# Question #0cf9a

Jul 3, 2016

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:
$0.001 = \frac{1}{1000}$, and
$0.0001 = \frac{1}{10000}$
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:
$0.0001234567891011121314 \ldots$ is irrational and between $0.0001$ and $0.001$. And, of course, the number $0.0001334567891011121314 \ldots$ is irrational and between $0.0001$ and $0.001$. Etc.