250 CHAPTER 5 Number Theory and the Real Number System A football field is 160 ft wide and 360 ft long. A football player catches the football from a kickoff in the northwest corner of one end zone and runs across the field in a straight line to the southeast corner of the other end zone, as shown by the dashed line in the diagram. 360 ft 160 ft How far did the football player run? The answer to this question is an example of an irrational number. In this section, we will introduce irrational numbers along with the operations of addition, subtraction, multiplication, and division of irrational numbers. The answer to the above question is 40 97 feet, or about 394 feet. The Irrational Numbers SECTION 5.4 LEARNING GOALS Upon completion of this section, you will be able to: 7 Simplify radicals. 7 Add and subtract radicals. 7 Multiply radicals. 7 Divide radicals. 7 Rationalize the denominator. 7 Estimate square roots without a calculator. Why This Is Important Irrational numbers play an important role in many areas of mathematics, science, business, architecture, banking, and other fields. Pythagoras (ca. 585–500 b.c.), a Greek mathematician, is credited with providing a written proof that in any right triangle (a triangle with a 90° angle; see Fig. 5.7), the square of the length of one side a( )2 added to the square of the length of the other side b( )2 equals the square of the length of the hypotenuse c( ). 2 The formula a b c 2 2 2 + = is now known as the Pythagorean theorem. The Pythagorean theorem is discussed in more detail in Section 8.3. Pythagoras discovered that when using the formula, with a 1 = and b 1, = the value of c was not a rational number. a b c c c c 1 1 1 1 2 2 2 2 2 2 2 2 2 + = + = + = = There is no rational number that when squared will equal 2. This fact prompted a need for a new set of numbers, the irrational numbers. In Section 5.2, we introduced the real number line. The points on the number line that are not rational numbers are referred to as irrational numbers. Recall that every rational number can be expressed as a ratio of two integers. This leads us to the definition of irrational number. Hypotenuse (longest side of a right triangle) Legs a c b a2 1 b2 5 c2 Figure 5.7 Definition: Irrational Number An irrational number is a real number that cannot be written as the ratio of two integers. Also recall that every rational number can be written as either a terminating or a repeating decimal number. Therefore, an irrational number has a decimal representation that is a nonterminating, nonrepeating decimal number. For example, a JoeSAPhotos/Shutterstock
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