All rational numbers
fall into two categories, terminating
. Terminating decimals are those said to have a definite last digit, such as -10, 2.5 or 1.793. Repeating decimals have a digit or series of digits that repeats ad infinitum
, and include 0.673333... and 2.142857142857....
Although repeating decimals can be expressed in the ellipsis notation that I just used, there are easier ways. One such way is to put a vinculum bar over the repeating digits. Thus, 0.67333... and 2.142857... become
0.673 and 2.142857
respectively. Another involves placing a dot over the first and last digits in the sequence. Using this format, the same two examples become:
. . .
0.673 and 2.142857
In spoken English, the decimal would be called "two point one four two eight five seven repeating." Or, of course, you could just write out the fraction.
To find the fraction, follow these steps:
- Subtract the decimal's integral part. (Thus, 2.142857 repeating becomes 0.142857 repeating.) We will call this number n.
- Multiply n by a power of ten until the difference between n and it is a terminating decimal. This number can be expressed as a multiple of n. (In the example above, we would have 142,857.142857 repeating = 1,000,000n.)
- Subtract the two values to get a number equal to a multiple of n. (We now have 142,857 = 999,999n.)
- Now divide to find the value of n. (In the example, n = 142,857/999,999 = 1/7.)
- Finally, add the integer you subtracted at the beginning. The decimal can be expressed as a fraction or mixed number. (We now have 2+1/7 or 15/7 as our final answer.)
What makes a decimal repeating? A decimal is repeating when the denominator of its simplest fractional form cannot be decomposed into 2's and 5's; it is terminating, in other words, if its denominator is a factor of a power of ten. Thus, 387/1000 is terminating, but 2801/3600 is not. Of course (as N-Wing reminds me), in other bases than decimal, an n-mal will terminate with a denominator which is a factor of a power of n. Thus, a bimal will only terminate with a denominator which is a power of 2
But be warned! All terminating decimals can be expressed as a repeating decimal: even benign 2.5 can become 2.5000000... or 2.4999999.... While this may seem surprising, it is quite true and even useful—Cantor himself used it to show that the real numbers are uncountable.