Transcript Ch 2-1 Rational Numbers - San Elijo Middle School

2-1 Rational Numbers
California
Evaluating
Algebraic Expressions
Standards
NS1.5 Know that every rational
number is either a terminating or a
repeating decimal and be able to convert
terminating decimals into reduced
fractions.
NS1.3 Convert fractions to decimals
and percents and use representations in
estimations, computations, and
applications.
2-1 Rational Numbers
Evaluating Algebraic Expressions
Vocabulary
rational number
terminating decimal
repeating decimal
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.
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.
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
Check It Out! Example 1A
Write
the fraction
as a decimal.
Evaluating
Algebraic
Expressions
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
Check It Out! Example 1B
Write the fraction as a decimal.
Evaluating Algebraic Expressions
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
5.37 = 5
100
B. 0.622
622
0.622 =
1000
311
=
500
7 is in the hundredths place,
so write hundredths as the
denominator.
2 is in the thousandths
place, so write thousandths
as the denominator.
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
Check It Out! 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
__
Check It Out! 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.