3.03 Converting between fractions and decimals
What is an irrational number?
A rational number is a number which can be written as a fraction where both the numerator and denominator are integers. An irrational number is a number which cannot be written as a fraction of two integers.
We can write any terminating or recurring decimal as a fraction, therefore these are rational numbers. However, decimals which are neither terminating nor recurring are irrational numbers.
You might be familiar with one irrational number already: $\pi$π. Like all other irrational numbers, $\pi$π really does go on forever without repeating itself. We say therefore that it doesn't terminate, or repeat.
Another number that is famously irrational is $\sqrt{2}$√2. In fact, the square root of most numbers are irrational. If a root is irrational it is called a surd. The square roots of perfect squares are rational, $\sqrt{1},\sqrt{4},\sqrt{9},\dots$√1,√4,√9,….
Practice question
Question 1
Is $\sqrt{35}$√35 rational or irrational?
Rational
AIrrational
B
But how can we write irrational numbers as decimals?
Converting fractions to decimals
Think about the names of the columns in our place value table: tenths, hundredths, thousandths and so on. To change a fraction to a decimal, it's easiest if we have a denominator that is a power of $10$10 so that it matches the fractions in the place value table.
Worked examples
Example 1
Write the fraction $\frac{7}{10}$710 as a decimal.
Think: This fraction can be read as "seven tenths". We can think about how we would write this in a place value table:| Ones | $\cdot$· | Tenths |
|---|---|---|
| $\cdot$· |
Example 2
Write the fraction $\frac{123}{100}$123100 as a decimal.
Think: We can split this fraction up into a sum of ones, tenths, and hundredths, to make it easier to fill in a place value table.
Do: We can rewrite $\frac{123}{100}$123100 as the sum $\frac{100}{100}+\frac{20}{100}+\frac{3}{100}$100100+20100+3100 which simplifies to $1+\frac{2}{10}+\frac{3}{100}$1+210+3100. We can now show this using a place value table.
| Ones | $\cdot$· | Tenths | Hundredths |
|---|---|---|---|
| $1$1 | $\cdot$· | $2$2 | $3$3 |
What if we have a denominator that is not a power of 10?
If we have a fraction that does not have a power of $10$10 in the denominator, then we want to find an equivalent fraction that does. We can follow these simple steps:
- Find a suitable number to multiply the denominator by to make it a power of $10$10 if it isn't already
- Multiply both the numerator and denominator by this number
- Using a place value table or otherwise, write out the number in decimal form
Worked example
example 2
Write the fraction $\frac{4}{5}$45 as a decimal.
Think: We first want to convert $\frac{4}{5}$45 to a fraction with a denominator of $10$10. Notice that we can multiply $5$5 by $2$2 to make $10$10.
Do: First we will multiply the numerator and denominator by $2$2:
| $\frac{4}{5}$45 | $=$= | $\frac{4\times2}{5\times2}$4×25×2 |
| $=$= | $\frac{8}{10}$810 |
Now that we have a value of $8$8 tenths, we can easily convert to the decimal $0.8$0.8.
Reflect: By converting the fraction to an equivalent fraction with a denominator of $10$10, we can easily convert the fraction to a decimal using our knowledge of the place value table.
Converting decimals to fractions
We know how to convert between decimals and fractions with powers of $10$10 in the denominator by using a place value table. This means we can very easily convert any decimal to a fraction with a denominator of $10,100,1000$10,100,1000 and so on. We normally want to write our fractions in simplified form, so we may need to then further simplify the fraction by cancelling common factors from the numerator and denominator.
Worked example
example 3
Write the decimal $0.32$0.32 as a simplified fraction.
Think: We can write $0.32$0.32 in a place value table. Since there are two numbers after the decimal point, we will have a denominator of $100$100. Once we have written the number as a fraction over $100$100 we can see if it can be further simplified.
Do: First we want to write $0.32$0.32 in a place value table:
| Ones | $\cdot$· | Tenths | Hundredths |
|---|---|---|---|
| $0$0 | $\cdot$· | $3$3 | $2$2 |
We can see that it is made up of $3$3 tenths and $2$2 hundredths. This is equivalent to $32$32 hundredths. This means $0.32=\frac{32}{100}$0.32=32100.
Now, we can see that as both the numerator and denominator are even we could divide them both by $2$2 to simplify the fraction. In fact, $32$32 and $100$100 have a highest common factor of $4$4 so we can divide both the numerator and denominator by $4$4.
| $\frac{32}{100}$32100 | $=$= | $\frac{32\div4}{100\div4}$32÷4100÷4 |
| $=$= | $\frac{8}{25}$825 |
So the simplified fraction is $\frac{8}{25}$825.
Reflect: By considering the place values of the digits in $0.32$0.32 we can convert the decimal to a fraction with a denominator of $100$100, and then we can simplify the fraction to arrive at the conversion $0.32=\frac{8}{25}$0.32=825.
Benchmarks
A benchmark is a reference point which we can use to make converting between various forms easier. There are a few common fractions and decimals that we use as benchmarks that are worth keeping in mind.
From the place value table, we know that $0.1=\frac{1}{10}$0.1=110, but what is the decimal equivalent of $\frac{1}{2}$12 and $\frac{1}{4}$14? Well, $\frac{1}{2}$12 we know is equivalent to $\frac{5}{10}=0.5$510=0.5.
$\frac{1}{4}$14 or "one quarter" is exactly half of one half, so if one half is $50$50 hundredths, then half of that must be $25$25 hundredths, or $\frac{25}{100}=0.25$25100=0.25.
Using these benchmarks we can see that $\frac{3}{4}=0.75$34=0.75 as it is three groups of one quarter, or $3\times0.25=0.75$3×0.25=0.75.
Practice questions
Question 2
Write the fraction $\frac{47}{100}$47100 as a decimal.
Question 3
Write the fraction $\frac{17}{25}$1725 as a decimal.
Question 4
Write the decimal $0.29$0.29 as a simplified fraction.
Question 5
Write the decimal $0.535$0.535 as a simplified fraction.