Rational and Irrational Numbers

I have a sixth grader now.

A rational number is a number that can be expressed as a fraction where the denominator ≠ 0.

That's fine, but numbers expressed as decimals may not intuitively qualify or disqualify as rational numbers, so I'm here to clear this up in a jiffy.

If the decimal stops, the number in question is a rational number.

For example, 12.34567890000 stops after the digit 9.

That number is the same as 12 and 3,456,789/10,000,000.

We can all see that "12 and 3,456,789/10,000,000" could be expressed as a fraction.

If the decimal ends in a repeating pattern, the number in question is a rational number.

For example, .9̅ is a rational number.

This is how we know.

We can define .9̅ as some variable.  Let's choose x.

Then we can construct an equation to subtract all of the repeating digits.

In this case, (10x - x) does the trick.

To elaborate, 10x - x = 9.9̅ - .9̅ = 9.

And that means that 9x = 9.

And x = 1, which is rational.

Does that mean .9̅ = 1?

You bet.

Here's another example.

12.34567̅8̅9̅ is a rational number.

Let's call it y.

(10,000,000y - 10,000y) = 123,456,789.7̅8̅9̅ - 123,456.7̅8̅9̅

So 9,990,000y = 123,333,333

And y = 123,333,333/9,990,000.

Rational.

If the decimal does not stop and does not end in a repeating pattern, then it cannot be expressed as a fraction, and this is why we have the set called "irrational numbers."  The most famous irrational numbers are the square root of 2 (√2), pi (π), the golden ratio (φ), and Euler's number (e), but you could start making up any irrational number you wanted, and you could make it exactly as big or as small as you wanted to, so long as you never stopped defining it.  (Once you stop, it becomes rational.)

They don't have to be famous to be irrational, and there are just as many irrational numbers as you can imagine there are.  Depending on your imagination, maybe even more.