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7.08 Rational and irrational numbers

Lesson

Irrational numbers

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. Like all other irrational numbers, \pi 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}. 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},\,...

A diagram that shows the real number system with different sets of numbers. Ask your teacher for more information.

Examples

Example 1

Is \sqrt{35} rational or irrational?

Worked Solution
Create a strategy

Check whether the number can be written as a fraction.

Apply the idea

35 is not a perfect square, so \sqrt{35} will not equal a whole number.

Using a calculator, we can get\sqrt{35}=5.916079783099616...

The decimals are neither terminating nor recurring. This means that \sqrt{35} cannot be written as a fraction.

So \sqrt{35} is irrational.

Idea summary

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.

A surd is a square root which is irrational.

Convert decimals to fractions

What if we were given the recurring decimal 0.0\overline{12}=0.012\,121\,212\,... and asked to convert it into a fraction? We could type 0.012\,121\,212 into our calculator and press the button to convert it into a fraction, but most calculators won't do it, and even if they do, it will give you the fraction for the terminating decimal 0.012\,121\,212, not for the recurring decimal 0.0\overline{12}.

Fortunately, we can use some simple algebra to convert any recurring decimal to a fraction. To do this, we use a nice trick to get rid of the endless recurring part of the decimal.

Examples

Example 2

Let x=0.58\overline{3}.

By considering the value of 100x, write x as a fully simplified fraction.

Worked Solution
Create a strategy

Find 100x then multiply x by another power of 10 to get the same decimal places after the decimal point, then subtract to cancel out the decimal places.

Apply the idea

To find 100x:

\displaystyle x\times100\displaystyle =\displaystyle 0.58\overline{3}\times100Multiply both sides by 100
\displaystyle 100x\displaystyle =\displaystyle 58.\overline{3}Evaluate

So 100x=58.\overline{3} will be our first equation.

Now, we will multiply x=0.58\overline{3} by 1000 to get the same repeating digits after the decimal point.

\displaystyle x\times1000\displaystyle =\displaystyle 0.58\overline{3}\times1000Multiply both sides by 1000
\displaystyle 1000x\displaystyle =\displaystyle 583.\overline{3}Evaluate

So the second equation is 1000x=583.\overline{3}.

Subtracting the two equations, we have:

\displaystyle 1000x-100x\displaystyle =\displaystyle 583.\overline{3}-58.\overline{3}Set up the equation
\displaystyle 900x\displaystyle =\displaystyle 525Evaluate
\displaystyle \dfrac{900x}{900}\displaystyle =\displaystyle \dfrac{525}{900}Divide both sides by 900
\displaystyle x\displaystyle =\displaystyle \dfrac{7}{12}Simplify
Idea summary

A recurring decimal, also known as a repeating decimal, is a number containing an infinitely repeating number or series of numbers occurring after the decimal point.

Convert fractions to decimals

Consider one third. This can be written as a fraction \dfrac{1}{3}, and as a decimal we know it repeats forever as 0.333\,333... So if we want to do an exact calculation that includes \dfrac{1}{3}, we should keep it as a fraction throughout the calculation.

If we type 1\div3 into a calculator, and it would show us around 8 or 9 digits on the screen. This is now an approximation. 0.333\,333\,333\,3 is a good approximation of \dfrac{1}{3}, but even this has been rounded to fit on your calculator screen, so it is no longer the exact value.

If we were given \dfrac{7}{9} and asked to express it as a decimal, we could enter it into our calculator and get 0.777\,777\,777. Our calculator eventually runs out of space, but we know that we have a recurring decimal 0.\overline{7} and that the sevens in 0.777\,777\,777\,... go on forever.

To convert from a fraction to a decimal, we can rewrite the fraction as division expression. For example, \dfrac{3}{7} is 3 divided by 7. Then we can use short division, adding extra zeros as required to the numerator.

Examples

Example 3

Write the fraction \dfrac{7}{9} as a recurring decimal.

Worked Solution
Create a strategy

Use short division, and adding extra zeros as required to the numerator.

Apply the idea
A long division with 7 as the dividend and 9 as the divisor. Ask your teacher for more information.

Set up the short division including some trailing zeros.

A long division with 7 as the dividend and 9 as the divisor. Ask your teacher for more information.

9 does not go into 7 so we put a 0 above the 7 and copy the decimal point to the top.

9 goes into 70 seven times since 9\times 7 =63, so we put a 7 above the first zero and carry the remaining 7 to the next zero.

A long division with 7 as the dividend and 9 as the divisor. Ask your teacher for more information.

Again, 9 goes into 70 seven times so we put a 7 above the second zero and carry the remaining 7 to the next zero.

A long division with 7 as the dividend and 9 as the divisor. Ask your teacher for more information.

Again, 9 goes into 70 five times so we put a 7 above the third zero. We can see that this pattern will continue forever, so this is a recurring decimal.

\dfrac{7}{9}=0.\overline{7}

Idea summary

A number has an exact value. In the case of fractions and roots, the exact value must be a fraction or root.

Numbers also have approximations. These are numbers which are close but not equal to the exact value. We usually find approximations by rounding the exact value.

For example, if \dfrac{2}{3} is the exact value, then 0.667 is an approximation.

Outcomes

MA4-5NA

operates with fractions, decimals and percentages

MA4-8NA

generalises number properties to operate with algebraic expressions

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