1.1.5 Square Roots and Real Numbers

Key Objectives • Find square roots of perfect squares and of fractions. • Classify real numbers. • Approximate square roots. Key Terms • The square root of a number is one of the two equal factors of the number. • A real number is a rational or irrational number. Every point on the number line represents a real number. • An irrational number is a real number that cannot be written as a ratio of integers. • A rational number is a number that can be written as a fraction with integers for its numerator and denominator (denominator cannot be 0). • A terminating decimal is a rational number in decimal form with a finite number of decimal places. • A repeating decimal is a rational number in decimal form that has a block of one or more digits that repeat continuously. • Integers are the set of all whole numbers and their opposites.

• Whole numbers are the set of natural numbers and 0. • Natural numbers are the set of counting numbers. Example 1 Finding Square Roots of Perfect Squares

The product of a whole number and itself is called a perfect square. For example, 25 is a perfect square because 5 · 5 = 25, and 49 is a perfect square because 7 · 7 = 49. The first 11 perfect squares are 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100. The square root of a perfect square is the number that when multiplied by itself is equal to that perfect square. For example, the square root of 49 is 7 because 7 · 7 = 49. However, note that − 7 · − 7 = 49 as well. Therefore, there are two square roots of 49, 7 and − 7. In fact, every positive real number has two square roots that are opposites of each other. The symbol is used to represent the square root of a number. The nonnegative square root of a real number n is represented by n . The negative square root of a real number n is represented by n − .

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