Question

Determine whether the number 0.121212… is rational or irrational. What is its fractional equivalent?

Answer 1

`12/99` and the number is rational

Explanation:

we can show `0.121212` as `0.(12)` It means that `12` after the point repeat again and again. It implies that this number is not rational.
Its fractional equivalent is:
`0.(12) = 0 + 12/99`
We have to write the repeated number `(12)` on the numerator and we have to write `9` as many as the number of the repeated number on the denominator.
The answer is `12/99=4/33`

Answer 2

Rational. `12/99=4/33`

Explanation:

We can find a fraction for the decimal this way:

First let's take the original decimal:

`1=>0.bar12`

and then multiply it by 100:

`100=>12.bar12`

and now subtract the two:

`100=>color(white)(0)12.bar12`
`ul(-1=>-0.bar12`
`99=>12`

So up to this point what I've said is that when there is 1 of the decimal, it's `0.bar12`, when there's 100 of them, it's `12.bar12` and when there's 99 of them, it's 12. And so we can divide the 12 by the 99 to get a fractional representation of the repeating decimal:

`12/99`

We can then reduce this:

`12/99=(3xx4)/(3xx33)=3/3xx4/33=4/33`

Now that we have a fraction, let's talk about what it means for a number to be rational and what it means for it to be irrational.

A rational number, quite simply, is one that can be expressed using a fraction of integers. For example, `1/2` is rational because we have the integers 1 and 2 in a fraction. The number 2 is also rational because we can express it as a fraction of two integers: `2/1`. For that same reason, `12/99` is rational.

An irrational number is one that cannot be expressed as a fraction of integers. `sqrt2 and pi` are both well known examples (for `pi`, people do use a decimal or a fraction as an approximation, but it's an approximation and not the actual number).