Download 1-5 Square Roots and Real Numbers 1

1-5
1-5 Square
SquareRoots
Rootsand
andReal
RealNumbers
Numbers
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 1
1-5 Square Roots and Real Numbers
Bell Quiz 1-5
Evaluate
2 pts 1. 52
2 pts
2. 62
2 pts
3. 72
2 pts
4. 82
2 pts
5.
92
Holt Algebra 1
10 pts
possible
1-5 Square Roots and Real Numbers
Questions on 0-4/0-10/0-11
Holt Algebra 1
1-5 Square Roots and Real Numbers
Objectives
Evaluate expressions containing square roots.
Classify numbers within the real number
system.
Holt Algebra 1
1-5 Square Roots and Real Numbers
Vocabulary
square root
perfect square
real numbers
natural numbers
whole numbers
integers
rational numbers
Holt Algebra 1
terminating decimal
repeating decimal
irrational numbers
1-5 Square Roots and Real Numbers
A square root is the number that is multiplied by
itself to form a 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)(–4) = (–4)2 = 16
Holt Algebra 1
–
=4
Positive square
root of 16
= –4
Negative square
root of 16
1-5 Square Roots and Real Numbers
The nonnegative square root is represented
by
. The negative square root is
represented by – .
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 1
4
9 16 25 36 49 64 81 100
02
Holt Algebra 1
12
22 32
42 52
62
72
82
92 102
1-5 Square Roots and Real Numbers
Reading Math
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.
Holt Algebra 1
1-5 Square Roots and Real Numbers
Example 1: Finding Square Roots of
Perfect Squares
Find each square root.
A.
Think: What number squared equals 16?
Positive square root
B.
Think: What is the opposite of the
square root of 9?
Negative square root
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positive 4.
negative 3.
1-5 Square Roots and Real Numbers
Example 1C: Finding Square Roots of
Perfect Squares
Find the square root.
Think: What number squared
equals 25 ?
81
Positive square root
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positive 5 .
9
1-5 Square Roots and Real Numbers
Check It Out! Example 1
Find the square root.
1a.
Think: What number squared
equals 4?
Positive square root
positive 2.
1b.
Think: What is the opposite of the
square root of 25?
Negative square root
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negative 5.
1-5 Square Roots and Real Numbers
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.
Holt Algebra 1
1-5 Square Roots and Real Numbers
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?
1
Understand the problem
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.
Holt Algebra 1
1-5 Square Roots and Real Numbers
Example 2 Continued
2
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.
Holt Algebra 1
1-5 Square Roots and Real Numbers
Example 2 Continued
Because 13 is closer to 16 than to 9,
is closer to 4 than to 3.
3
4
You can use a guess-and-check
method to estimate
.
Holt Algebra 1
1-5 Square Roots and Real Numbers
Example 2 Continued
3
Solve
Guess 3.6: 3.62 = 12.96
too low
Guess 3.7:
too high
3
3.72 = 13.69
3.6
is greater than 3.6.
is less than 3.7.
3.7
Because 13 is closer to 12.96 than to
13.69,
is closer to 3.6 than to 3.7.
Holt Algebra 1
4
≈ 3.6
1-5 Square Roots and Real Numbers
Example 2 Continued
4
Look Back
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.
Holt Algebra 1
1-5 Square Roots and Real Numbers
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 a ,
b
where a and b are both integers and b ≠ 0:
1, 7, 9 .
2 1 10
Holt Algebra 1
1-5 Square Roots and Real Numbers
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 a . They include square roots of whole
b
numbers that are not perfect squares and
nonterminating decimals that do not repeat: ,
,π
Holt Algebra 1
1-5 Square Roots and Real Numbers
All numbers that can be represented on a
number line are called real numbers and can
be classified according to their characteristics.
Holt Algebra 1
1-5 Square Roots and Real Numbers
Flow Chart to Find All
Classifications for All Real Numbers
Start Here
Look at the
decimal
representation
of the #
Term Dec?
Rational
Neither?
Irrational
Rep Dec?
Yes
Rational
Yes
Integer?
Whole?
No
No
Holt Algebra 1
Yes
Natural?
No
1-5 Square Roots and Real Numbers
Example 3: Classifying Real Numbers
Write all classifications that apply to each
Real number.
A. –32
32 can be written as a
fraction and a decimal.
rational number, integer, terminating decimal
B. 5
5 can be written as a
fraction and a decimal.
rational number, integer, whole number, natural
number, terminating decimal
Holt Algebra 1
1-5 Square Roots and Real Numbers
Check It Out! Example 3
Write all classifications that apply to each real
number.
4
4
7 9 can be written as a
3a. 7
repeating decimal.
9
rational number, repeating decimal
3b. –12
32 can be written as a
fraction and a decimal.
rational number, terminating decimal, integer
3c.
irrational number
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The digits continue with no
pattern.
1-5 Square Roots and Real Numbers
HOMEWORK
Sec 1-5 (Pg 35) 1-23 all,
56, 59
Holt Algebra 1
1-5 Square Roots and Real Numbers
HOMEWORK
Holt Algebra 1
1-5 Square Roots and Real Numbers
HOMEWORK
Holt Algebra 1
1-5 Square Roots and Real Numbers
HOMEWORK
Holt Algebra 1