Recurring Decimals as Exact Fractions
A decimal that repeats forever equals an exact fraction. Multiplying by a power of ten and then subtracting lets you find it. This gives you a way to use the claim about rational numbers and repeating decimals instead of only accepting it.
What a learner can do afterwards
- Converts a recurring decimal such as 0.4545... to a fraction by subtracting a shifted copy of it
- Handles a decimal with a non-recurring part before the repeat, such as 0.16333...
- Predicts whether a fraction gives a terminating or non-terminating recurring decimal by reducing it to lowest terms and checking the denominator's prime factors
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You have seen that rational numbers have decimals that stop or repeat. Each rational number equals a fraction of two integers with a nonzero denominator. A terminating decimal stops. A recurring decimal has a block that repeats forever. Here, 0.1666... means that only the 6 repeats forever. Without dots or a stated repeat, 0.1666 is a finite decimal.
Reduce a fraction to lowest terms before testing its denominator. A positive denominator with only 2 and 5 as prime factors gives a terminating decimal. Any other prime factor gives a non-terminating recurring decimal. A denominator of 1 gives an integer. For example, 14/35 reduces to 2/5, so it terminates even though 35 contains a factor of 7.
To convert a terminating fraction to a decimal, make an equivalent fraction with denominator 10, 100, 1000 or another power of ten. Multiply the numerator and denominator by the same number. For 9/40, multiply both by 25: 9/40 = 225/1000 = 0.225. The denominator 1000 tells you to read the numerator as thousandths.
Let x stand for the entire recurring decimal. Here, 10x means 10 times x. If the repeat starts just after the point, multiply by 10 for a one-digit block, 100 for two digits, or 1000 for three. For x = 0.444..., 10x = 4.444... . Subtract the original equation from the shifted equation. The equal decimal tails cancel, leaving 10x - x = 4. Since 10x - x = 9x, divide both sides by 9: x = 4/9. Reduce the fraction if needed.
If digits come before the repeat, first move those digits left of the point. Count any zeros too. Then shift one full repeating block further. Subtract these two equations, whose tails now match. For x = 0.23555..., only 5 repeats. Use 100x = 23.555... and 1000x = 235.555... . Subtract: 900x = 212. Divide by 900 and reduce: x = 212/900 = 53/225.
The dots name the entire infinite decimal. They do not describe a finite calculation that eventually stops. Subtracting equal tails gives an exact equation, so the resulting fraction is exact. A finite decimal and an endless repeat can share starting digits without being equal. For example, 0.312 is less than 0.312312..., where 312 repeats forever.
Name it x, shift by the repeat, subtract the tail away, and simplify what remains.
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More to read
Watch the remainders
Division explains why a fraction's decimal can repeat. Consider 7/12. To find decimal digits, divide 70 by 12. This gives a digit of 5 and a remainder of 10. The next step divides 100 by 12, giving a digit of 8 and a remainder of 4. Then 40 divided by 12 gives a digit of 3 and another remainder of 4.
That remainder of 4 returns every time. Each following step therefore produces another 3. The result is 7/12 = 0.58333..., where only the 3 repeats forever.
A zero remainder ends the division. A returning remainder repeats the same calculations and digits. Keep the notation clear: 0.58333 is finite, while 0.58333... with the stated repeat includes endlessly many threes.
Why the reduced denominator matters
Every terminating decimal can be written as a fraction over a power of ten. For example, 0.25 = 25/100. Powers of ten have only 2 and 5 as prime factors.
Reduce the fraction before applying the denominator test. If its positive denominator is 1 or contains only factors of 2 and 5, its decimal terminates. If another prime remains, that denominator cannot divide a power of ten. Division does not end; its remainders eventually repeat.
For example, 21/84 reduces to 1/4. The factors 3 and 7 in the original denominator cancel. The reduced denominator is 4 = 2 x 2, so the decimal terminates: 1/4 = 25/100 = 0.25.
Checking an unreduced denominator can give the opposite prediction. Cancel common factors first.
A two-digit repeating block
Let x = 0.939393..., with 93 repeating forever. Since the block has two digits, multiply by 100:
x = 0.939393... 100x = 93.939393...
The decimal parts are equal. Subtract the original equation from the shifted one. The left side becomes 100x - x = 99x. The equal decimal tails cancel on the right, leaving 93.
Thus 99x = 93. Divide both sides by 99 to obtain x = 93/99. Dividing the numerator and denominator by 3 gives 31/33.
This is an exact fraction, not a rounded value. For a repeat starting just after the point, the shift moves one full block: one place for one digit, two places for two digits, and so on.
Count the digits before the repeat
In 0.038888..., only the 8 repeats forever. Two digits come before it: 0 and 3. The zero counts as a decimal place.
Let x equal this number. Multiply by 100 to move both non-recurring digits left of the point. Then shift the one-digit repeat one place further:
100x = 3.888... 1000x = 38.888...
The tails now match. Subtracting gives 1000x - 100x = 38 - 3, so 900x = 35. Divide by 900 to get 35/900. Reduce by 5 to obtain 7/180.
For a two-digit repeating block, the second shift would move two further places. Count the non-recurring digits and repeating digits separately, including zeros in either part.
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