What makes a decimal repeating?

Answer 1

When you divide a number (like in long division) and end up with a remainder that keeps coming up the same remainder as you carry the numbers into the decimal portion.

For example: 1 divide by 3: You start out 1.000 divide by 3, and 1 divided by 3 is zero, bring down the 1 and a zero and then you have 10 divided by 3 is 3 remainder 1, so now you have 0.3 remainder 1. If you continue this process you will keep getting more 3's and more remainder 1, until you realize that the 3's will repeat forever. (You will always get a 3 remainder 1, no matter how far you carry it out)

Some numbers give 2 or more digits that repeat, like dividing by 11 gives a pair of digits that repeat, while dividing by 7 gives a set of six digits that repeat.

Answer 2

Suppose you are dividing N by D. Then either D goes into N or it does not and you are left with a remainder. The remainder can only be 1, 2, 3, ... , D-1 = R1, say

At the next stage (in decimal numbers) you divide 10*R1 by D. Again, the remainder can only be 1, 2, 3, ... , D-1 = R2 say.

If R2 = R1 then you are repeating the division, and will get the same result. And if R2 <> R1 then continue to R3. And R4 etc. But there are only D-1 possible reminders and you must repeat Rk at or before k = D-1.

Answer 3

Suppose the simplified form of the fraction is p/q. Division by q can only leave q-1 different remainders: 1, 2, 3, ..., q-1. So, as you carry forward the division, when you reach the qth division, the remainder is either new or repeats one of those that appeared before.

The only new remainder is 0, in which case the decimal terminates. Otherwise, the remainder repeats one that appeared before, and then you are in a repeating cycle.

Answer 4

Any denominator which has a prime factor other than 2 or 5 will create a repeating decimal.