Back in Ms. Stanton’s class, what stopped us from looking at decimals that neither terminate nor repeat periodically? It’s easy to cook up such stomach-churners. Here’s an example: 0.12122122212222 . . . By design, the blocks of 2 get progressively longer as we move to the right. There’s no way to express this decimal as a fraction. Fractions always yield decimals that terminate or eventually repeat periodically — that can be proven — and since this decimal does neither, it can’t be equal to the ratio of any whole numbers. It’s irrational.
