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).