Is the Fraction a Terminating or Repeating Decimal.notebook
November 19, 2013
Is the Fraction a Terminating or Repeating Decimal?
Write each of these terminating decimals as a decimal
fraction:
a) 0.3
b) 0.34
c) 0.342
d) 0.3425
e) 0.32457
f) 0.324579
Apr 30­2:31 PM
Answers:
a) 3
10
b)
d) 3425
10 000
e) 32 457
100 000
34
100
c) 324
1000
f)
342 579
1 000 000
Aug 8­12:55 PM
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Is the Fraction a Terminating or Repeating Decimal.notebook
November 19, 2013
A terminating decimal can be very long, for example:
0.123 456 789 876 543 421
Write this terminating decimal as a decimal fraction.
Apr 30­2:35 PM
How did you know what to write in the denominator?
Apr 30­2:36 PM
2
Is the Fraction a Terminating or Repeating Decimal.notebook
November 19, 2013
Any terminating decimal can be written as a decimal
fraction.
For example, consider a decimal with two decimal
places, such as 3.12.
Since there are two decimal places, we can read the
number of hundredths: there are 312 hundredths, so
3.12 = 312 .
100
Similarly, 0.045 has 3 decimal places, so we can read
the number of thousandths: there are 45 thousandths, so
0.045 = 45 .
1000
Apr 30­2:37 PM
Terminating decimals can also look very much like the
start of a repeating decimal.
For example, 0.323232323232 is a terminating decimal.
Write that decimal as a decimal fraction.
Apr 30­2:37 PM
3
Is the Fraction a Terminating or Repeating Decimal.notebook
November 19, 2013
Can any decimal fraction be written as a
terminating decimal?
Apr 30­2:38 PM
The smallest place value in the decimal will be
determined by the denominator of the decimal
fraction.
For example, if the decimal fraction has
denominator 10 000, then the smallest place
value in the decimal will be ten thousandths,
which means the decimal has only 4 place
values.
Apr 30­2:39 PM
4
Is the Fraction a Terminating or Repeating Decimal.notebook
November 19, 2013
Write these decimal fractions as terminating decimals:
a)
36
100
b)
5
1000
c)
2341
1000
d)
23 456
100 000
Apr 30­2:40 PM
Answers:
a) 0.36
b) 0.005
c) 2.341
d) 0.23456
Aug 8­1:01 PM
5
Is the Fraction a Terminating or Repeating Decimal.notebook
November 19, 2013
Calculate these quotients using a calculator:
1 =1÷25 = ?
25
1 =1÷27 =?
27
1 =1÷29 =?
29
Which of the fractions looks like it has a decimal
equivalent that terminates?
Can you tell for sure from the calculator display for all
three fractions?
Apr 30­2:42 PM
My calculator says that
1 = 0.037037037037037037037037037037.
27
This could either be a very long terminating decimal or
it could be the repeating decimal 0.037 .
My calculator says that
1 = 0.034482758620689655172413793103448.
29
This also could be a terminating decimal or it could be
the repeating decimal
0.0344827586206896551724137931.
Apr 30­2:46 PM
6
Is the Fraction a Terminating or Repeating Decimal.notebook
November 19, 2013
If after dividing a numerator by a denominator, the
digits do not go to the end of the calculator display—
the most digits that the calculator can show—then you
can be sure that the equivalent decimal terminates,
since the calculator would have shown more
digits if there were any more to show.
If the digits do go to the end of the calculator display,
you won’t be able to tell for sure whether the equivalent
decimal repeats or not.
Guess by looking at the digits whether there is a
repeating pattern or not.
Apr 30­2:49 PM
A fraction will have a terminating decimal equivalent if
it has an equivalent fraction that is a decimal fraction.
Look at the fractions you said were repeating.
Try to find an equivalent decimal fraction.
Apr 30­2:51 PM
7
Is the Fraction a Terminating or Repeating Decimal.notebook
November 19, 2013
Articulate a reason why 7 has no equivalent decimal
15
fraction.
Apr 30­2:52 PM
7
The fraction 15 is in lowest terms, so to make any
equivalent fraction we have to multiply the numerator
and denominator by the same number.
Since we want to find an equivalent decimal fraction,
we’re looking for a fraction that has a denominator
power of 10.
Apr 30­2:52 PM
8
Is the Fraction a Terminating or Repeating Decimal.notebook
November 19, 2013
Extra Practice:
1
14
1. Write out all the fifteenths from 15 to 15, and
then write them in lowest terms. Decide from
the denominators which fractions will terminate
as decimals, then check on a calculator.
May 1­12:50 PM
2. Write these fractions in lowest terms and then
decide if their decimal equivalents will terminate or
not. Check your answers on a calculator.
a) 7
25
b) 13
20
c) 9
150
5
7
e) 6
21
f) 4
110
d)
Aug 8­1:09 PM
9
Is the Fraction a Terminating or Repeating Decimal.notebook
A calculator shows
November 19, 2013
50
= 0.5154639175257732.
97
Is this an exact value or an approximation?
How do you know? May 1­12:54 PM
Create a fraction with a terminating decimal equivalent
that:
a) has a 1­digit denominator,
b) has a 2­digit denominator,
c) has a 3­digit denominator,
d) has a 4­digit denominator. May 1­12:57 PM
10
Is the Fraction a Terminating or Repeating Decimal.notebook
November 19, 2013
Extension:
Why is the decimal representation of every
fraction either repeating or terminating? Why
can’t it be like pi, which doesn’t terminate or
repeat?
Aug 8­1:13 PM
AP Book: 8:2:1 pg. 5
Apr 30­2:32 PM
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