureau statistics, people with a bachelor’s degree earn nearly twice as much as those with only a high school diploma. If x represents the earnings of a high school graduate, write an expression that represents the earnings of a person with a bachelor’s degree. To answer this question, we need to know how to translate phrases into algebraic expressions. We will learn that and more in this section. The Identity and Inverse Properties The set of real numbers contains some very interesting properties. The first one 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. Zero is called the additive identity. • When we multiply numbers, the only number by which another number can be multiplied without changing its value is one. One is called the multiplicative identity. Identity Properties Identity Elements Examples (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 by its reciprocal is 1. Inverse Properties Inverse Elements Examples (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 and 5 + (-5) = (-5) + 5 = 0 Multiplication For all real numbers a 2 0, a · 1 a = 1 a · a = 1 1 a is the multiplicative inverse (or reciprocal) of a, a 2 0 The reciprocal of 5 is 1 and 5 5 · 1 5 = 1 5 · 5 = 1 ▶ OBJECTIVES As a result of completing this section, you will be able to 1. Apply the identity and inverse properties. 2. Apply the commutative, associative, and distributive properties. 3. Simplify algebraic expressions. 4. Translate phrases or statements into algebraic expressions, equations, or inequalities. 5. Translate phrases or statements related to applications. 6. Troubleshoot common errors. Objective 1 ▶ Apply the identity and inverse properties.
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