Exact and approximate values | Year 10

01 / LEARN

Understand the idea.

Exact means unchanged

An exact value represents the intended number without rounding. A surd such as $\sqrt{2}$ is exact, while $1.414$ is only an approximation to it. A finite decimal can itself be exact, such as $0.5 = \frac{1}{2}$ .

Use $=$ between equal values and $\approx$ when a value has been rounded. Give the exact form first if requested, then state the decimal to the specified number of decimal places with the correct units.

Round once at the end

Keep exact expressions or full calculator precision during intermediate steps. Round only the final requested quantity: rounding a side length before finding a perimeter creates an avoidable error in every side.

To judge an approximation, compare it with the exact result. The absolute difference measures the size of the error; dividing by the magnitude of the nonzero exact reference gives the relative error.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Exact perimeter

A square has area $18 {cm}^{2}$ . Find the exact perimeter and a decimal to 2 decimal places.

Worked solution

1
Find the square side length.
$s = \sqrt{18} = 3 \sqrt{2} cm$
A square of side $s$ has area $s^{2}$ , and a length takes the positive root.
2
Form the exact perimeter.
$P = 4 s = 4 (3 \sqrt{2}) = 12 \sqrt{2} cm$
All four sides have equal length.
3
Round the perimeter at the end.
$P = 12 \sqrt{2} cm \approx 16.97 cm$
Evaluating the exact expression before rounding gives two decimal places accurately.

Answer

$12 \sqrt{2} cm \approx 16.97 cm$

EXAMPLE 02

Compare early rounding

Compare ${(\sqrt{2})}^{2}$ with ${1.4}^{2}$ .

Worked solution

1
Square the exact value.
${(\sqrt{2})}^{2} = 2$
The square and principal square root undo one another for a positive radicand.
2
Square the rounded value.
${1.4}^{2} = 1.96$
Replacing the root before calculating changes the input.
3
Measure the difference.
$2 - 1.96 = 0.04$
The rounded-input result is below the exact result.
4
Compare relative to the exact result.
$\frac{0.04}{2} \times 100 % = 2 %$
The exact result $2$ is the reference for this percentage error.

Answer

2 versus 1.96; difference 0.04

03 / YOUR TURN

Try it yourself.

Check your understanding

Give $\sqrt{50}$ exactly and to 1 decimal place.

Show answer & reasoning

$5 \sqrt{2} \approx 7.1$

Extract the square factor to get $\sqrt{50} = 5 \sqrt{2}$ . Its decimal is $7.071 \dots$ , so the one-decimal-place approximation is $7.1$ .

Keep practising

6 worksheet questions · 14 marks

Open worksheet Worked solutions

Read the questions online

Write $\sqrt{32}$ in simplest exact form.
2 marks
Round $\sqrt{7}$ to 3 decimal places.
2 marks
Find exactly ${(3 \sqrt{5})}^{2}$ .
2 marks
Compare $π \times 10$ using exact $π$ with using $3.14$ . Give the two decimals to 3 places.
3 marks
A square has diagonal $10$ cm. Find its side exactly and to 2 decimal places.
3 marks
Is $0.333$ exactly equal to $\frac{1}{3}$ ? Explain.
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Write $\sqrt{32}$ in simplest exact form.
2 marks
$32 = 16 \times 2$ .
$4 \sqrt{2}$
Round $\sqrt{7}$ to 3 decimal places.
2 marks
$\sqrt{7} = 2.645751 \dots$ .
2.646
Find exactly ${(3 \sqrt{5})}^{2}$ .
2 marks
$9 \times 5 = 45$ .
45
Compare $π \times 10$ using exact $π$ with using $3.14$ . Give the two decimals to 3 places.
3 marks
$10 π = 31.415926 \dots$ ; $10 (3.14) = 31.4$ .
31.416 and 31.400
A square has diagonal $10$ cm. Find its side exactly and to 2 decimal places.
3 marks
$s^{2} + s^{2} = 100$ , so $s = \sqrt{50} = 5 \sqrt{2}$ .
$5 \sqrt{2} cm \approx 7.07 cm$
Is $0.333$ exactly equal to $\frac{1}{3}$ ? Explain.
2 marks
$\frac{1}{3} = 0.3333 \dots$ continues forever; $0.333$ is a finite approximation.
No.

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics

Integer and fractional index laws; connecting fractional powers with roots; simplifying surds; operations with surds; rationalising denominators; exact and approximate values; exponential tables and graphs; growth/decay factors; comparing linear/quadratic/exponential patterns; exponential equations; logarithms as inverses; logarithm laws; interpreting logarithmic scales; graph transformations and parameter experiments; choosing and evaluating models.

Official curriculum source ↗

In this lesson

Understand the idea.

Exact means unchanged

Round once at the end

See the method in action.

Exact perimeter

Find the square side length.

Form the exact perimeter.

Round the perimeter at the end.

Compare early rounding

Square the exact value.

Square the rounded value.

Measure the difference.

Compare relative to the exact result.

Try it yourself.

Check your understanding

Keep practising