Irrational / Rational Numbers and Recurring Decimals
Every number is either rational or irrational, whether large or small, negative or positive.
Rational Numbers and recurring decimals
A rational number can be written exactly in the form \(\frac{a}{b}\), where 饾憥 and 饾憦 are integers, while an irrational number cannot.
Examples of rational numbers
| 袄(鈥冣赌冣赌僜蹿谤补肠调7皑调8皑鈥冣赌冣赌僜) | 鈥冣赌冣赌冣赌87鈥冣赌冣赌 | 鈥冣赌冣赌冣赌0.21鈥冣赌冣赌 | 鈥冣赌冣赌僜(-袄蹿谤补肠调19皑调4皑袄)鈥冣赌冣赌 | 袄(鈥冣赌冣赌0.调袄诲辞迟调6皑皑袄) |
| 鈥冣赌冣赌冣赌冣赌冣赌 | Can be written as \(\frac{87}{1}\) | Can be written as \(\frac{21}{100}\) | 鈥冣赌冣赌冣赌冣赌冣赌 | Can be written as \(\frac{2}{3}\) |
All Recurring decimalNumbers repeat forever after the decimal point can be written as fractions and are therefore rational.
A Terminating decimalsDecimal numbers that have a finite number of digits after the decimal point. can be easily written as a fraction (as in the examples above), but a recurring decimal is not so easily converted to a fraction.
Changing a recurring decimal to a fraction
Example
Write \(0.\dot{6}\) as a recurring decimal. (We know that the answer should be \(\frac{2}{3}\)).
Let r = the recurring decimal.
r = 0.666666鈥
Multiply r by 10
10r = 6.666666666鈥
r = 0.666666666鈥
Subtract
9r = 6
Divide
\(r = \frac{6}{9} = \frac{2}{3} \)
In general, for any recurring decimal
- Let r = the recurring decimal
- Multiply r by a suitable power of ten - multiply by 10 if one digit repeats, 100 for 2 digits, 1000 for 3 digits etc.
- Subtract to get a multiple of r
- Divide to give the required fraction
Example
Write the recurring decimal 0.1818181818鈥 as a fraction
Solution
- Let r = the recurring decimal
r = 0.18181818181鈥 - Multiply r by 100 as two digits are repeating
100r = 18.181818181鈥 - Subtract to get a multiple of r
100r = 18.181818181鈥
r = 0.18181818181鈥
99r = 18 - Divide to give the required fraction
\(r= \frac{18}{99} = \frac{2}{11}\)
Answer
\(\mathbf{\frac{2}{11}}\)
Question
Write the recurring decimal 0.3888888888888888鈥 as a fraction
100r = 38.8888888888鈥
10r = 3.88888888888鈥
\(90r = 35\)
\(r = \frac{35}{90} = \frac{7}{18}\)
Irrational numbers
Irrational numbers cannot be written as a fraction.
The types of irrational numbers that should be known are:
- Square roots (of numbers that are not square) e. g. 鈭2, 鈭5, 鈭3.65, etc.
- Multiples and powers of 蟺 e. g. \(\frac {\pi}{2}, 5\pi, 2\pi^2\) etc.
Example
Which of these numbers are irrational?
0.635 鈥 18蟺 鈥 鈭400 鈥 \(\mathbf{0.\dot{1}\dot{2}}\) 鈥 鈭28
Solution
18蟺, 鈭28 are irrational. The others can be written in the form \(\frac{a}{b}\)
\(0.635 = \frac{635}{1000}\) 鈥 \(\sqrt{400} = 20 = \frac{20}{1}\) 鈥 \(0.\dot{12}鈥 = \frac{4}{33}\)
Question
Which of these are irrational?
A: \((5\sqrt{3})^2\) 鈥B: \(\frac{2}{\pi}(3.6\pi^2)\)鈥C: \(\sqrt{3} \times \sqrt{12}\)鈥D: \(2\sqrt{3} \times 3\sqrt{2} \times \sqrt{24}\)
Solution
A: \((5\sqrt{3})^2 = 25 \times 3 = 75\)鈥冣冣冣僐ational
B: \(\frac{2}{2\pi}(3.6\pi^2) = 7.2\pi\)鈥冣冣冣冣Irrational
C: \(\sqrt{3} \times \sqrt{12} = \sqrt{36} = 6\)鈥冣冣冣僐ational
D: \(2\sqrt{3} \times 3\sqrt{2} \times \sqrt{24}= 6\sqrt{144} = 72\)鈥冣僐ational
Test yourself
More on M8: Algebra
Find out more by working through a topic
- count4 of 7
- count6 of 7
