Questions from middle schoolers VI: Irrational numbers

How do they know irrational numbers never repeat?

What a lovely question. When you are told that “irrational numbers have decimal representations that never repeat,” it’s a good instinct to say, “That’s just because you haven’t looked hard enough.”

There are only 10 digits, right? So they must repeat eventually. There are only so many ways to arrange 10 digits. So it makes sense that they don’t repeat very often, but they must repeat eventually, right?

Wrong.

And the way we know they don’t repeat isn’t what you expect. You expect that mathematicians have looked, but not found patterns in the digits of irrational numbers, and that from this they conclude that the digits don’t repeat. You are correct in being suspicious of this argument.

But it’s not the one mathematicians use.

Instead, we know that all repeating (or terminating) decimals are rational. And we know that rational numbers have other properties (like that they can be simplified-or reduced-to a fraction with whole number numerator and denominator). And then we know that a number such as pi or the square root of 2 cannot be reduced to a fraction with whole number numerator and denominator. Therefore pi and the square root of 2 must have decimals that don’t repeat (or terminate…because if they did, they would be rational, which would mean they could be reduced which they cannot).

This is common in mathematics-knowing that something is true in a roundabout way.

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