The Real Number System
Theory
The real numbers are all the numbers on the number line. Each one is either rational, so it can be written as a fraction of two integers, or irrational, so it cannot. A surd is an irrational root, such as \(\sqrt{7}\) or \(\sqrt[3]{9}\). This Year 9 maths page shows how to sort real numbers, write recurring decimals as fractions, estimate and order surds, and use them in problems. Curriculum: VC2M9N01 / AC9M9N01.
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.
Together, the rational and irrational numbers make up the real numbers. Every point on the number line is a real number.
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:
How to decide if a number is rational or a surd
- Simplify first: evaluate roots of perfect squares and cubes, and write decimals as fractions.
- 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 decide whether \(\sqrt{an}\) is rational for a whole number \(n\), check whether \(an\) is a perfect square.
- 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 \(\sqrt{75}\) lies
(ii) find every whole number \(n\) from \(1\) to \(30\) for which \(\sqrt{3n}\) is rational.
(i) Find the perfect squares either side of \(75\):
| \(8^2 = 64\) | \(\lt \) | \(75 \lt 81 = 9^2\) |
So \(\sqrt{75}\) lies between \(\mathbf{8}\) and \(\mathbf{9}\).
(ii) \(\sqrt{3n}\) is rational only when \(3n\) is a perfect square. For \(n\) from \(1\) to \(30\), \(3n\) goes from \(3\) to \(90\). The perfect squares in that range that are multiples of \(3\) are:
| \(3n\) | \(=\) | \(9,\ 36,\ 81\) |
| \(n\) | \(=\) | \(\boldsymbol{3,\ 12,\ 27}\) |
Check: \(\sqrt{9} = 3\), \(\sqrt{36} = 6\) and \(\sqrt{81} = 9\). The other perfect squares up to \(90\) (\(4, 16, 25, 49, 64\)) are not multiples of \(3\).
(i) \(0.\dot{8}\)
(ii) \(0.\dot{3}\dot{6}\)
(i) One digit repeats, so multiply by \(10\). Subtracting removes the repeating tail:
| \(x\) | \(=\) | \(0.888\ldots\) |
| \(10x\) | \(=\) | \(8.888\ldots\) |
| \(10x - x\) | \(=\) | \(8\) |
| \(9x\) | \(=\) | \(8\) |
| \(x\) | \(=\) | \(\mathbf{\dfrac{8}{9}}\) |
(ii) Two digits repeat, so multiply by \(100\):
| \(x\) | \(=\) | \(0.363636\ldots\) |
| \(100x\) | \(=\) | \(36.363636\ldots\) |
| \(99x\) | \(=\) | \(36\) |
| \(x\) | \(=\) | \(\dfrac{36}{99} = \mathbf{\dfrac{4}{11}}\) |
Both are fractions of two integers, so both are rational. Every recurring decimal can be written this way.
(ii) A square tile has an area of \(30~\text{cm}^2\), so its side is \(\sqrt{30}\) cm. Find its perimeter, correct to \(2\) decimal places.
(i) Write each number as a decimal to four places:
| \(\sqrt{7}\) | \(=\) | \(2.6457\ldots\) |
| \(2.65\) | \(=\) | \(2.6500\) |
| \(\dfrac{8}{3}\) | \(=\) | \(2.6666\ldots\) |
So the order is \(\boldsymbol{\sqrt{7},\ 2.65,\ \dfrac{8}{3}}\). Note that \(\dfrac{8}{3} = 2.\dot{6}\) is rational, but \(\sqrt{7}\) is not.
(ii) Four equal sides. Keep the exact value \(4\sqrt{30}\) and round only at the end:
| \(P\) | \(=\) | \(4\sqrt{30}\) |
| \(\) | \(=\) | \(21.9089\ldots\) |
| \(\) | \(\approx\) | \(\mathbf{21.91}\ \textbf{cm}\) |
Rounding the side first, to \(5.48\), would give \(21.92\) cm, which is wrong.
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.
What are the real numbers?
All the rational numbers and all the irrational numbers together. Every point on the number line is a real number. The square root of a negative number, such as \(\sqrt{-9}\), is not a real number.