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

2 Answers
Apr 17, 2018

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

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).