Surds and Rational Numbers
Every real number is either rational, meaning it can be written as a fraction of two integers, or irrational. Recurring decimals such as \(0.\dot{4}\dot{5}\) are rational, while surds such as \(\sqrt{20}\) and the number \(\pi\) are not. In NSW Year 10 Advanced maths you sort numbers into these groups and convert recurring decimals to fractions (NESA outcome MA5‑IND‑P‑02).
You will decide whether roots, percentages and decimals are rational or irrational, pick out surds, and change recurring decimals such as \(0.2\dot{3}\) and \(0.\dot{1}3\dot{5}\) into fractions in simplest form. You will place surds between whole numbers and decimals, order mixed lists, count how many roots in a list are irrational, and prove facts about sums of recurring decimals. It is part of Year 10 Advanced Surds.
Every question with a fully worked solution.
- Surds and Rational Numbers - Video - What is the Difference Between Rational and Irrational Numbers Watch
Theory
A rational number can be written as a fraction of two integers; an irrational number, such as a surd or \(\pi\), cannot. This Year 10 Advanced maths page covers classifying numbers, converting recurring decimals to fractions, estimating surds without a calculator and putting surds, decimals and fractions in order.
A rational number can be written as \(\dfrac{a}{b}\) where \(a\) and \(b\) are integers and \(b \ne 0\). Integers, fractions, terminating decimals and recurring decimals are all rational.
An irrational number cannot be written as a fraction. Its decimal goes on forever and never settles into a repeating block. Examples are \(\pi\) and surds: roots such as \(\sqrt{2}\) or \(\sqrt[3]{10}\) that do not simplify to a rational number.
A root is rational exactly when the number under it is a perfect power of the right kind: \(\sqrt{49} = 7\), \(\sqrt[3]{-27} = -3\), \(\sqrt{\dfrac{4}{25}} = \dfrac{2}{5}\) and \(\sqrt{0.16} = 0.4\). So \(\sqrt{n}\) for a whole number \(n\) is irrational unless \(n\) is a perfect square.
A dot over a digit marks a repeat: \(0.\dot{3} = 0.333\ldots\), and dots over the first and last digits mark a repeating block: \(0.\dot{1}2\dot{4} = 0.124124\ldots\). Rational and irrational numbers together make up the real numbers.
| Recurring decimal | Multiply by | Result |
|---|---|---|
| one digit repeats, \(0.\dot{a}\) | \(10\) | \(\dfrac{a}{9}\) |
| two digits repeat, \(0.\dot{a}\dot{b}\) | \(100\) | \(\dfrac{ab}{99}\) |
| three digits repeat | \(1000\) | \(\dfrac{abc}{999}\) |
| a digit before the repeat, \(0.a\dot{b}\) | \(100\) and \(10\), subtract | denominator \(90\) |
(Here \(ab\) means the two-digit number, not \(a\times b\).)
How to decide if a number is rational
- Fractions, integers, terminating or recurring decimals: rational.
- Roots: simplify. Write decimals and mixed numerals as fractions first. If every root disappears, the number is rational; if a surd is left, it is irrational.
- \(\pi\), and non-repeating patterns such as \(0.12112111\ldots\): irrational.
- Counting: among \(\sqrt{1}, \sqrt{2}, \ldots, \sqrt{n}\), only the perfect squares give rational values; subtract their count from \(n\).
How to convert a recurring decimal to a fraction
- Let \(x\) equal the decimal and write out several repeats.
- Multiply by \(10\), \(100\) or \(1000\) (one zero per repeating digit) so a copy of the repeat lines up.
- If digits come before the repeat, also multiply so the point sits at the start of the repeat, and subtract those two equations instead.
- Subtract, solve for \(x\) and simplify the fraction.
- Sums and differences of recurring decimals: convert each one, then add or subtract the fractions. A whole-number part is added separately.
(i) \(\sqrt{2.25}\)
(ii) \(\sqrt[3]{-64}\)
(iii) \(\sqrt{0.9}\)
(iv) \(3\pi\)
(v) \(1.\dot{2}\)
Try to write each one as a fraction of two integers.
| \(\text{(i)}\ \sqrt{2.25}\) | \(=\) | \(1.5 = \dfrac{3}{2}\) |
| \(\) | \(\) | \(\textbf{rational}\) |
| \(\text{(ii)}\ \sqrt[3]{-64}\) | \(=\) | \(-4\) |
| \(\) | \(\) | \(\textbf{rational}\) |
| \(\text{(iii)}\ \sqrt{0.9}\) | \(=\) | \(\sqrt{\dfrac{9}{10}} = \dfrac{3}{\sqrt{10}}\) |
| \(\) | \(\) | \(\textbf{irrational}\) |
In (iii), \(10\) is not a perfect square, so \(\sqrt{10}\) is a surd. Careful: \(\sqrt{0.9}\) is not \(0.3\), because \(0.3^2 = 0.09\).
(iv) \(3\pi\) is irrational: a non-zero whole number times an irrational number stays irrational.
(v) \(1.\dot{2} = 1.222\ldots\) recurs, so it is rational (it equals \(\dfrac{11}{9}\)).
(i) \(0.\dot{4}\dot{5}\)
(ii) \(0.2\dot{3}\)
(iii) \(0.\dot{1}3\dot{5}\)
(i) Two digits repeat, so multiply by \(100\).
| \(100x\) | \(=\) | \(45.4545\ldots\) |
| \(99x\) | \(=\) | \(45\) |
| \(x\) | \(=\) | \(\dfrac{45}{99} = \mathbf{\dfrac{5}{11}}\) |
(ii) Only the \(3\) repeats. Move the point to the start of the repeat (\(\times 10\)) and past one full repeat (\(\times 100\)), then subtract.
| \(100x\) | \(=\) | \(23.333\ldots\) |
| \(10x\) | \(=\) | \(2.333\ldots\) |
| \(90x\) | \(=\) | \(21\) |
| \(x\) | \(=\) | \(\dfrac{21}{90} = \mathbf{\dfrac{7}{30}}\) |
(iii) The block \(135\) repeats, so multiply by \(1000\).
| \(999x\) | \(=\) | \(135\) |
| \(x\) | \(=\) | \(\dfrac{135}{999} = \mathbf{\dfrac{5}{37}}\) |
\(135 = 27\times 5\) and \(999 = 27\times 37\).
(i) find the whole number closest to \(\sqrt{90}\)
(ii) find a one-decimal-place number between \(\sqrt{11}\) and \(\sqrt{12}\).
(i) Find the squares either side of \(90\).
| \(9^2\) | \(=\) | \(81\) |
| \(10^2\) | \(=\) | \(100\) |
So \(9\lt \sqrt{90}\lt 10\). Comparing \(90\) with \(81\) and \(100\) is not a reliable test, because square roots are not evenly spaced. Square the halfway value instead:
| \(9.5^2\) | \(=\) | \(90.25\) |
\(90 \lt 90.25\), so \(\sqrt{90} \lt 9.5\). The closest whole number is \(\mathbf{9}\), only just.
(ii) A number lies between \(\sqrt{11}\) and \(\sqrt{12}\) if its square lies between \(11\) and \(12\).
| \(3.3^2\) | \(=\) | \(10.89\ (\text{too small})\) |
| \(3.4^2\) | \(=\) | \(11.56\ (\text{between})\) |
So \(\mathbf{3.4}\) works.
Write each one as a decimal to enough places to compare.
| \(2.\dot{2}\) | \(=\) | \(2.2222\ldots\) |
| \(2.23\) | \(=\) | \(2.2300\) |
| \(\dfrac{9}{4}\) | \(=\) | \(2.2500\) |
For \(\sqrt{5}\), compare squares: \(2.23^2 = 4.9729 \lt 5\) and \(2.25^2 = 5.0625 \gt 5\), so \(2.23 \lt \sqrt{5} \lt 2.25\).
| \(\) | \(\) | \(\mathbf{2.\dot{2},\ 2.23,\ \sqrt{5},\ \dfrac{9}{4}}\) |
Squaring works because, for positive numbers, a larger number always has a larger square.
Common pitfalls
Frequently asked questions
How do you know if a square root is rational or irrational?
Simplify it. If the number under the square root is a perfect square, or a fraction of two perfect squares, the root is rational. Otherwise a surd is left over and the number is irrational.
Is a recurring decimal rational?
Yes. Every recurring decimal can be written as a fraction of two integers, so it is rational. For example \(0.\dot{4}\dot{5} = \dfrac{5}{11}\).
How do you change a recurring decimal into a fraction?
Let \(x\) equal the decimal. Multiply by \(10\), \(100\) or \(1000\) so the repeating part lines up, subtract the two equations so the repeats cancel, then solve for \(x\) and simplify.
Is pi a surd?
No. \(\pi\) is irrational, but a surd is a root of a rational number, such as \(\sqrt{2}\). \(\pi\) cannot be written as a root like that, so it is irrational but not a surd.
How do you estimate a square root without a calculator?
Find the perfect squares either side: \(81 \lt 90 \lt 100\), so \(9 \lt \sqrt{90} \lt 10\). To see which is closer, square \(9.5\): \(90.25\), so \(\sqrt{90}\) is just below \(9.5\) and closest to \(9\).