5.4 The Irrational Numbers 251 nonrepeating decimal number such as 5.12639537… can be used to indicate an irrational number. Notice that no number or set of numbers repeat on a continuous basis, and the three dots at the end of the number indicate that the number continues indefinitely. Nonrepeating number patterns can be used to indicate irrational numbers. For example, 6.1011011101111… and 0.525225222… are both irrational numbers. The expression 2 is read “the square root of 2” or “radical 2.” The symbol is called the radical sign , and the number or expression inside the radical sign is called the radicand . In 2, 2 is the radicand. The square roots of some numbers are rational, whereas the square roots of other numbers are irrational. The principal (or positive ) square root of a number n, written n, is the positive number that, when multiplied by itself, gives n. Whenever we mention the term “square root” in this text, we mean the principal square root. For example, = ⋅ = = ⋅ = 9 3, since 33 9 36 6, since 6 6 36 Both 9 and 36 are examples of numbers that are rational numbers because 3 and 6, respectively, are terminating decimal numbers. We return to the problem faced by Pythagoras: If c 2, 2 = then it can be shown that c has a value of 2, but what is 2 equal to? The 2 is an irrational number, and it cannot be expressed as a terminating or repeating decimal number. It can only be approximated by a decimal number: 2 is approximately 1.4142136 (to seven decimal places). Later in this section, we will discuss using a calculator to approximate irrational numbers. Other irrational numbers include 3, 5, and 37. Another important irrational number used to represent the ratio of a circle’s circumference to its diameter is pi, symbolized .π Pi is approximately 3.1415927. We have discussed procedures for performing the arithmetic operations of addition, subtraction, multiplication, and division with rational numbers. We can perform the same operations with the irrational numbers. Before we can proceed, however, we must understand the numbers called perfect squares. Any number that is the square of a natural number is said to be a perfect square . Some perfect squares are shown in the following chart. Profile in Mathematics Pythagoras of Samos Pythagoras of Samos founded a philosophical and religious school in the sixth century B.C. The school was located in the Greek city of Kroton, which is in modern day Italy. The scholars at the school, known as Pythagoreans, produced important works of mathematics, astronomy, and the theory of music. Although the Pythagoreans are credited with proving the Pythagorean theorem, it was known to the ancient Babylonians 1000 years earlier. The Pythagoreans were a secret society that formed a model for many secret societies in existence today. One practice was that students were to spend their first three years of study in silence, while their master, Pythagoras, spoke to them from behind a curtain. Among other philosophical beliefs of the Pythagoreans was “that at its deepest level, reality is mathematical in nature.” Natural numbers 1, 2, 3, 4, 5, 6,… Squares of the natural numbers 1 ,2 2 ,2 3 ,2 4 ,2 5 ,2 6 ,2 … Perfect squares 1, 4, 9, 16, 25, 36,… The numbers 1, 4, 9, 16, 25, and 36 are some of the perfect square numbers. The square root of a perfect square number will be a natural number. For example, 1 1, 4 2, 9 3, 16 4, 25 5, = = = = = and so on. The number that multiplies a radical is called the radical’s coefficient . For example, in 3 5, the 3 is the coefficient of the radical. Simplify Radicals Some irrational numbers can be simplified by determining whether there are any perfect square factors in the radicand. If there are, the following rule can be used to simplify the radical. Simplifying Radicals Product Rule for Radicals a b a b a b , 0, 0 ⋅ = ⋅ ≥ ≥ Lanmas/Alamy Stock Photo
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