2.1 find square roots and compare real numbers you will find square roots and compare real numbers....
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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?
Warm-Up ExercisesFind square rootsEXAMPLE 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.
Warm-Up ExercisesFind square rootsEXAMPLE 1
– 1. 9 –= 3
2. 25 = 5
GUIDED PRACTICE for Example 1
Evaluate the expression.
64 3. –+ = 8–+
– 4. 81 = 9–
Warm-Up ExercisesApproximate a square rootEXAMPLE 2
FURNITURE
The top of a folding table is a square whose area is 945 square inches. Approximate the side length of thetabletop 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.
Warm-Up ExercisesApproximate a square rootEXAMPLE 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.
Warm-Up ExercisesApproximate a square rootEXAMPLE 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.
Warm-Up ExercisesApproximate 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
Warm-Up ExercisesClassify numbersEXAMPLE 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
Warm-Up ExercisesGraph and order real numbersEXAMPLE 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 . , , ,
Warm-Up ExercisesGraph and order real numbersEXAMPLE 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
Warm-Up ExercisesGraph 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
Warm-Up ExercisesRewrite a conditional statement in if-then formEXAMPLE 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.
Warm-Up ExercisesRewrite a conditional statement in if-then formEXAMPLE 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.
Warm-Up ExercisesRewrite a conditional statement in if-then formEXAMPLE 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.
Warm-Up ExercisesRewrite a conditional statement in if-then formEXAMPLE 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
Warm-Up ExercisesRewrite a conditional statement in if-then formEXAMPLE 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.
Warm-Up ExercisesDaily Homework Quiz
Evaluate the expression.
ANSWER 316
Approximate the square root to the nearest integer.
1.
–+ 289
ANSWER –+17
2. 36–
3. 21–
Warm-Up ExercisesDaily Homework Quiz
ANSWER – 5
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
4. 620
ANSWER 25
• 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.