Does an irrational number have to go on forever? - Answers

Yes. Any number with a decimal representation that terminates or repeats must be a rational number. If it terminates, it's obviously rational. For example, 0.12345 is clearly 12345/100000, and the same is true for any other decimal; simply multiply it by a power of ten large enough to convert it to an integer, and then for the fraction put that integer over that power of ten. Repeating is a little trickier; it may be best to illustrate with a couple of examples. Take the decimal 0.3333.... This multiplied by ten is 3.333...., and subtracting the original number obviously gives 3 exactly, so 10x - 1x = 9x = 3; x = 3/9 which reduces to 1/3. If it repeats in an alternating sequence (0.545454...) it's necessary to use a larger power of ten: 100x - 1x = 99x = 54; x = 54/99 = 6/11. One or the other of these can be done for any terminating or repeating decimal expansion, therefore they must be rational.