How can a repeating decimal be a rational number?

1 Answer
Mar 24, 2018

Answer:

A rational number is any number that can be written as a fraction with an integer over a denominator which is also an integer.

Explanation:

For example...

Irrational number is #pi ~~ 3.1415926535...,# and since nothing repeats, it cannot be written as one integer over a denominator as a fraction.

A rational number is #39.3939393939...or 369.3693693693... #

Any #2# numbers that repeat go over #99# in a fraction.

Any #3# numbers that repeat go over #999# in a fraction.

So #39.3939393939... #as a fraction would be #39 39/99#,

and #369.3693693693...# as a fraction would be #369 369/999#.

The simplest example is #1/3# which gives #0.333333...#

All recurring decimals come from dividing a fraction, so they are rational.