1.8 Degree of accuracy

Learning objectives

1.8A Round integers to a given power of 10round integers to a given power of 10 1.8B Rounding to significant figures or decimal placesround to a given number of significant figures or decimal places 1.8C Upper and lower boundsidentify upper and lower bounds where values are given to a degree of accuracy 1.8D Estimating numerical calculationsUse estimation, including rounding values to one significant figure, to evaluate numerical approximations.

Rounding to a power of 10 means choosing the nearest multiple of that place value: $10^1$ for tens, $10^2$ for hundreds, $10^3$ for thousands, and so on.

Round 6739 to	Look at	Result
nearest 10	units digit 9	6740
nearest 100	tens digit 3	6700
nearest 1000	hundreds digit 7	7000

Locate the rounding digit, inspect the digit immediately to its right, increase the rounding digit if that next digit is 5 or more, then replace later integer digits with zeros.

For negative integers, choose the nearest multiple on the number line; for example, $-6739$ rounds to $-6700$ to the nearest hundred.

Do not count digits from the left without identifying place value. Rounding to $10^2$ means the nearest hundred, not two significant figures.

Decimal places count digits after the decimal point. Significant figures begin at the first non-zero digit and describe the precision of the whole value.

Number	Instruction	Result
45.621	nearest whole number	46
45.621	2 decimal places	45.62
0.004786	2 significant figures	0.0048
58 749	3 significant figures	58 700

Mark the last digit to keep, inspect the next digit, round up for 5–9, and retain placeholder zeros when they communicate magnitude or required decimal places.

Leading zeros are not significant; zeros between non-zero digits are significant. Trailing zeros after a decimal can show stated precision.

Two decimal places and two significant figures usually give different answers. Identify which counting system the question states.

A rounded value represents an interval of possible original values. Half a rounding unit lies below the stated value and half lies above it.

Stated value	Rounding unit	Interval
4.3 kg to nearest 0.1 kg	0.1	$4.25\le w<4.35$
125 cm to nearest cm	1	$124.5\le l<125.5$
2400 to nearest 100	100	$2350\le n<2450$

The lower bound is included because it rounds up to the stated value. The upper bound is excluded because that exact value rounds to the next result.

For a stated number of decimal places or significant figures, first identify the value of the last retained digit; that is the rounding unit.

The bound offset is half a rounding unit, not half the rounded value and not always $0.5$ .

An estimate replaces values with nearby easy numbers so a calculation can be checked mentally and its order of magnitude judged.

Exact expression	Suitable estimate
$68.3 imes42.8\div0.021$	$70 imes40\div0.02=140000$
$19.8 imes4.13$	$20 imes4=80$
$598\div0.31$	$600\div0.3=2000$

Rounding each value to one significant figure is a reliable default, but choose compatible numbers that keep the approximation easy and reasonably close.

Compare the estimate with the calculator result. A large disagreement in size or decimal position signals an input or operation error.

An estimate is not expected to equal the exact result. It must be simple enough to evaluate and close enough to test plausibility.

To maximise or minimise an expression, choose the combination of input bounds that makes the entire expression largest or smallest.

Positive quantities	Upper value uses	Lower value uses
product $ab$	$a_Ub_U$	$a_Lb_L$
quotient $a/b$	$a_U/b_L$	$a_L/b_U$
difference $a-b$	$a_U-b_L$	$a_L-b_U$

For an outer rectangle $8.3 imes7.2$ minus an inner rectangle $6.2 imes5.3$ , all lengths correct to 0.1 cm, the upper shaded area is $8.35 imes7.25-6.15 imes5.25=28.25 ext{ cm}^2$ .

To give a result to a suitable degree of accuracy, calculate both outcome bounds and round only to a precision for which both bounds produce the same stated value.

Using every upper bound does not always maximise a composite expression. A subtracted area or denominator may need its lower bound.

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—1.9 Standard formRead

1.8 Degree of accuracy

Learning objectives

Round integers to powers of 10

Round to decimal places and significant figures

Identify upper and lower bounds

Estimate numerical calculations

Solve upper- and lower-bound problems

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