Is 0.121212... rational or irrational

Step by step

The question

Question: is 0.121212… rational or irrational?

Rational means: it can be written as a fraction, with whole numbers on top and at the bottom.

Then its decimal stops, or repeats for ever.

Irrational means: no fraction works.

Its decimal never stops and never repeats.

1 Look at the decimal

The block 1 2 repeats for ever.

A decimal that repeats is rational.

Careful!

It goes on for ever, but it repeats.

Irrational decimals never repeat.

2 Write it as a fraction

A repeating block of 2 digits goes over 99.

So 0.121212 repeating is 12/99, which makes simpler: 4/33.

So 0.121212… is rational.

It is a fraction of whole numbers.

3 Check

Let's check: 4 divided by 33.

We get 0.121212, and so on.

The same decimal.

So the answer is right!

Let's look back.

First: look at the decimal.

Then: write it as a fraction.

Last: check.