Rounding, Significant Figures and Bounds Calculator

How to use it

The first tab rounds a single number. Type the number, choose the accuracy — significant figures, decimal places, or the nearest whole number, 10, 100 or 1,000 — and the tool highlights the digit it kept last, underlines the digit that decided the rounding, and fades the digits it threw away.

The second tab is the one that carries marks at grade 6 and above. Enter two values that have already been rounded, say how each was rounded, choose an operation, and it works out the largest and smallest the answer could be, naming which bound of each value it used to get there.

• Rounding a measurement — use the first tab and read the error interval underneath
• Answering a "calculate the upper bound of" question — use the second tab
• Checking a bounds answer you already have — enter the same values and compare the pairing

How the calculation is worked out

Every rounding rule here is the same operation underneath: round to the nearest multiple of a power of ten. Two decimal places is the nearest 0.01. The nearest 100 is 10². Significant figures are the same thing with the power chosen from wherever the first non-zero digit sits, which is why 0.00408 to two significant figures rounds to the nearest 0.0001 and gives 0.0041.

That single power of ten also produces the error interval. Half a step below the rounded value and half a step above it are the smallest and largest measurements that would round to it, so 4.3 given to one decimal place means 4.25 ≤ x < 4.35.

For a calculation, the extremes always sit on a corner of the two intervals, so the tool tries all four combinations and reports which one won. The pairing rules fall out of that rather than being recited: the maximum of a ÷ b takes the largest a with the smallest b, and the minimum of a − b takes the smallest a with the largest b.

The arithmetic runs on the digits themselves rather than on floating-point numbers. A computer stores 4.35 as 4.3499999999999996, and a tool that named the deciding digit as 5 and then rounded down would be teaching you to distrust your own working.

What it deliberately does not do

It does not decide what degree of accuracy your answer should be given to. That is set by the question, or by the least accurate figure you were handed, and no calculator can read the question for you.

It does not take a calculation with more than two rounded values, or one where the same value appears twice. Bounds for an expression such as a(b + c) need the intervals combined in stages, and flattening that into a single answer would hide the step being assessed.

It does not round the final bound for you, and it does not turn a recurring bound into a fraction. Rounding partway through a bounds calculation moves the bound, so the figures are shown to ten significant figures and the last rounding is left to you.

Where students lose the marks

The commonest error by a distance is counting leading zeros as significant. In 0.00408 the first three zeros are placeholders — they fix the size of the number and do nothing else — so counting starts at the 4.

The second is dropping a trailing zero. 0.0999 to two significant figures is 0.100, not 0.1: those zeros state the accuracy being claimed, and removing them quietly changes the answer.

The third is the pairing in a bounds question. Almost everybody gets addition and multiplication right, then reaches for the largest value of b to find the largest value of a − b or a ÷ b. Subtracting or dividing by a bigger number makes the answer smaller, so the maximum needs the smallest b.

• Leading zeros are placeholders, never significant figures
• Trailing zeros after a decimal point are significant and must be written
• The maximum of a − b and of a ÷ b uses the lower bound of b
• Upper bounds are never reached — quote 4.35, but the value is strictly below it