Year 10AC9M10N01

Approximation and Exact Values

Recognise the effect of using approximations of real numbers in repeated calculations and compare the results when using exact representations.

Rounding early is the most common way to get a right method and a wrong answer. The rule is simple: keep everything exact until the very last line.

Method

Round once, at the end

Every intermediate rounding introduces an error, and the next calculation multiplies it.

Exact forms are exact

\pi,\ \sqrt{2},\ \tfrac{1}{3}

Leave them as symbols. $3{,}14$ is not $\pi$, and $0{,}33$ is not a third.

Error compounds

A small error repeated ten times is not a small error. It grows with every operation.

Know which you are asked for

Decimal places counts after the point. Significant figures counts from the first non-zero digit.

e.g.

\text{Find the area of a circle of radius } 2{,}7 \text{ cm, then use it to find the volume of a cylinder of height } 11{,}4 \text{ cm.}

Write the formula and keep $\pi$ as a symbol. $A = \pi r^2 = \pi(2{,}7)^2 = 7{,}29\pi$
Carry the exact form into the next calculation. Do not round yet. $V = 7{,}29\pi \times 11{,}4 = 83{,}106\pi$
Round only now, at the final answer. $V = 261{,}1 \text{ cm}^3$
Now compare with rounding early, to see the damage. $A \approx 22{,}9 \ \Rightarrow\ V \approx 22{,}9 \times 11{,}4 = 261{,}06$
Look at where the answers separate. Two roundings have already moved the third figure. $261{,}1 \ \text{vs} \ 261{,}1 \text{ — close here, but}$
Repeat the same rounding ten times and the gap becomes visible. $\text{a } 0{,}2\% \text{ error applied ten times is about } 2\%$

Keep exact forms until the last line.

\text{Round } 0{,}004\,916 \text{ to 2 significant figures.}

Answer

0{,}0049

\text{Round } 0{,}004\,916 \text{ to 2 decimal places.}

Answer

0{,}00

\text{Why should } \sqrt{50} \text{ be left as } 5\sqrt{2} \text{ during working?}

Answer

\text{It is exact. Any decimal for } \sqrt{2} \text{ is already wrong, and the error grows.}

\text{A value is rounded to } 3 \text{ s.f. at each of 4 stages. Is the final answer accurate to 3 s.f.?}

Answer

\text{Not reliably. Repeated rounding can move the third figure.}

\text{Evaluate } \tfrac{1}{3} \times 3 \text{ exactly, and using } 0{,}33

Answer

1 \quad \text{vs} \quad 0{,}99