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UNIT 1
Study Guide
?
1
Real Numbers
How can you use real numbers to solve real-world problems?
EXAMPLE 1
_
_
Estimate the value of √5, and estimate the position of √5 on a
number line.
4<5<9
5 is between the perfect squares 4 and 9.
_
_
_
√4 < √5 < √9
_
√ 5 is between 2 and 3.
2.2 = 4.84
2
2π is approximately equal to 2 × 3.14, or 6.28.
_
_
_
_
√ 36 < √ 38 < √ 49
6 < √38 < 7
_
√ 38 is approximately 6.15.
Key Vocabulary
ESSENTIAL QUESTION
Take the square root of each number.
EXAMPLE 3
_
Order 6, 2π, and √38 from least to greatest.
_
2 < √5 < 3
irrational number (número
irracional)
perfect square (cuadrado
perfecto)
principal square root (raíz
cuadrada principal)
square root (raíz cuadrada)
terminating decimal
(decimal finito)
(Lesson 1.1)
6.5
2 _
_
, - 25
5
4
2. __
25
1 _
_
, - 17
7
1
4. __
49
5.
_
√ 10
15, -15
3. 225
3.15, -3.15
6.
_
√ 18
4.25, -4.25
Write all names that apply to each number. (Lesson 1.2)
2.5
_
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4, -4
1. 16
15
9. __
5
_
A 5.4
rational, real
5.4 is a repeating decimal.
B _84
whole, integer, rational, real
8
__
=2
4
_
7. _23
3
EXAMPLE 2
Write all names that apply to each number.
8. -√100
rational, real
integer, rational, real
10.
_
√ 21
irrational, real
whole, integer, rational, real
Compare. Write <, >, or =. (Lesson 1.3)
11.
_
√7 +
5
<
7+
_
√5
12. 6 +
_
√8
<
_
√6
+8
13.
_
√4 -
2
<
4-
_
√2
Order the numbers from least to greatest. (Lesson 1.3)
13 is a whole number
that is not a perfect
square.
14.
_
72
√ 81, __
, 8.9
7
_
72
8.9, √81 , __
7
Unit 1
8_MTXESE052888_U1EM.indd 47
6.4
EXERCISES
Find the two square roots of each number. If the number is not a
perfect square, approximate the values to the nearest 0.05.
real number (número real)
repeating decimal (decimal
periódico)
2.3 = 5.29
irrational, real
6.3
rational number (número
racional)
A good estimate is 2.25.
_
√ 13
2π
6.2
_
_
√ 5 is between 2.2 and 2.3.
C
6.1
6.22 = 38.44
From least to greatest, the numbers are 6, √38, and 2π.
2
2
√38
6
6
6.12 = 37.21
47
12/24/12 7:19 PM
48
15.
_
7
√ 7, 2.55, _
3
_
7
_
, 2.55, √7
3
© Houghton Mifflin Harcourt Publishing Company
MODULE
Review
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Expressions and the Number System 48
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2
Scientific Notation
Unit 1 Performance Tasks
Key Vocabulary
scientific notation
(notación científica)
ESSENTIAL QUESTION
1.
How can you use scientific notation to solve real-world problems?
CAREERS IN MATH Astronomer An astronomer is studying
Proxima Centauri, which is the closest star to our Sun. Proxima Centauri
is 39,900,000,000,000,000 meters away.
a. Write this distance in scientific notation.
EXAMPLE 1
The diameter of Earth at the equator is approximately
12,700 kilometers. Write the diameter of Earth in scientific notation.
3.99 × 1016 m
b. Light travels at a speed of 3.0 × 108 m/s (meters per second). How
can you use this information to calculate the time in seconds it takes
for light from Proxima Centauri to reach Earth? How many seconds
does it take? Write your answer in scientific notation.
Move the decimal point in 12,700 four places to the left: 1.2 7 0 0.
12,700 = 1.27 × 104
Divide the distance Proxima Centauri is from Earth by the speed of
light; 1.33 × 108 s
EXAMPLE 2
The diameter of a human hair is approximately 0.00254 centimeter.
Write the diameter of a human hair in scientific notation.
c. Knowing that 1 year = 3.1536 × 107 seconds, how many years does
it take for light to travel from Proxima Centauri to Earth? Write your
answer in standard notation. Round your answer to two decimal
places.
Move the decimal point in 0.00254 three places to the right: 0.0 0 2.5 4
4.22 years
0.00254 = 2.54 × 10-3
2. Cory is making a poster of common geometric shapes. He draws a
3
square
_with a side length of 4 cm, an equilateral triangle with a height
of √200 cm, a circle with a circumference of 8π cm, a rectangle with
122
length ___
5 cm, and a parallelogram with base 3.14 cm.
EXERCISES
Write each number in scientific notation. (Lessons 2.1, 2.2)
3 × 103
1. 3000
3. 25,500,000
1.5 × 10−5
2. 0.000015
2.55 × 10
7
7.34 × 10
4. 0.00734
a. Which of these numbers are irrational?
_
√ 200 and 8π
−3
52,300
6. 1.05 × 106
1,050,000
b. Write the numbers in this problem in order from least to greatest.
Approximate
_π as 3.14.
122
3.14, √200 , ___
, 8π, 43
0.47
8. 1.33 × 10
0.0000133
c. Explain why 3.14 is rational, but π is not.
Write each number in standard notation. (Lessons 2.1, 2.2)
© Houghton Mifflin Harcourt Publishing Company
5. 5.23 × 104
7. 4.7 × 10
-1
-5
5
314
3.14 is a terminating decimal that can be written in the form __ba: ____
1000
Use the information in the table to write each weight in
scientific notation. (Lessons 2.1, 2.2)
Animal
Weight (lb)
9. Ant
10. Butterfly
11. Elephant
ant
butterfly
elephant
0.000000661
0.00000625
9900
157
. π is a nonrepeating, nonterminating decimal that cannot be
or ___
500
written in the form __ba .
6.61 × 10-7 lb
6.25 × 10-6 lb
9.9 × 103 lb
Unit 1
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© Houghton Mifflin Harcourt Publishing Company
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Expressions and the Number System 50
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Personal
Math Trainer
Texas Test Prep
Selected Response
A between 20 and 21 millimeters
B between 64 and 65 millimeters
C between 204 and 205 millimeters
D between 649 and 650 millimeters
2. Which of the following numbers is rational
but not an integer?
B -4.3
C 0
D 3
3. Which statement is false?
A No integers are irrational numbers.
B All whole numbers are integers.
C All rational numbers are real numbers.
D All integers are whole numbers.
4. Which set best describes the numbers
displayed on a telephone keypad?
A whole numbers
© Houghton Mifflin Harcourt Publishing Company
Online
Assessment and
Intervention
11. A quarter weighs about 0.025 pound.
What is this weight written in scientific
notation?
B rational numbers
C real numbers
D integers
5. In 2011, the population of Laos was about
6.586 × 106 people. What is this number
written in standard notation?
A 6,586 people
B 658,600 people
C 6,586,000 people
_
B 2.5 × 101 pound
_
A √ 16 + 4 > √ 4 + 5
C 2.5 × 10-1 pound
B 4π > 12
_
15
C √ 18 + 2 < __
2
_
D 6 - √ 35 < 0
7. Which number is between
22
A __
3 _
B 2 √8
D 2.5 × 102 pound
12. Which of the following is the number
3.0205 × 10-3 written in standard notation?
_
5π
√ 50 and __
?
2
C 6
D π+3
7.2
7.4
7.6
7.8
8
15. Jerome is writing a number in scientific
notation. The number is greater than one
million and less than ten million. What will
be the exponent in the number Jerome
writes?
6
.
C 3.0205
0
0
0
0
0
0
B 0.0030205
D 3020.5
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
A 0.0000042 meter
6
6
6
6
6
6
B 0.000042 meter
7
7
7
7
7
7
8
8
8
8
8
8
9
9
9
9
9
9
C 0.00042 meter
A π+4
152
B ___
20_
C √ 14 + 4
Underline key words given in
the test question so you know
for certain what the question is
asking.
A 0.00030205
13. A human fingernail has a thickness of about
4.2 × 10−4 meter. What is this width written
in standard notation?
8. What number is indicated on the
number line?
7
Hot !
Tip
A 2.5 × 10-2 pound
6. Which of the following is not true?
1. A square on a large calendar has an area of
4220 square millimeters. Between which
two integers is the length of one side of the
square?
A -9
my.hrw.com
D 0.0042 meter
16. Write the number 3.3855 × 10 in standard
notation.
2
Gridded Response
_
D 7.8
14. The square root of a number is -18. What is
the other square root?
9. Which of the following is the number
5.03 × 10-5 written in standard form?
A 503,000
1
8
.
.
3
3
8
5
5
0
0
0
0
0
0
1
1
1
1
1
1
2
2
2
2
2
2
0
0
0
0
0
0
3
3
3
3
3
3
1
1
1
1
1
1
4
4
4
4
4
4
C 0.00503
2
2
2
2
2
2
5
5
5
5
5
5
D 0.0000503
3
3
3
3
3
3
6
6
6
6
6
6
4
4
4
4
4
4
7
7
7
7
7
7
5
5
5
5
5
5
8
8
8
8
8
8
6
6
6
6
6
6
9
9
9
9
9
9
7
7
7
7
7
7
8
8
8
8
8
8
9
9
9
9
9
9
B 50,300,000
10. In a recent year, about 20,700,000
passengers traveled by train in the United
States. What is this number written in
scientific notation?
A 2.07 × 101 passengers
B 2.07 × 104 passengers
© Houghton Mifflin Harcourt Publishing Company
UNIT 1 MIXED REVIEW
C 2.07 × 107 passengers
D 2.07 × 108 passengers
D 65,860,000 people
Unit 1
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Expressions and the Number System 52