Exact Values and Approximations
Theory
A number is rational if it can be written as a fraction of two integers and irrational if it cannot, like \(\pi\) or a surd such as \(\sqrt{7}\). Irrational numbers can be written exactly only as symbols; every decimal of them is an approximation. This Year 10A maths page shows how to classify real numbers, change recurring decimals to fractions, estimate and order surds, compare surds using entire surds, and compare exact answers with rounded ones. In the Year 10A course (Victorian Curriculum 10A) this is Level 10 content. Curriculum: VC2M10N01 / AC9M10N01.
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.
An exact value is the number itself, written without rounding: \(\sqrt{7}\), \(\dfrac{2}{3}\) or \(25\pi\). A calculator display such as \(2.645\,751\) is an approximation: the digits of a surd or of \(\pi\) never end, so any decimal of it has been rounded or cut off (truncated).
The absolute error of an approximation is the size of the difference between the approximation and the exact value. The percentage error is the absolute error as a percentage of the exact value.
| 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.
How to order or compare real numbers
- A mix of types: write each one as a decimal to three or four places. For \(1.\dot{6},\ \sqrt{3},\ \sqrt[3]{5},\ 1\dfrac{5}{7}\): \(1.667,\ 1.732,\ 1.710,\ 1.714\), so the order is \(1.\dot{6},\ \sqrt[3]{5},\ 1\dfrac{5}{7},\ \sqrt{3}\).
- Surds only: write each one as an entire surd and compare the numbers under the roots.
- A surd against a decimal: square both. For positive numbers, the larger number has the larger square.
- Between two numbers: to find a rational number between two surds, write both as decimals and choose a terminating decimal in between.
How to work with exact values
- Keep the exact form (\(\sqrt{\ }\), \(\pi\), fractions) through the working.
- Round only the final answer, to the accuracy asked for. If the answer must be exact, leave it in terms of the surd or \(\pi\).
- Compare an approximate answer with the exact one by finding the absolute and percentage error.
(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) decide which is larger: \(2\sqrt{7}\) or \(3\sqrt{3}\)
(i) Find the squares either side of \(90\).
| \(9^2\) | \(=\) | \(81\) |
| \(10^2\) | \(=\) | \(100\) |
So \(9\lt \sqrt{90}\lt 10\). Square roots are not evenly spaced, so square the halfway value:
| \(9.5^2\) | \(=\) | \(90.25\) |
\(90 \lt 90.25\), so \(\sqrt{90} \lt 9.5\). The closest whole number is \(\mathbf{9}\).
(ii) Write both as entire surds, then compare the numbers under the roots.
| \(2\sqrt{7}\) | \(=\) | \(\sqrt{4 \times 7} = \sqrt{28}\) |
| \(3\sqrt{3}\) | \(=\) | \(\sqrt{9 \times 3} = \sqrt{27}\) |
\(28 > 27\), so \(\boldsymbol{2\sqrt{7}}\) is larger.
(ii) A square has an area of \(72~\text{cm}^2\). Write its side length exactly, in simplest surd form, and correct to \(1\) decimal place.
(i) Keep \(\pi\) as a symbol for the exact value:
| \(A\) | \(=\) | \(\pi \times 5^2 = \mathbf{25\pi}~\textbf{cm}^2\) |
| \(3.14 \times 25\) | \(=\) | \(78.5\) |
| \(25\pi\) | \(=\) | \(78.5398\ldots\) |
| \(\text{absolute error}\) | \(=\) | \(|78.5 - 78.5398\ldots|\) |
| \(\) | \(\approx\) | \(\mathbf{0.04}~\textbf{cm}^2\) |
| \(\text{percentage error}\) | \(=\) | \(\dfrac{0.0398\ldots}{78.5398\ldots} \times 100\%\) |
| \(\) | \(\approx\) | \(\mathbf{0.05\%}\) |
(ii) The side is the square root of the area. Take out the largest square factor, \(36\):
| \(s\) | \(=\) | \(\sqrt{72} = \sqrt{36} \times \sqrt{2}\) |
| \(\) | \(=\) | \(\mathbf{6\sqrt{2}}~\textbf{cm}\) |
| \(\) | \(\approx\) | \(\mathbf{8.5}~\textbf{cm}\) |
\(6\sqrt{2}\) is exact; \(8.5\) cm is a rounded approximation of it.
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 \(9\).
What is the difference between an exact value and an approximation?
An exact value is the number itself, such as \(\sqrt{3}\) or \(25\pi\). An approximation is a rounded or truncated decimal of it, such as \(1.732\) or \(78.54\). Squaring \(1.732\) gives \(2.999\,824\), not \(3\), so \(1.732\) is only close to \(\sqrt{3}\).
How do you find the percentage error of an approximation?
Find the absolute error, the size of the difference between the approximation and the exact value, then divide it by the exact value and multiply by \(100\%\).