Rational numbers, when written as decimals, are either terminating or non-terminating, repeating decimals. Converting terminating decimals into fractions is straightforward: multiplying and dividing by an appropriate power of ten does the trick. For example, 2.556753=10000002556753. However, when the decimals are repeating, things are a little more difficult. Repeating decimals occur very frequently both when doing simple arithmetic and when solving competition problems, so being able to convert them to fractions is a valuable skill.
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
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
False
True or False?
0.9999…=1
Note: The "…" indicates that there are infinitely many 9's.
The correct answer is: True
Intuition:
Visualize a number line. If two real numbers x and y on the number line are different, then we should see some space between them. In fact, there would be room to place another real number, namely their average 2x+y. Since no number exists between 0.999… and 1, it must be that they are the same.
Easy Way to Convert Irrational Decimals to Fractions
Problem Solving
0.70.7ˉ97134997
None of the above choices
0.42−0.35=?
Note:0.ab=0.abababab…
The correct answer is: 997
A=0.19+0.199,B=0.19×0.199
Recall that 0.19, for example, stands for the repeating decimal 0.19191919... and that the period of a repeating decimal is the number of digits in the repeating part. In this case, the period of 0.19 is 2.
Find the sum of the periods of A and B.
The correct answer is: 60
810n=0.9D5=0.9D59D59D5...
Suppose n and D are integers with n>0 and 0≤D≤9 satisfying the above equation. Determine the value of n+D.