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

Apr 17, 2018

$\frac{12}{99}$ and the number is rational

#### Explanation:

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

Rational. $\frac{12}{99} = \frac{4}{33}$

#### Explanation:

We can find a fraction for the decimal this way:

First let's take the original decimal:

$1 \implies 0. \overline{12}$

and then multiply it by 100:

$100 \implies 12. \overline{12}$

and now subtract the two:

$100 \implies \textcolor{w h i t e}{0} 12. \overline{12}$
ul(-1=>-0.bar12
$99 \implies 12$

So up to this point what I've said is that when there is 1 of the decimal, it's $0. \overline{12}$, when there's 100 of them, it's $12. \overline{12}$ 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:

$\frac{12}{99}$

We can then reduce this:

$\frac{12}{99} = \frac{3 \times 4}{3 \times 33} = \frac{3}{3} \times \frac{4}{33} = \frac{4}{33}$

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, $\frac{1}{2}$ 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: $\frac{2}{1}$. For that same reason, $\frac{12}{99}$ is rational.

An irrational number is one that cannot be expressed as a fraction of integers. $\sqrt{2} \mathmr{and} \pi$ 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).