A number has infinite decimal expansion 0.45445444544445, where the number of fours
between the fives continues to increase. Is this a rational number?
The decimal form of rational numbers either terminates, that means they stop, or they
repeat. First of all, this decimal expansion never terminates. We know that is an infinite decimal expansion. The only way this could be a rational number is if it is repeating.
It’s really tempting to look at all these fours and fives and think that this is
repeating. It’s not repeating four five. You could think it was repeating the value four five four. Then again, it breaks that pattern.
This decimal value has no set of repeating values. And that means it’s not a rational number.