Some examples of non-terminating repeating decimals are 0.12121212121212… and 1.2354354354354…. We can represent these decimals in short as 0.12 and 1.2354, respectively.
To convert these types of decimals to fractions, we can view the decimal as the sum of (infinite) terms in a geometrical progression. This can be easily understood by some examples.
Write 0.34 as a fraction.
Proof 1:
We can write 0.34 as 0.3434343434…. Now let x=0.34, then
x=0.34+0.0034+0.000034+⋯=10034+1000034+100000034+⋯=34×(10011+10021+10031+⋯).
Recognize that this is the sum of infinite terms of a GP which has initial term a=1001 and common ratio r=1001. Since the sum of infinite terms is 1−ra, substituting the values of a and r gives x=34×1−10011001=34×991=9934. □
Proof 2:
Here is an alternative way to solve this problem: Let x=0.3434343434…, then 100x=34.343434…. On subtracting the first equation from the second, we have 99x=34⟹x=9934. □
Write 0.1 as a fraction.
We can write 0.1 as 0.1111111111…. Let x=0.1, then
x=0.1+0.01+0.001+⋯=101+1001+10001+⋯=1−101101=91. □
Write 0.023 as a fraction.
We can write 0.023 as 0.02323232323…. Let x=0.023, then
x=0.023+0.00023+0.0000023+⋯=100023+10000023+1000000023+⋯=100023×(1+1001+10021+⋯)=100023×1−10011=100023×99100=99023. □
Write 4.1454 as a fraction.
We can write 4.1454 as 4.1454454454454…. Let x=4.1454, then
x=4.1+0.0454+0.0000454+0.0000000454+⋯=1041+10000454+10000000454+10000000000454+⋯=1041+10000454×(1+10001+100021+⋯)=1041+10000454×1−100011=1041+10000454×9991000=1041+9990454=999041413. □
Which of the following is equal to 0.5+0.7?
(a) 1.2(b) 1.3(c) 1.23(d) 1.32
We can write 0.5 as 0.55555555…. Let x=0.5, then
x=0.5+0.05+0.005+0.0005+⋯=105+1025+1035+1045+⋯=105×(1+101+1021+1031+⋯)=105×1−1011=105×910=95.
Similarly, if we let y=0.7, then we can get y=97. Thus,
0.5+0.7=x+y=95+97=912=1+93=1.3.
Therefore, the answer is 1.3. □
The non-terminating, repeating decimal 3.91 can be written as a fraction a176. What is a?
Observe that
100x10x=391.1111111=39.1111111.(1)(2)
Taking (1)−(2) gives
90x=352⟹x=45176,
which implies a=45. □
True or False?
0.9999…=1
Note: The "…" indicates that there are infinitely many 9's.