How to predict how long the repeating decimal and the non-repeating part are going to be?
- Wednesday Oct 28,2009 12:59 PM
- By diddy
- In Others
With a repeating decimal (recurring decimal), how can you predict how long the the repeating decimal and the non-repeating part are going to be?
Like, with 0,0142857142857 the repeating part is 6 digits long (142857) and the nonrepeating part 1 digit long (the 0 behind the ,).
I only know how to predict the repeating part of a cyclic number.
142857, 6 Digits, Recurring Decimal, Repeating Decimal
2 Comments
Since you know how to predict the repeating part of a repeating decimal:
For the non-repeating part, look at the powers of 2 and 5 in the denominator. The larger of theses exponents will tell you how long the non-repeating part of the decimal is. (Why 2 and 5? They are factors of 10, which is the base for the decimal system.)
Example: 1/720
720 = 2^4 * 3^2 * 5.
Since there are four 2’s and 1 5, there must be 4 nonrepeating decimal places in the decimal expansion.
Double check: 0.001388888…
I hope this helps!
I’m not sure that I follow what you’re asking. The example you have given is the fraction 1/70. In this case the six digits will recur indefinitely whilst the 0 will appear just once. This is because any zero’s merely decide where the recurring six digits will start. So if you had picked the fraction 1/7 then we would have 0.142857. If you had picked 1/700 then the decimal equivalent would be 0.00142857. In this case the two zero’s would occur just once, at the start and the six digits (142857) would recur indefinitely.
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