Surds and Irrational Numbers
A surd is a root, such as \(\sqrt{7}\), that cannot be written as a fraction, so its decimal never ends or repeats. In NSW Year 9 Advanced maths, students learn to tell rational and irrational numbers apart (NESA outcome MA5‑IND‑P‑02).
You will decide whether roots like \(\sqrt{324}\), \(\sqrt[3]{9}\) and \(\sqrt{0.56}\) are surds, explain why \(\sqrt{-9}\) is not a real number and why \(\pi\) is irrational but not a surd, and place \(3\sqrt{7}\) between two whole numbers without a calculator. You will order surds, show that \(1.414\) only approximates \(\sqrt{2}\), and test whether sums and products of irrational numbers must be irrational. It is part of Year 9 Advanced Pythagoras Theorem.
Every question with a fully worked solution.
- Surds and Irrational Numbers - Video - Surds and Irrational Numbers Watch
Theory
A rational number can be written as a fraction of two integers; an irrational number cannot. A surd is an irrational root, such as \(\sqrt{7}\) or \(\sqrt[3]{9}\). This Year 9 Advanced maths page shows how to recognise surds, estimate and order them without a calculator, and decide whether an answer built from surds is rational.
A rational number can be written as \(\dfrac{a}{b}\), where \(a\) and \(b\) are integers and \(b \ne 0\). Integers, fractions, terminating decimals such as \(0.3\) and recurring decimals such as \(0.\dot{4}\dot{5}\) are all rational.
An irrational number cannot be written as a fraction. Its decimal goes on forever without repeating. \(\pi\) is irrational: \(\dfrac{22}{7}\) and \(3.14\) are only approximations.
A surd is a root that is irrational, such as \(\sqrt{7}\), \(\sqrt{50}\) or \(\sqrt[3]{9}\). A root is a surd only when it does not come out exactly: \(\sqrt{36} = 6\) and \(\sqrt[3]{8} = 2\) are rational, so they are not surds. \(\pi\) is irrational but is not a surd, because it is not a root of a rational number.
A surd is an exact value. A decimal such as \(\sqrt{7} \approx 2.65\) is only a rounded approximation. The square root of a negative number, such as \(\sqrt{-4}\), is not a real number, because no real number squares to a negative.
A number is rational if it can be written as \(\dfrac{a}{b}\), with \(a\) and \(b\) integers and \(b \ne 0\). Otherwise it is irrational.
For a whole number \(n\):
Rules for working with roots of positive numbers:
Expanding brackets with a surd:
How to decide if a number is rational or a surd
- Simplify first: evaluate roots of perfect squares and cubes, turn decimals into fractions, and multiply, add or expand any surds.
- Check the number under the root. For \(\sqrt{n}\), is \(n\) a perfect square? For \(\sqrt[3]{n}\), a perfect cube? For a fraction, check top and bottom.
- Decide. If the result is a whole number, fraction, terminating or recurring decimal, it is rational. An irrational root is a surd. A square root of a negative number is not real.
- To estimate \(k\sqrt{n}\), write it as \(\sqrt{k^2 n}\) and find the perfect squares on either side.
- To show a decimal is only an approximation, square it: \(1.73^2 = 2.9929\), not \(3\), so \(\sqrt{3} \ne 1.73\).
(i) \(\sqrt{0.09}\)
(ii) \(\sqrt{\dfrac{7}{25}}\)
(iii) \(\sqrt[3]{40}\)
(iv) \(\sqrt{-16}\)
Try to write each one as a fraction of two integers.
| \(\sqrt{0.09}\) | \(=\) | \(\sqrt{\dfrac{9}{100}} = \dfrac{3}{10}\) |
| \(\sqrt{\dfrac{7}{25}}\) | \(=\) | \(\dfrac{\sqrt{7}}{5}\) |
(i) \(0.3\) is a fraction, so \(\sqrt{0.09}\) is rational.
(ii) \(7\) is not a perfect square, so \(\sqrt{7}\) is irrational and \(\sqrt{\dfrac{7}{25}}\) is a surd.
(iii) \(3^3 = 27\) and \(4^3 = 64\), so \(40\) is not a perfect cube. \(\sqrt[3]{40}\) lies between \(3\) and \(4\) and is a surd.
(iv) Any real number squared is \(0\) or positive, so no real number squares to \(-16\). \(\sqrt{-16}\) is not a real number.
(i) find the two consecutive whole numbers between which \(4\sqrt{3}\) lies
(ii) write \(4\sqrt{3},\ 7,\ 3\sqrt{5},\ \sqrt{46}\) in ascending order.
(i) Move the \(4\) inside the root as \(\sqrt{16}\):
| \(4\sqrt{3}\) | \(=\) | \(\sqrt{16 \times 3} = \sqrt{48}\) |
| \(6^2 = 36\) | \(\lt \) | \(48 \lt 49 = 7^2\) |
So \(4\sqrt{3}\) lies between \(\mathbf{6}\) and \(\mathbf{7}\).
(ii) All four are positive, so compare their squares. A bigger square means a bigger number.
| \((4\sqrt{3})^2\) | \(=\) | \(48\) |
| \(7^2\) | \(=\) | \(49\) |
| \((3\sqrt{5})^2\) | \(=\) | \(9 \times 5 = 45\) |
| \((\sqrt{46})^2\) | \(=\) | \(46\) |
\(45 \lt 46 \lt 48 \lt 49\), so the order is \(\boldsymbol{3\sqrt{5},\ \sqrt{46},\ 4\sqrt{3},\ 7}\).
Two digits repeat, so multiply by \(100\). Subtracting then removes the repeating tail.
| \(x\) | \(=\) | \(0.454545\ldots\) |
| \(100x\) | \(=\) | \(45.454545\ldots\) |
| \(100x - x\) | \(=\) | \(45\) |
| \(99x\) | \(=\) | \(45\) |
| \(x\) | \(=\) | \(\dfrac{45}{99} = \mathbf{\dfrac{5}{11}}\) |
It is a fraction of two integers, so \(0.\dot{4}\dot{5}\) is rational. Every recurring decimal can be written this way. An irrational number such as \(\sqrt{2}\) never ends and never repeats.
(i) \(\sqrt{8} \times \sqrt{2}\)
(ii) \(\sqrt{8} + \sqrt{2}\)
(iii) \((\sqrt{5} + 2)(\sqrt{5} - 2)\)
| \(\sqrt{8} \times \sqrt{2}\) | \(=\) | \(\sqrt{16} = 4\) |
| \(\sqrt{8} + \sqrt{2}\) | \(=\) | \(2\sqrt{2} + \sqrt{2} = 3\sqrt{2}\) |
| \((\sqrt{5} + 2)(\sqrt{5} - 2)\) | \(=\) | \((\sqrt{5})^2 - 2^2\) |
| \(\) | \(=\) | \(5 - 4 = 1\) |
(i) \(4\) is rational. (ii) \(3\sqrt{2}\) is irrational: a non-zero whole number times a surd is still a surd. (iii) \(1\) is rational. The middle terms \(-2\sqrt{5}\) and \(+2\sqrt{5}\) cancel.
Two irrational numbers can combine to give a rational one, so always simplify before you decide.
Common pitfalls
Frequently asked questions
What is a surd?
A surd is a root that cannot be simplified to a rational number, such as \(\sqrt{7}\) or \(\sqrt[3]{9}\). Its decimal goes on forever without repeating, so a surd is left in root form as an exact value.
What is the difference between rational and irrational numbers?
A rational number can be written as a fraction of two integers, so its decimal ends or repeats. An irrational number cannot, so its decimal never ends and never repeats.
Is pi a surd?
No. \(\pi\) is irrational, but a surd is an irrational root of a rational number, and \(\pi\) is not one. \(\dfrac{22}{7}\) is only an approximation of \(\pi\).
How do you tell if a square root is rational?
Simplify what is under the root. For a whole number, the square root is rational only if the number is a perfect square, like \(49\). For a fraction or decimal, write it as a fraction and check the top and bottom.
Is the product of two irrational numbers always irrational?
No. \(\sqrt{8} \times \sqrt{2} = \sqrt{16} = 4\). Always simplify before deciding.