Answer: No, it's not rational. It's irrational.
Reason:
There isn't a repeating pattern. The number of zeros grows from 1 zero to 2 zeros, to 3 zeros, and so on. And we also have 4s show up every once in a while. But there isn't a repeating pattern like 0.23232323...
This would make the number 0.040040004.... an irrational value.
All rational numbers convert to either
- a decimal number that has a repeating pattern of some kind, or,
- a terminating decimal
Some examples of rational numbers:
- 0.232323... = 23/99
- 0.1666666... = 1/6
- 0.7 = 7/10
Examples of irrational numbers:
- pi = 3.1415926535898..., this goes on forever without a repeated pattern
- sqrt(2) = 1.4142135623731..., same idea as above