Question

Question #0cf9a

Answer 1

Rational numbers have a finite or periodic decimal expansion, while irrational numbers have an infinite, non-periodic expansion

Explanation:

First let's answer the second part of the question:
`0.001=1/1000`, and
`0.0001=1/10000`
But the larger the denominator, the smaller the number, so the first number is greater than the second. In mathematical notation you have `0.0001<0.001`

What we want then is rational numbers `r` such that `0.0001 < r <0.001`. But the numbers:
`0.00012`
`0.00013`
`0.00014`
`0.00015`
`0.00016`
`0.00017`
are all rational because their decimal expansion is finite, and they are all between `0.0001` and `0.001`

Similarly, the number:
`0.0001234567891011121314 ...` is irrational and between `0.0001` and `0.001`. And, of course, the number `0.0001334567891011121314 ...` is irrational and between `0.0001` and `0.001`. Etc.