Transcript An-introduction-to-Rational

2-1 Rational Numbers
Warm Up
Evaluating Algebraic Expressions
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
Warm Up
Evaluating Algebraic Expressions
Patricia works twice as many days as
Laura works each month. Laura works
3 more days than Jaime. If Jaime works
10 days each month, how many days
does Patricia work?
2-1 Rational Numbers
Evaluating Algebraic Expressions
Vocabulary
rational number
terminating decimal
mixed number
repeating decimal
bar notation
2-1 Rational Numbers
Evaluating
Algebraic
Expressions
A rational
number
is any number
that can
n
be written as a fraction
, where n
d
and d are integers and d  0.
Any fraction can be written as a decimal
by dividing the numerator by the
denominator. If the division ends or
terminates, because the remainder is
zero, then the decimal is a terminating
decimal.
2-1 Rational Numbers
Evaluating Algebraic Expressions
If the division leads to a repeating block of
one or more digits (where all digits are not
zeros) after the decimal point, then the
decimal is a repeating decimal. A
repeating decimal can be written with a
bar over the digits that repeat. So
0.13333… = 0.13 (bar notation)
2-1 Rational Numbers
Additional Example 1A: Writing Fractions as
Decimals
Evaluating Algebraic Expressions
Write the fraction as a decimal.
11
9
1 .2
9 11 .0
–9
20
–1 8
2
The fraction
The pattern repeats.
This is a repeating decimal.
11
is equivalent to the decimal 1.2
9
2-1 Rational Numbers
Additional Example 1B: Writing Fractions as
Decimals
Evaluating Algebraic Expressions
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.
7
The fraction
is equivalent to the decimal 0.35
20
2-1 Rational Numbers
Partner Share! Example 1A
Write the fraction as a decimal. Check your
Evaluating
Algebraic Expressions
answer
with a calculator.
15
9
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
The fraction
is equivalent to the decimal 1.6.
9
2-1 Rational Numbers
Partner Share! Example 1B
Write the fraction as a decimal. Check your
Evaluating
Algebraic Expressions
answer
with a calculator.
9
40
0.2 2 5 This is a terminating
40 9.0 0 0 decimal.
–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
Evaluating Algebraic Expressions
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
Additional Example 2: Writing Terminating
Decimals as Fractions
WriteEvaluating
each decimal Algebraic
as a fraction Expressions
in simplest form.
A. 5.37
37 7 is in the hundredths place, so
5.37 = 5
100 write hundredths as the
denominator.
B. 0.622
2 is in the thousandths place, so
622
0.622 =
write thousandths as the
1000
denominator.
311
=
500
Simplify by dividing by the
greatest common divisor.
2-1 Rational Numbers
Evaluating Algebraic Expressions
Remember!
A fraction is in reduced, or simplest, form when
the numerator and the denominator have no
common divisor other than 1.
2-1 Rational Numbers
Partner Share! Example 2
Write each decimal as a fraction in simplest form.
Evaluating Algebraic Expressions
A. 8.75
5 is in the hundredths place,
75 so write hundredths as the
8.75 = 8
100 denominator.
3
Simplify by dividing by the
= 8
4
greatest common divisor.
B. 0.2625
5 is in the
2625
0.2625 =
10,000 ten-thousandths place.
Simplify by dividing by the
21
=
greatest common divisor.
80
2-1 Rational Numbers
Additional Example 3: Writing Repeating Decimals
as Fractions
_
Algebraic
Expressions
WriteEvaluating
0.4 as a fraction
in simplest
form.
x = 0.44444…
10x = 10(0.44444…)
10x = 4.444444…
-x = -0.44444…
9x = 4
9x = 4
9
9
4
x=
9
Let x represent the number.
Multiply both sides by 10
because 1 digit repeats.
Subtract x from both sides to
eliminate the repeating part.
Since x = 0.44444…, use
0.44444… for x on the right side
of the equation.
Since x is multiplied by 9,
divide both sides by 9.
2-1 Rational Numbers
__
Partner Share! Example 3
Write 0.36 as a fraction in simplest form.
Evaluating Algebraic Expressions
x = 0.363636…
100x = 100(0.363636…)
100x = 36.363636…
-x = -0.363636…
99x = 36
99x = 36
99
99
x = 36 = 4
99 11
Let x represent the number.
Multiply both sides by 100
because 2 digits repeat.
Subtract x from both sides to
eliminate the repeating part.
Since x = 0.363636…, use
0.363636… for x on the right
side of the equation.
Since x is multiplied by 99,
divide both sides by 99.
Write in simplest form.
2-1 Rational Numbers
Lesson Review!
Write each decimal as a fraction in
Evaluating Algebraic Expressions
simplest form.
3
7
1. 0.35
2.
0.600
5
20
13
3. Write
as a decimal. 2.16
6
4. Tommy 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