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2-1 Rational Numbers
Word & Letter Play
The Value of Months
If March = 43 and May = 39, then by
the same logic, what does July equal?
STRATEGY:Look for a pattern – think of
ways letters are represented by numbers
2-1 Rational Numbers
Word & Letter Play
The Value of Months
Each letter is replaced by the number of its
position in the English alphabet. Then the
numbers are added together.
68
STRATEGY:Look for a pattern – think of
ways letters are represented by numbers
2-1 Rational Numbers
Warm Up
Problem of the Day
Lesson Presentation
Lesson Quizzes
2-1 Rational Numbers
Warm Up
Divide.
1. 36  3
3. 68  17
12
4
5. 1024  64 16
2. 144  6
4. 345  115
24
3
2-1 Rational Numbers
Problem of the Day
An ice cream parlor has 6 flavors of ice
cream. A dish with two scoops can have
any two flavors, including the same flavor
twice. How many different double-scoop
combinations are possible?
21
2-1 Rational Numbers
Learn to write rational numbers in
equivalent forms.
2-1 Rational Numbers
Vocabulary
rational number
relatively prime
2-1 Rational Numbers
A rational number is any number that can
be written as a fraction n , where n
d
and d are integers and d  0.
2-1 Rational Numbers
The goal of simplifying fractions is to make
the numerator and the denominator
relatively prime. Relatively prime
numbers have no common factors other
than 1.
2-1 Rational Numbers
You can often simplify fractions by dividing
both the numerator and denominator by
the same nonzero integer. You can
12
simplify the fraction 15
to 45 by dividing
both the numerator and denominator by 3.
12 of the 15 boxes
are shaded.
12
15
4 of the 5 boxes
are shaded.
=
4
5
The same total area is shaded.
2-1 Rational Numbers
Additional Example 1A: Simplifying Fractions
Simplify.
16
80
16 = 1 • 4 • 4;16 is a common factor.
80 = 5 • 4 • 4
16 ÷ 16
16
=
80
80 ÷ 16
1
=
5
Divide the numerator
and denominator by 16.
Remember!
a = 1 for a ≠ 0
0 = 0 for a ≠ 0
a
a
–7= 7 = – 7
8
–8
8
2-1 Rational Numbers
Additional Example 1B: Simplifying Fractions
Simplify.
–18
29
18 = 2 • 9
29 = 1 • 29
–18
–18
=
29
29
;There are no common
factors.
18 and 29 are relatively prime.
2-1 Rational Numbers
Check It Out: Example 1A
Simplify.
18 = 3 • 3 • 2 ; 9 is a common factor.
27 = 3 • 3 • 3
18
27
18 = 18 ÷ 9
27 27 ÷ 9
=
2
3
Divide the numerator
and denominator by 9.
2-1 Rational Numbers
Check It Out: Example 1B
Simplify.
17
–35
17 = 1 • 17 ; There are no common
factors.
35 = 5 • 7
17
17
=–
17 and 35 are relatively prime.
–35
35
2-1 Rational Numbers
Decimals that terminate or repeat are rational
numbers.
To write a terminating decimal as a fraction,
identify the place value of the digit farthest to
the right. Then write all of the digits after the
decimal point as the numerator with the place
value as the denominator.
2-1 Rational Numbers
2-1 Rational Numbers
Additional Example 2: Writing Decimals as Fractions
Write each decimal as a fraction in simplest form.
A. 5.37
37
5.37 = 5
100
7 is in the hundredths place.
B. 0.622
0.622 =
622
1000
311
=
500
2 is in the thousandths
place.
Simplify by dividing by the
common factor 2.
2-1 Rational Numbers
Check It Out: Example 2
Write each decimal as a fraction in simplest form.
A. 8.75
8.75 = 8
75
5 is in the hundredths place.
100
3
= 8
4
B. 0.2625
Simplify by dividing by the
common factor 25.
5 is in the
2625
0.2625 =
10,000 ten-thousandths place.
Simplify by dividing by
21
=
the common factor 125.
80
2-1 Rational Numbers
To write a fraction as a decimal, divide the
numerator by the denominator. You can
use long division.
numerator
denominator
denominator numerator
2-1 Rational Numbers
Additional Example 3A: Writing Fractions as
Decimals
Write the fraction as a decimal.
11
9
The fraction
1 .2
9 11 .0
–9
20
–1 8
2
The pattern repeats.
Writing Math
A repeating decimal can be written
with a bar over the digits_that
repeat. So 1.2222… = 1.2.
11
is equivalent to the decimal 1.2.
9
2-1 Rational Numbers
Additional Example 3B: Writing Fractions as
Decimals
Write the fraction as a decimal.
7
20
0.3 5 This is a terminating decimal.
20 7.0 0
–0
70
–6 0
1 00
–1 0 0
0 The remainder is 0.
The fraction
7
is equivalent to the decimal 0.35.
20
2-1 Rational Numbers
Check It Out: Example 3A
Write the fraction as a decimal.
15
9
The fraction
1 .6
9 15 .0
–9
60
–5 4
6
The pattern repeats, so
draw a bar over the 6 to
indicate that this is a
repeating decimal.
15
is equivalent to the decimal 1.6.
9
2-1 Rational Numbers
Check It Out: Example 3B
Write the fraction as a decimal.
9
40
0.2 2 5 This is a terminating decimal.
40 9.0 0 0
–0
90
–8 0
1 00
– 80
200
– 2 00
0 The remainder is 0.
9
The fraction
is equivalent to the decimal 0.225.
40
2-1 Rational Numbers
Rational Numbers
The real number system consists of rational and
irrational numbers.
Rational numbers can be expressed in fractional
form, a , where a (the numerator) and b (the
b
denominator)
are both integers and b = 0.
The decimal form of the number either terminates
or repeats.
Counting numbers, whole numbers, integers, and
non-integers are all rational numbers.
2-1 Rational Numbers
Counting numbers are the natural numbers.
{1, 2, 3, 4, 5, 6, …}
Whole numbers consist of the counting
numbers and zero.
{0, 1, 2, 3, 4, 5, …}
Integers consist of the counting numbers, their
opposites, and zero.
{…, -3, -2, -1, 0, 1, 2, 3, …}
2-1 Rational Numbers
Non-integers consist of fractions that can be
written as terminating or repeating decimals.
– A terminating decimal comes to a complete stop.
– A repeating decimal continues the same digit or
block of digits forever.
23
7
5.2
5
1
3
0.
6
-9.261
2-1 Rational Numbers
Irrational Numbers
Irrational numbers are numbers that cannot be
written as a ratio of two integers.
Irrational numbers are non-repeating and
non-terminating decimals because the decimal form
of the number never ends and never repeats.
The most common irrational number is pi (п).
The value of п is 3.141592654…
2-1 Rational Numbers
Example
1
Tell whether each real number is rational or irrational.
-23.75
rational
decimal terminates
4.750918362… irrational
5
9
rational
√15 irrational
decimal does not terminate
number is in fraction
form
decimal form does not terminate
2-1 Rational Numbers
2-1 Rational Numbers
Lesson Quizzes
Standard Lesson Quiz
Lesson Quiz for Student Response Systems
2-1 Rational Numbers
Lesson Quiz: Part I
Simplify.
18
1.
42
3
7
15
2.
21
5
7
Write each decimal as a fraction in
simplest form.
5
27
–
3. 0.27
4.
–0.625
8
100
13
5. Write
as a decimal
6
2.16
2-1 Rational Numbers
Lesson Quiz: Part II
6. Alex had 13 hits in 40 at bats for his
baseball team. What is his batting
average? (Batting average is the
number of hits divided by the number
of at bats, expressed as a decimal.)
0.325
2-1 Rational Numbers
Lesson Quiz for Student Response Systems
1. Simplify
16
.
28
5
7
A. 3
7
C.
B. 4
7
D. 6
7
2-1 Rational Numbers
Lesson Quiz for Student Response Systems
2. Simplify
24
.
30
6
5
A. 4
6
C.
B. 4
5
D. 5
4
2-1 Rational Numbers
Lesson Quiz for Student Response Systems
3. Which of the following represents the given
decimal as a fraction in simplest form?
0.43
A. 43
100
B. 100
43
C. 43
10
D.
43
1000
2-1 Rational Numbers
Lesson Quiz for Student Response Systems
4. Which of the following represents the given
decimal as a fraction in simplest form?
–0.875
A. – 5
7
C. – 7
8
B. – 4
5
D. – 9
10
2-1 Rational Numbers
Lesson Quiz for Student Response Systems
5. Which of the following represents the
given fraction as a decimal?
51
9
A. 4.3
B. 4.6
C. 5.6
D. 6.5