This document provides information about square roots and real numbers. It includes: 1) Examples of finding square roots of perfect squares and estimating other square roots. 2) Classifying different types of real numbers such as rational numbers, irrational numbers, integers, natural numbers, and more. 3) Examples of classifying given numbers as rational, irrational, integers, and other categories.

1. Square Roots/Rational
Numbers

2. 1-5 Square Roots and Real Numbers
Holt Algebra 1
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz
Warm UpWarm Up

3. Warm Up
Simplify each expression.
1. 62
36 2. 112 121
3. (–9)(–9) 81 4. 25
36
Write each fraction as a decimal.
5.
2
5
5
96.
7. 5 3
8
8. –1 5
6
0.4
5.375
0.5
–1.83

4. Evaluate expressions containing square roots.
Classify numbers within the real number
system.
Objectives

5. square root terminating decimal
perfect square repeating decimal
real numbers irrational numbers
natural numbers
whole numbers
integers
rational numbers
Vocabulary

6. A number that is multiplied by itself to form a
product is called a square root of that product.
The operations of squaring and finding a square
root are inverse operations.
The radical symbol , is used to represent
square roots. Positive real numbers have two
square roots.
4 • 4 = 42
= 16 = 4 Positive square
root of 16
(–4)(–4) = (–4)2
= 16 = –4 Negative square
root of 16
–

7. A perfect square is a number whose positive
square root is a whole number. Some examples
of perfect squares are shown in the table.
0
02
1
12
1004
22
9
32
16
42
25
52
36
62
49
72
64
82
81
92
102
The nonnegative square root is represented
by . The negative square root is
represented by – .

8. The expression does not represent
a real number because there is no real
number that can be multiplied by itself to
form a product of –36.
Reading Math

9. Example 1: Finding Square Roots of
Perfect Squares
Find each square root.
42
= 16
32
= 9
Think: What number squared equals 16?
Positive square root positive 4.
Think: What is the opposite of the
square root of 9?
Negative square root negative 3.
A.
= 4
B.
= –3

10. Find the square root.
Think: What number squared
equals ?25
81
Positive square root positive .
5
9
Example 1C: Finding Square Roots of
Perfect Squares

11. Find the square root.
Check It Out! Example 1
22
= 4 Think: What number squared
equals 4?
Positive square root positive 2.= 2
52
= 25 Think: What is the opposite of the
square root of 25?
1a.
1b.
Negative square root negative 5.

12. The square roots of many numbers like , are not
whole numbers. A calculator can approximate the
value of as 3.872983346... Without a calculator,
you can use square roots of perfect squares to help
estimate the square roots of other numbers.

13. Example 2: Problem-Solving Application
As part of her art project, Shonda will
need to make a square covered in glitter.
Her tube of glitter covers 13 square
inches. What is the greatest side length
Shonda’s square can have?
The answer will be the side length of the
square.
List the important information:
• The tube of glitter can cover an area of
13 square inches.
Understand the problem11

14. 22 Make a Plan
The side length of the square is because
13. Because 13 is not a perfect
square, is not a whole number. Estimate
to the nearest tenth.
=•
Find the two whole numbers that is
between. Because 13 is between the perfect
squares 0 and 16. is between and
, or between 3 and 4.
Example 2 Continued

15. 3 4
Because 13 is closer to 16 than to 9,
is closer to 4 than to 3.
You can use a guess-and-check
method to estimate .
Example 2 Continued

16. Guess 3.6: 3.62
= 12.96
too low
Guess 3.7: 3.72
= 13.69
too high
is greater than 3.6.
is less than 3.7.
Solve33
Example 2 Continued
3.6 3.7 43
Because 13 is closer to 12.96 than to
13.69, is closer to 3.6 than to 3.7. ≈ 3.6

17. A square with a side length of 3.6 inches
would have an area of 12.96 square inches.
Because 12.96 is close to 13, 3.6 inches
is a reasonable estimate.
Look Back44
Example 2 Continued

18. What if…? Nancy decides to buy more
wildflower seeds and now has enough to cover
38 ft2
. What is the side length of a square
garden with an area of 38 ft2
?
Check It Out! Example 2
Guess 6.2 6.22
= 38.44 too high
Guess 6.1 6.12
= 37.21 too low
Use a guess and check method to estimate .
is greater than 6.1.
is less than 6.2.
A square garden with a side length of 6.2 ft
would have an area of 38.44 ft2
. 38.44 ft is
close to 38, so 6.2 is a reasonable answer.

19. All numbers that can be represented on a
number line are called real numbers and can
be classified according to their characteristics.

20. Natural numbers are the counting numbers: 1, 2, 3, …
Whole numbers are the natural numbers and zero:
0, 1, 2, 3, …
Integers are whole numbers and their opposites:
–3, –2, –1, 0, 1, 2, 3, …
Rational numbers can be expressed in the form ,
where a and b are both integers and b ≠ 0:
, , .
a
b
1
2
7
1
9
10

21. Terminating decimals are rational numbers in
decimal form that have a finite number of digits:
1.5, 2.75, 4.0
Repeating decimals are rational numbers in
decimal form that have a block of one or more
digits that repeat continuously: 1.3, 0.6, 2.14
Irrational numbers cannot be expressed in the
form . They include square roots of whole
numbers that are not perfect squares and
nonterminating decimals that do not repeat: ,
, π
a
b

22. Example 3: Classifying Real Numbers
Write all classifications that apply to each
Real number.
A. –32
–32 = – = –32.0
32
1
32 can be written as a
fraction and a decimal.
rational number, integer, terminating decimal
B. 5
5 = = 5.0
5
1
5 can be written as a
fraction and a decimal.
rational number, integer, whole number, natural
number, terminating decimal

23. Write all classifications that apply to each real
number.
. 7
4
9
rational number, repeating decimal
3b. –12
rational number, terminating decimal, integer
irrational number
Check It Out! Example 3
3c.
67 ÷ 9 = 7.444… = 7.4
7 can be written as a
repeating decimal.
4
9
–12 = – = –12.0
12
1
32 can be written as a
fraction and a decimal.
= 3.16227766… The digits continue with no
pattern.

24. Find each square root.
1. 2. 3. 4.12 -8 3
7
–
1
2
5. The area of a square piece of cloth is 68 in2
.
How long is each side of the piece of cloth?
Round your answer to the nearest tenth of an
inch. 8.2 in.
Lesson Quiz
Write all classifications that apply to each
real number.
6. 1
7. –3.89
8.
rational, integer, whole number, natural
number, terminating decimal
rational, repeating decimal
irrational