Rational and Irrational Numbers
Theory
A rational number can be written as a fraction \(\dfrac{a}{b}\) of two integers (with \(b \ne 0\)). An irrational number cannot: its decimal never ends and never repeats, like \(\sqrt{2}\) and \(\pi\). This Year 8 maths page shows how to tell them apart and how to work with each.
A number is rational if it can be written as \(\dfrac{a}{b}\), where \(a\) and \(b\) are integers and \(b \ne 0\). Every integer, fraction, percentage, terminating decimal and recurring decimal is rational.
Making the fraction. A terminating decimal goes over \(10\), \(100\) or \(1000\), then simplifies: \(0.35 = \dfrac{35}{100} = \dfrac{7}{20}\); a negative keeps its sign: \(-1.6 = -\dfrac{8}{5}\). A recurring decimal with a one-digit block goes over \(9\), a two-digit block over \(99\): \(0.\dot{7} = \dfrac{7}{9}\). A mixed decimal keeps its whole number: \(2.\dot{5} = 2\dfrac{5}{9}\).
A number is irrational if it cannot be written that way. Its decimal never ends and never repeats. A pattern is not a repeating block: \(0.2020020002\ldots\) has a pattern but no block repeats, so it is irrational.
Square roots. For a whole number \(n\), \(\sqrt{n}\) is irrational unless \(n\) is a perfect square: \(\sqrt{49} = 7\) is rational, \(\sqrt{45}\) is irrational. Decimals can work too: \(\sqrt{0.49} = 0.7\) is rational.
\(\pi\) is irrational. \(\dfrac{22}{7}\) and \(3.14\) are rational numbers that are close to \(\pi\), not equal to it.
Exact or approximate. \(\sqrt{5}\) and \(12\pi\) are exact. Any decimal written for them is an approximation, so use \(\approx\) and round only at the last step.
Mixing them. \(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\), so two irrational numbers can multiply to a rational one: \(\sqrt{3} \times \sqrt{12} = \sqrt{36} = 6\). Between any two different numbers there is a rational number: choose a short decimal between them. To order a list, write each to the same number of decimal places, then compare.
Rational or irrational?
| Rational | Irrational |
|---|---|
| \(7 = \dfrac{7}{1}\) | \(\sqrt{2} = 1.41421356\ldots\) |
| \(0.35 = \dfrac{7}{20}\) | \(\sqrt{3} = 1.73205080\ldots\) |
| \(0.\dot{7} = \dfrac{7}{9}\) | \(\sqrt{7} = 2.64575131\ldots\) |
| \(\sqrt{49} = 7\) | \(\pi = 3.14159265\ldots\) |
How to decide if a number is rational
- Try to write it as a fraction of two integers (decimals over \(10\), \(100\), \(\ldots\); recurring blocks over \(9\), \(99\), \(\ldots\)).
- For a square root, check whether the number is a perfect square.
- Decide: a fraction of integers means rational; a decimal that never ends or repeats means irrational.
How to place an irrational square root
- Trap it between square numbers: \(4 < 7 < 9\), so \(2 < \sqrt{7} < 3\).
- Round with a calculator and mark it on the number line.
(i) \(0.85\)
(ii) \(24\%\)
(iii) \(0.\dot{8}\)
| \(0.85\) | \(=\) | \(\dfrac{85}{100} = \dfrac{17}{20}\) |
| \(24\%\) | \(=\) | \(\dfrac{24}{100} = \dfrac{6}{25}\) |
| \(0.\dot{8}\) | \(=\) | \(\dfrac{8}{9}\) |
(i) \(\dfrac{17}{20}\)
(ii) \(\dfrac{6}{25}\)
(iii) \(\dfrac{8}{9}\). Check: \(8 \div 9 = 0.888\ldots\)
(i) \(\sqrt{45}\)
(ii) \(\sqrt{1.44}\)
(iii) \(3\pi\)
(iv) \(0.\dot{5}\dot{8}\)
(i) \(36 < 45 < 49\), so \(45\) is not a perfect square: irrational
(ii) \(\sqrt{1.44} = 1.2 = \dfrac{6}{5}\): rational
(iii) \(3\pi = 9.4247\ldots\) never ends or repeats: irrational
(iv) \(\dfrac{58}{99}\): rational
(ii) Write \(\sqrt{7}\) correct to \(2\) decimal places.
(iii) Mark it on a number line.
(i) \(4 < 7 < 9\), so \(\sqrt{4} < \sqrt{7} < \sqrt{9}\).
(ii) A calculator gives \(\sqrt{7} = 2.6457\ldots\); the third decimal digit is \(5\), so round up.
(i) \(2\) and \(3\)
(ii) \(2.65\) (but \(2.65 \times 2.65 = 7.0225\), not exactly \(7\))
(iii) Between \(2.6\) and \(2.7\), as shown.
(i) Write the circumference exactly.
(ii) Find it correct to \(1\) decimal place.
(iii) Is the rounded answer exact?
(i) \(40\pi\) cm
(ii) \(40\pi = 125.66\ldots \approx 125.7\) cm
(iii) No. \(\pi\) is irrational, so only \(40\pi\) cm is exact.
(ii) Find a rational number between \(\sqrt{5}\) and \(\sqrt{6}\).
(i) To \(4\) places: \(\sqrt{8} \approx 2.8284\), \(2.\dot{8} = 2.8888\ldots\), \(\dfrac{17}{6} = 2.8333\ldots\), \(2.83 = 2.8300\).
(ii) \(\sqrt{5} \approx 2.236\) and \(\sqrt{6} \approx 2.449\).
(i) \(\sqrt{8}\), \(2.83\), \(\dfrac{17}{6}\), \(2.\dot{8}\)
(ii) \(2.3 = \dfrac{23}{10}\) (any short decimal between them works)
| \(4 \times \sqrt{50}\) | \(=\) | \(28.284\ldots\) |
\(28.3\) cm. Rounding \(\sqrt{50}\) to \(7.1\) first would give \(4 \times 7.1 = 28.4\), which is wrong.
Common pitfalls
Frequently asked questions
What is a rational number?
A number that can be written as a fraction of two integers, such as \(0.75 = \dfrac{3}{4}\) or \(-5 = \dfrac{-5}{1}\).
What is an irrational number?
A number whose decimal never ends and never repeats, such as \(\sqrt{2}\) or Ο.
Is the square root of every whole number irrational?
No. It is rational when the number is a perfect square: \(\sqrt{64} = 8\).
Is pi equal to 22/7?
No. \(\dfrac{22}{7}\) is a rational approximation of the irrational number Ο.
Are recurring decimals rational?
Yes: \(0.\dot{6}\dot{3} = \dfrac{63}{99} = \dfrac{7}{11}\).