2.1 Find Square Roots and Compare Real Numbers You will find square roots and compare real numbers. Essential Questions How do you evaluate a square root and compare real numbers?

Evaluate the expression. EXAMPLE 1 Find square roots Evaluate the expression. a.  – + 36 = 6 The positive and negative square are 6 and – 6. roots of 36 b.  49 = 7 The positive square root of 49 is 7. The negative square root of 4 is – 2. c.  4 – = 2

Evaluate the expression. EXAMPLE 1 GUIDED PRACTICE Find square roots for Example 1 Evaluate the expression.  – 1. 9 – = 3 2. 25  = 5  64 3. – + = 8 – +  – 4. 81 = 9 –

Approximate a square root EXAMPLE 2 Approximate a square root FURNITURE The top of a folding table is a square whose area is 945 square inches. Approximate the side length of the tabletop to the nearest inch. SOLUTION 2 = You need to find the side length s of the tabletop such that s 945. This means that s is the positive square root of 945. You can use a table to determine whether 945 is a perfect square.

Approximate a square root Number 28 29 30 31 32 Square of number 784 EXAMPLE 2 Approximate a square root Number 28 29 30 31 32 Square of number 784 841 900 961 1024 As shown in the table, 945 is not a perfect square. The greatest perfect square less than 945 is 900. The least perfect square greater than 945 is 961. 900 < 945 < 961 Write a compound inequality that compares 945 with both 900 and 961. <  961 900 945 Take positive square root of each number. 30  945 < 31 Find square root of each perfect square.

EXAMPLE 2 Approximate a square root The average of 30 and 31 is 30.5 and (30.5)2 = 930.25. Because 945 > 930.25, is closer to 31 than 30. ANSWER The side length of the tabletop is about 31 inches.

Approximate the square root to the nearest integer. EXAMPLE 2 GUIDED PRACTICE Approximate a square root for Example 2 Approximate the square root to the nearest integer. 5.  32 6 6.  103 10 7.  48 – – 7 8.  350 – – 19

EXAMPLE 3 Classify numbers Tell whether each of the following numbers is a real number, a rational number, an irrational number, an integer, or a whole number: , , – .  24 81 100 No Yes Real Number? Whole Number? Integer? Irrational Number? Rational Number? Number  24 100 –  81

EXAMPLE 4 Graph and order real numbers , Order the numbers from least to greatest: 4 3 –  5 13 –2.5 9 . SOLUTION Begin by graphing the numbers on a number line. 4 3 ANSWER Read the numbers from left to right: –2.5, –  5 9 13 . ,

EXAMPLE 4 GUIDED PRACTICE Graph and order real numbers for Examples 3 and 4 Tell whether each of the following numbers is a real number, a rational number, an irrational number, an integer, or a whole number: , 5.2, 0, , 4.1, Then order the numbers from least to greatest. 9.  7 9 2 – –  20 9 2 – ,  7 20 4.1 5.2. ANSWER

Graph and order real numbers for Examples 3 and 4 GUIDED PRACTICE Graph and order real numbers for Examples 3 and 4 No Yes Real Number? Whole Number? Integer? Irrational Number? Rational Number? Number  20 7 9 2 4.1 – 5.2

EXAMPLE 5 Rewrite a conditional statement in if-then form Rewrite the given conditional statement in if-then form. Then tell whether the statement is true or false. If it is false, give a counterexample. SOLUTION a. Given: No integers are irrational numbers. If-then form: If a number is an integer, then it is not an irrational number. The statement is true.

EXAMPLE 5 Rewrite a conditional statement in if-then form b. Given: All real numbers are rational numbers. If-then form: If a number is a real number, then it is a rational number. The statement is false. For example, is a real number but not a rational number.  2

EXAMPLE 5 GUIDED PRACTICE Rewrite a conditional statement in if-then form for Example 5 Rewrite the given conditional statement in if-then form. Then tell whether the statement is true or false. If it is false, give a counterexample. All square roots of perfect squares are rational numbers. 10. If-then form: If a number is the square root of perfect square, then it is a rational number. The statement is true.

EXAMPLE 5 GUIDED PRACTICE Rewrite a conditional statement in if-then form for Example 5 Rewrite the given conditional statement in if-then form. Then tell whether the statement is true or false. If it is false, give a counterexample. All repeating decimals are irrational numbers. 11. If-then form: If a number is a repeating decimal, then it is an irrational number. The statement is false. For example, 0.333… is a repeating decimal and can be written as , so it is a rational number. 1 3

EXAMPLE 5 GUIDED PRACTICE Rewrite a conditional statement in if-then form for Example 5 Rewrite the given conditional statement in if-then form. Then tell whether the statement is true or false. If it is false, give a counterexample. No integers are irrational numbers. 12. If-then form: If a number is an integer, then it is not an irrational number. The statement is true.

Evaluate the expression. Daily Homework Quiz Evaluate the expression. 1. – +  289 ANSWER – + 17 2.  36 – ANSWER 3 16 Approximate the square root to the nearest integer. 3.  21 –

Daily Homework Quiz ANSWER – 5 4.  620 ANSWER 25 A square courtyard has an area of 272 square feet. What is the side length of the courtyard to the nearest foot? 5. ANSWER 16 ft

You will find square roots and compare real numbers. Essential Questions How do you evaluate a square root and compare real numbers? All positive numbers have a positive and a negative square root. Square roots of positive integers or rational numbers that are not perfect squares are irrational numbers that can be approximated by nonrepeating decimals. To evaluate the square root of a, you need to find the number b such that b2 = a. To compare real numbers, you can graph the numbers on a number line, suing approximations for any square roots that are irrational numbers.