2.1 Find Square Roots and Compare Real Numbers

Essential Question: How do you find the square roots of numbers and compare real numbers?

Warm-up: Find the square root of the number.1. 14 2. 16

Complete the statements using <, >, or =.

25

21- ___

6

58

19 ___ 5.2

3.

4.

5. A square room has an area of 625 ft. What is the length of one of it’s sides?

Common Core9.12.N.Q.1Use units asa way to understand problems and to guide the solution to mult-step problems; choose and interpret units consistently in formulas.

9-8-14

Square Root of a Number

• Words: If , then b is a square root of a.

• Example:

ab 2

9. of roots square

are 3- and 3 so ,9(-3) and 93 22

Radical Symbol• All positive real numbers have two square

roots, a positive square root and a negative square root.

• A square root is written with the radical symbol .

• The number or expression inside the radical symbol is the radicand.

• Zero has only one square root, 0.• Negative real numbers do not have real

square roots because the square of every real number is either positive or zero.

a

Example 1 Find square roots

• Evaluate the expression.

a.

b.

c.

4

49

36

Find square roots

EXAMPLE 1

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.

c. 4– = 2– The negative square root of 4 is – 2.

Evaluate the expression.

Guide Practice

• Evaluate the expression.

• 1.• 2.• 3.• 4. 81

64

25

9

Find square roots

EXAMPLE 1

– 1. 9 –= 3

2. 25 = 5

GUIDED PRACTICE for Example 1

Evaluate the expression.

64 3. –+ = 8–+

– 4. 81 = 9–

Perfect Squares

• The square of an integer is called a perfect square.

• The square root of a perfect square is an integer.

Approximate a square root

EXAMPLE 2

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 this to find the side length s of the tabletop such that s 945. 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

EXAMPLE 2

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.

Approximate a square root

EXAMPLE 2

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 a square rootEXAMPLE 2GUIDED PRACTICE for Example 2

Approximate the square root to the nearest integer.

5. 32 6

6. 103 10

7. 48– – 7

8. 350– – 19

Approximate a square rootEXAMPLE 2GUIDED PRACTICE for Example 2

Approximate the square root to the nearest integer.

5. 32 6

6. 103 10

7. 48– – 7

8. 350– – 19

Irrational Numbers

• The square root of a whole number that is not a perfect square is an example of an irrational number.

• An irrational number is a number that cannot be written as a quotient of two integers.

• The decimal form of an irrational number neither terminates nor repeats.

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Real Numbers• The set of real numbers is the set of all

rational and irrational numbers. Every point on the real number line represents a real number.

Real Numbers

• Rational fraction or decimal Irrational

• Integers {…, -3, -2, -1, 0, 1, 2, 3,…}• Whole numbers {0, 1, 2, 3….}• Natural numbers (1, 2, 3…)

Example 3 Classify numbers• Tell whether each of the following numbers is a real

number, a rational number, an irrational number, and integer, or a whole number: 81,100,24

Number Real Number?

Rational number?

Irrational number?

Integer? Whole number?

81

100

24

Classify numbers

EXAMPLE 3

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

NoNoYesNoYes

NoYesNoYesYes

Yes

Real Number?

YesYesNoYes

Whole Number?Integer?

Irrational Number?

Rational Number?

Number

24

100

– 81

Graph and order real numbers

EXAMPLE 4

,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 . , , ,

Graph and order real numbers

EXAMPLE 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.

792

– – 20

GUIDED PRACTICE for Examples 3 and 4

9 2

– , 7, , ,– 20 0, 4.1 5.2.

ANSWER

Graph and order real numbersEXAMPLE 4GUIDED PRACTICE for Examples 3 and 4

NoNoYesNoYes

YesYesNoYesYes

Yes

Real Number?

NoNoNoYes

Whole Number?Integer?

Irrational Number?

Rational Number?

Number

20

7

9 2

4.1

0

NoNoYesNoYes

NoNoNoYesYes

–

–

5.2 NoNoNoYesYes

Rewrite a conditional statement in if-then form

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.

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.

Rewrite a conditional statement in if-then form

EXAMPLE 5

b.

Given: All real numbers are rational numbers.

The statement is false. For example, is a real number but not a rational number.

2

If-then form: If a number is a real number, then it is a rational number.

Rewrite a conditional statement in if-then form

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.

GUIDED PRACTICE for Example 5

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.

Rewrite a conditional statement in if-then form

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.

GUIDED PRACTICE for Example 5

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.

13

Rewrite a conditional statement in if-then form

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.

GUIDED PRACTICE for Example 5

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.

Classwork/Homework

•2.1 Exercises •2-52 even

•Pages 76-79