Question #0cf9a

1 Answer
Jul 3, 2016

Answer:

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.