Repetições de decimais

Converting Repeating Decimals to Fractions

As with converting a non-repeating (i.e., "normal") decimal to a fraction, there are a few simple steps to follow to convert repeating decimals to fractions.

Think of a repeating decimal like a caterpillar. It's pretty gross looking, you don't really want to touch it, and it's made up of repeating parts. Same thing as a repeating decimal, at least to me.

Then, think of the fraction like a beautiful butterfly. Definitely preferable, much easier to handle. Our job is to get the caterpillar to transform into a butterfly.

Let's walk through the steps of this conversion using the repeating decimal below:

Even though our decimal looks short here, you should think of it like $0.18181818181818181818...$. Don't be fooled.

Notice that we have $2$ equations now. They are the first and last lines in the blue boxes above:

Repeating Decimals

We have learned about decimals (or decimal numbers) in general, but what about repeating decimals? Repeating decimals are those which go on forever but have a pattern. They are also called recurring decimals.

Examples

Here are some examples of repeating decimals. The part that infinitely repeats is boxed. $$1.\boxed{01}01010101010\cdots$$ $$15.18\boxed{3}333333333\cdots$$ $$-213.233\boxed{78943}7894378943\cdots$$

Notation

Sometimes, you will see a bar above the infinitely-repeating component of the repeating decimal, signifying that it is infinitely-repeating. This notation makes it clear that you are dealing with a repeating decimal. For example: $$1.\boxed{01}01010101010\cdots = 1.\overline{01}$$ $$15.18\boxed{3}333333333\cdots = 15.18\bar{3}$$ $$-213.233\boxed{78943}7894378943\cdots = -213.233\overline{78943}$$

It's also easier to write the numbers this way. If only calculators always did that.

Terminating Decimals

The following number is not a repeating decimal: $$35.908\boxed{0}000000\cdots$$ Notice that the part that infinitely repeats is completely composed of $0\text{s}.$ We could have also written the number like this: $$35.908$$ In cases like this, the number is referred to as a terminating decimal. Think about it—terminating means ending. Terminating decimals do not go off into infinity...they terminate or end.

Converting Repeating Decimals to Fractions

Believe it or not, repeating decimals are rational numbers. We can convert these to fractions just like any other decimal. Here is an explanation on how to convert repeating decimals to fractions.

(Video) Converting Repeating Decimals to Fractions

by Art of Problem Solving

Here's a video by the Art of Problem Solving where he teaches you how to turn a repeating decimal into a fraction.

$\large{\text{Summary}}$

So you've mastered converting decimals into fractions but there's still one type of decimal that won't convert: repeating decimals.

Suppose we have the decimal $0.\bar{6}$. The bar over the $6$ tells us that the $6$ repeats forever. $$0.\bar{6}=0.666666666\dots$$

This can be challenging to work with since we can't really write it down fully. The first step is to set it equal to a variable. This way we can reference it easier. $$x=0.666666\dots$$

Next, we can multiply this by $10$. All that does is move the decimal point over, so we don't actually have to work through multiplying an infinite amount of decimal places. $$10x=6.666666\dots$$

Here's the trick! We can subtract these two. This will cancel out any repeating portion of our decimal. $$\begin{align} 10x&=6.666666\dots\\ -\quad x&=0.666666\dots\\ \hline (10x-x)&=(6.66{\dots}-0.66\dots)\\ 9x&=6 \end{align}$$

Finally, we just solve for our variable, $x$. This involves dividing both sides by $9$ to isolate the $x$. $$\begin{align} 9x&=6\\ x&=\frac69\\ x&=\boxed{\frac23} \end{align}$$

What if our decimal is now: $$0.\bar{36}=0.36363636\dots$$ If do the exact same steps above, it won't work. Instead we need to multiply the decimal by $100$. Why? We need the decimal point to move over so that the repeating portions end up canceling when we subtract.

Multiply by a power of $10$ with zeros equal to the number of digits being repeated.

Since our decimal has two digits being repeated, $36$, we multiply it by a power of ten with two zeros: $100$. $$100x=36.363636\dots\\ x=0.363636\dots$$ Now we can follow the steps from before. Subtract these equations. $$\begin{align} 100x&=36.3636\dots\\ -\quad x&=\phantom{3}0.3636\dots\\ \hline (100x-x)&=(36.3636{\dots}-0.3636\dots)\\ 99x&=36 \end{align}$$ Finally, we solve for $x$. $$\begin{align} 99x&=36\\ x&=\frac{36}{99}\\ x&=\boxed{\frac{4}{11}} \end{align}$$