Real numbers, rational numbers, fractions, decimals, repeating decimals, zero-repeating decimals, terminating decimals, non-terminating recurring decimals, non-terminating non-recurring decimals, fractions to decimals, decimals to fractions, convert fractions to decimals, convert decimals to fractions

1. Real Numbers

2. Rational Number
A rational number can be made by dividing two integers.
Given that 𝑎 and 𝑏 are two integers, then 𝑎
𝑏
is a rational number.
𝑎
𝑏
numerator
denominator
Writing fractions as decimals
Rational numbers are commonly referred to as fractions. A fraction can
be written as a decimal by dividing the numerator by the denominator.
For example,
2
5
can be written as a decimal by dividing 2 by 5 as follows:
25
0
.
.
0
4
.02
Hence,
2
5
= 0.4

3. From the preciously given method, the following fractions can be
written into decimals as follow:
3
2
= 1.5
4
25
= 0.16
19
20
= 0.95
21
8
= 2.625

4. 1
3
is written as a decimal by dividing 1 by 3 as follows:
13
0
.
.
0
3
9
1
0
0
9
1
3
0
0
3
9
1
0
0
3
9
1
…
From the above, if we repeatedly divide by 3, we will get
1
3
= 0.3333…

5. Using the shown method, we can write the following fractions in terms
of decimals as follow:
7
9
= 0.777 …
16
45
= 0.3555…
171
99
= 1.7272 …
9
37
= 0.243243243 …
3
7
= 0.428571428571 …
Repeating decimals
= 0. 7
zero point seven,
seven repeating
= 0. 35
zero point three five,
five repeating
= 1. 72
one point seven two,
seven two repeating
= 0. 243
zero point two four three,
two four three repeating
= 0. 428571
zero point four two eight five seven one,
four two eight five seven one repeating

6. Case 1: If there is one repeated digit, write ∙ above that digit such as
0.5888… = 0.58
Case 2: If there are two or more repeated digits, write∙ above the first∙
repeated digit and the last repeated digit such as
1.8383… = 1. 83
2.0367367… = 2.0367
4.2583198319… = 4.258319

7. Any decimals, such as 1.5 and 2.83, are repeating decimals because
1.5 = 1.5000… and 2.83 = 83000… are called zero repeating
decimals.

8. Writing repeating decimals as fractions
Case 1. Zero Repeating Decimals
1 3 6 . 4 2 7
integer part
decimal point
decimal part
Place Value
Integer Decimal
… Hundreds Tens Ones Tenths Hundredths Thousandths Ten-thousandths …
… 102 10 1 1
10
1
102
1
103
1
104
…

9. The number 136.427 can be written in the expanded form as follows:
136 . 427
1 is in the hundreds place 1 × 102
3 is in the tens place 3 × 10
6 is in the ones place 6 × 1
4 is in the tenths place 4 ×
1
10
2 is in the hundredths place 2 ×
1
102
7 is in the thousandths place 7 ×
1
103
= 1 × 102
+ 3 × 10 + 6 × 1 + 4 ×
1
10
+ 2 ×
1
102
+ 7 ×
1
103

10. Write the following numbers in the expanded form:
1) 125.2
2) 28.313
3) 0.576
= 1 × 102 + 2 × 10 + 5 × 1 + 2 ×
1
10
= 2 × 10 + 8 × 1 + 3 ×
1
10
+ 1 ×
1
102
+ 3 ×
1
103
= 5 ×
1
10
+ 7 ×
1
102
+ 6 ×
1
103

11. The number 0.37 can be converted into fraction as follows:
0.37 = 3 ×
1
10
+ 7 ×
1
102
=
3
10
+
7
100
=
30
100
+
7
100
=
37
100

12. Write the following numbers as fractions.
1) 0.5
2) 3.25
= 5 ×
1
10
=
5
10
=
1
2
= 3 × 1 + 2 ×
1
10
+ 5 ×
1
102
= 3 +
2
10
+
5
100
=
300
100
+
20
100
+
5
100
=
325
100

13. Write the following numbers as fractions.
3) 12.379 = 1 × 10 + 2 × 1 + 3 ×
1
10
+ 7 ×
1
102 + 9 ×
1
103
= 10 + 2 +
3
10
+
7
100
+
9
1000
=
10000
1000
+
2000
1000
+
300
1000
+
70
1000
+
9
1000
=
12379
1000

14. Write the following numbers as fractions.
1) 0.5
2) 3.25
= 5 ×
1
10
=
5
10
=
1
2
= 3 × 1 + 2 ×
1
10
+ 5 ×
1
102
= 3 +
2
10
+
5
100
=
300
100
+
20
100
+
5
100
=
325
100
0.5 is a decimal with one decimal place. When
it is written as fraction, the numerator is equal
to the same number without the point and the
denominator is 10.
3.25 is a decimal with two decimal
places. When it is written as
fraction, the numerator is equal to
the same number without the point
and the denominator is 100.

15. Write the following numbers as fractions.
0.8
0.37
1.382
2.2545
= 8
10
= 37
100
= 1382
1000
= 6951
500
= 22545
10000
= 4509
2000

16. Write the following numbers as fractions.
1. 3.7
2. 4.82
= 37
10
= 482
100
= 241
50
3. 0.1257 = 1257
10000

17. Writing repeating decimals as fractions
Case 2. Non-zero Repeating Decimals
Repeating decimals that are not zero repeating decimals can be
converted into fractions as follow:
1. Write 0.7 as a fraction.
Let 𝑁 = 0. 7 = 0.777 … (1)
Multiply both sides of equation (1) by 10
10𝑁 = 7.777 … (2)
From equations (2) and (1), we have,
10𝑁 − 𝑁 = 7.777 … − (0.777 … )
7.777…
0.777…
7.000…
9𝑁 = 7
𝑁 =
7
9
0. 7 =
7
9
0. 7 is a repeating decimal with one
digit repeated from the first
decimal place. The number used for
multiplication is 10.

18. 2. Write 0.42 as a fraction.
Let 𝑁 = 0. 42 = 0.4242 … (1)
Multiply both sides of equation (1) by 10
10𝑁 = 7.777 … (2)
From equations (2) and (1), we have,
10𝑁 − 𝑁 = 4.2422 … − (0.4242 … )
The difference is complicated and
cannot be written as a fraction easily.
Multiply both sides of equation (1) by 100
100𝑁 = 42.4242 … (3)
From equations (3) and (1), we have,
100𝑁 − 𝑁 = 42.4242 … − (0.4242 … )
42.4242…
0.4242…
42.0000…
99𝑁 = 42
𝑁 =
42
99
0. 42 =
42
99
=
14
33
0. 42 is a repeating decimal with
two digits repeated from the first
decimal place. The number used for
multiplication is 100.

19. 3. Write 0.348 as a fraction.
Let 𝑁 = 0.348 = 0.348348 … (1)
Multiply both sides of equation (1) by 10
10𝑁 = 3.48348 … (2)
From equations (2) and (1), we have,
10𝑁 − 𝑁 = 3.483483 … − (0.348348 … )
The difference is complicated and cannot be
written as a fraction easily.
Multiply both sides of equation (1) by 100
100𝑁 = 34.8348 … (3)
From equations (3) and (1), we have,
100𝑁 − 𝑁 = 34.834834 … − (0.348348 … )
The difference is complicated and cannot be
written as a fraction easily.
Multiply both sides of equation (1) by 1000
1000𝑁 = 348.348348 … (4)
From equations (4) and (1), we have,
1000𝑁 − 𝑁 = 348.348348 … − (0.348348 … )
348.348348…
0.348348…
348.000000…
999𝑁 = 348
𝑁 =
348
999
0.348 =
348
999
=
116
333
0.348 is a repeating decimal with
three digits repeated from the first
decimal place. The number used for
multiplication is 1000.

20. 0. 7 is converted into a fraction as
7
9
.
That is, if there is a repeating decimal with one digit repeated from the
first decimal place, when it is written as a fraction, the denominator is 9
and the numerator is the repeated digit.
0. 42 is converted into a fraction as
42
99
.
That is, if there is a repeating decimal with two digits repeated from the
first decimal place, when it is written as a fraction, the denominator is 99
and the numerator are the repeated digits.
0. 348 is converted into a fraction as
348
999
.
That is, if there is a repeating decimal with three digits repeated from the
first decimal place, when it is written as a fraction, the denominator is 999
and the numerator are the repeated digits.

21. From the previous method, the repeating decimals with non-zero
repeating digits can be written as fractions as follow:
0.2
0. 27
=
2
9
=
27
99
=
3
11
3.5
1. 257
= 3
5
9
= 1
257
999

22. Write the following repeating decimals as fractions.
1. 0.3
2. 0. 12
3. 0. 175
4. 1.6
5. 2. 163
6. 10.25
=
3
9
=
1
3
=
12
99
=
4
33
=
175
999
= 1
6
9
= 1
2
3
= 2
163
999
= 10
25
99

23. 4. Write 0.62 as a fraction.
Let 𝑁 = 0.62 = 0.6222 … (1)
Multiply both sides of equation (1) by 10
10𝑁 = 6.222 … (2)
Multiply both sides of equation (1) by 100
100𝑁 = 62.222 … (3)
From equations (3) and (2), we have,
100𝑁 − 10𝑁 = 62.222 … − (6.222 …)
90𝑁 = 56
𝑁 =
56
90

24. 5. Write 0.637 as a fraction.
Let 𝑁 = 0.637 = 0.63777 … (1)
Multiply both sides of equation (1) by 100
100𝑁 = 63.777 … (2)
Multiply both sides of equation (1) by 1000
1000𝑁 = 637.777 … (3)
From equations (3) and (2), we have,
1000𝑁 − 100𝑁 = 637.777 … − (63.777 …)
900𝑁 = 574
𝑁 =
574
900
=
287
450

25. 6. Write 0.625 as a fraction.
Let 𝑁 = 0.625 = 0.62525 … (1)
Multiply both sides of equation (1) by 10
10𝑁 = 6.2525 … (2)
Multiply both sides of equation (1) by 1000
1000𝑁 = 625.2525 … (3)
From equations (3) and (2), we have,
1000𝑁 − 10𝑁 = 625.2525 … − (6.2525 …)
990𝑁 = 619
𝑁 =
619
990

26. 625 − 6
990
637 − 63
900
From example 5, write 0.637 as a fraction.
62 − 6
90
From example 4, write 0.62 as a fraction.
We have
56
90
=
Observe that the subtrahend is 6 which
is the digit that is not repeated
It is found that there is one 9 when
there is only one digit that is repeated
It is found that there is only one 0 when
there is only one digit that is not repeated
We have
574
900
=
Observe that the subtrahend is 63
which is the digit that is not repeated
It is found that there is one 9 when
there is only one digit that is repeated
It is found that there are two 0s when
there are two digits that are not repeated
From example 6, write 0.625 as a fraction.
We have
619
990
=
Observe that the subtrahend is 6 which
is the digit that is not repeated
It is found that there are two 9s when
there are two digits that are repeated
It is found that there is only one 0 when
there is only one digit that is not repeated

27. Write the following repeating decimals as fractions.
0.74 =
74 − 7
90
=
67
90
0.234 =
234 − 23
900
=
211
900
1.3642 = 1
3642 − 36
9900
= 1
3606
9900
= 1
601
1650

28. Write the following repeating decimals as fractions.
0.63
0.572
0.572
1.34
2.237
3.534
=
63 − 6
90
=
57
90
=
572 − 57
900
=
515
900
=
572 − 5
990
=
567
990
= 1
34 − 3
90
= 1
31
90
= 2
237 − 23
900
= 2
214
900
= 3
534 − 4
990
= 3
529
990