Exact and approximate values | Year 10

01 / LEARN

The idea, step by step

Exact means unchanged

$\sqrt{2}$ is exact. Replacing it by the finite decimal $1.414$ gives an approximation to that root. A finite decimal can itself be an exact rational value. Use $=$ for equality and $\approx$ for a rounded value.

An exact answer is often shorter and more informative than many calculator digits. Give a decimal only to the requested precision, with units where needed.

Round once at the end

When several calculations depend on an irrational value, store its exact form or retain full calculator precision until the final step.

Small errors can grow when a rounded value is squared or used repeatedly. Checking the precision is part of solving the original problem, not a separate formula to memorise.

A common trap

Writing $\sqrt{3} = 1.73$ .

Instead: Use $\sqrt{3} \approx 1.73$ , because squaring $1.73$ does not give exactly $3$ .

02 / WORKED EXAMPLES

See the thinking

Example 1

Exact perimeter

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

The side is $\sqrt{18} = 3 \sqrt{2}$ cm.
The perimeter is $12 \sqrt{2}$ cm.
Only now evaluate: $12 \sqrt{2} \approx 16.97$ .

12 \sqrt{2} cm \approx 16.97 cm

Example 2

Compare early rounding

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

The exact calculation gives $2$ .
The rounded calculation gives $1.96$ .
The result is $0.04$ below the exact value, a $2 %$ difference relative to $2$ .

2 versus 1.96; difference 0.04

Check your understanding

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

Show answer

$5 \sqrt{2} \approx 7.1$

Factor $50 = 25 \times 2$ , then round $7.071 \dots$ at the end.

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF 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 $1 / 3$ ? 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 $1 / 3$ ? Explain.
2 marks
$1 / 3 = 0.3333 \dots$ 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 ↗

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