2.01 Convert fractions to decimals
Introduction
When we have a common fraction like \dfrac{1}{4} or \dfrac{1}{2}, we can often recall what the decimal form is. Such as \dfrac{1}{4} = 0.25 and \dfrac{1}{2} = 0.50. We also learned how to convert fractions with a denominator of 100 or 1000 . For example, we know that because 0.35 goes to the hundredths place, we can rewrite that as the fraction \dfrac{35}{100}.
Let's now look at what to do when we don't have a common fraction or a denominator that is a power of 10.
Convert fractions to decimals
Calculators can make converting fractions to decimals quite simple. In the following Exploration, divide the numerator by the denominator to do the conversions.
Exploration
Using the division button on a calculator, find the corresponding decimal representations of the following fractions:
| Fraction | \dfrac{1}{2} | \dfrac{1}{3} | \dfrac{1}{4} | \dfrac{1}{5} | \dfrac{1}{6} | \dfrac{1}{7} | \dfrac{1}{8} | \dfrac{1}{9} | \dfrac{1}{10} | \dfrac{1}{11} |
|---|---|---|---|---|---|---|---|---|---|---|
| Decimal |
What two types of decimals do you see?
It's important to note that when rational numbers are converted to decimals, one of two things happens:
The decimal terminates (ends in 0).
The decimal repeats (e.g. \dfrac{1}{3}=0.33333\ldots, repeating forever).
We use the ellipsis \left(\ldots\right) to represent repeating decimals. To use ellipsis, we usually write the first few decimal places to show which are repeating. Another way to write repeating decimals is to use a bar above the repeating numbers. Let's take a look at the following examples:
| \displaystyle 0.33333333 \ldots | \displaystyle = | \displaystyle 0.\overline{3} |
| \displaystyle 0.21111111 \ldots | \displaystyle = | \displaystyle 0.2\overline{1} |
| \displaystyle 0.45454545 \ldots | \displaystyle = | \displaystyle 0.\overline{45} |
| \displaystyle 0.03790379 \ldots | \displaystyle = | \displaystyle 0.\overline{0379} |
When we are not using a calculator, fractions that have powers of 10, or numbers which can easily be multiplied to be a power of 10, in the denominators, can easily be converted to decimals using the fact that \dfrac{1}{100} = 0.01.
But what do we do when we have a fraction that isn't a power of 10, like the fraction \dfrac{5}{6} ?
We can use long division to convert fractions into decimals by dividing the numerator by the denominator. Let's look at the following question to see an example.
Examples
Example 1
Convert \dfrac{5}{6} to a decimal. Round your answer to the nearest hundredth.
Example 2
Convert \dfrac{63}{28} to a decimal.
For fractions where the denominator is not a power of 10, nor can it be multiplied easily to be a power of 10, we can use long division to convert into a decimal.
When rational numbers are converted to decimals, one of two things happens:
The decimal terminates (ends in 0).
The decimal repeats.