For example, many beginner questions on Math SE about irrational numbers are based in the mistaken belief that an irrational number is one whose decimal represe

For example, many beginner questions on Math SE about irrational numbers are based in the mistaken belief that an irrational number is one whose decimal representation doesn't repeat. How is this a mistaken belief? Every rational number winds up in a repeating decimal representation and every number with a repeating decimal representation is a rational number. We learn algorithms to go back and forth between the two in elementary school. Therefore irrational numbers cannot have repeating decimal representations. Conversely numbers with decimal representations that don't wind up repeating cannot be rational and so must be irrational.