IB Maths AI HLNumber ToolkitPaper 1 & 2s.f. & d.p.~6 min read
Approximation
Approximation means rounding a number to a stated accuracy — significant figures or decimal places. The rule is always the same: keep the digits you want, then look at the next digit to decide whether to round up. Some real-world contexts also force you to round a particular way regardless of that digit.
The 1st significant figure is the first non-zero digit, reading left–to–right; every digit after it counts.
Decimal places (d.p.) count digits after the decimal point.
Rounding rule: look at the next digit — if it is 5 or more round up, if it is 4 or less round down.
If a question states an accuracy (e.g. “2 d.p.”), follow that — it overrides the 3 s.f. default.
In context you may have to round up or down regardless of the digit — e.g. enough coaches for every student.
Significant figures
The significant figures of a number are its meaningful digits, starting from the first non-zero digit. Leading zeros are not significant — they only fix the size of the number — but zeros between significant digits are.
Counting starts at the first non-zero digit. The deciding digit (7) sits just past the last figure kept — 7 ≥ 5, so the 8 rounds up.
Decimal places
The number of decimal places is simply how many digits sit after the decimal point. Rounding to d decimal places means keeping d digits there, then using the next digit to decide whether the last kept digit rounds up.
The rounding rule
next digit ≥ 5 ⇒ round up
next digit ≤ 4 ⇒ round down (leave it)the same rule applies to significant figures and to decimal places
Significant figures and decimal places are different counts: 0.0408 has 3 s.f. but 4 d.p. Always check which one the question asks for.
Rounding up in context
Sometimes the situation — not the deciding digit — tells you which way to round. If you need enough seats, containers or vehicles for everyone, you must round up, even when the decimal part is small.
Working accuracy: to be sure a final answer is correct to 3 s.f., carry 4 s.f. or more (or exact values) through your working — round only at the very end.
🧠Recipe — rounding a number
Read the accuracy asked for — s.f. or d.p.; if none is given, use 3 s.f.
Identify the last digit to keep — count significant figures from the first non-zero digit, or decimal places from the point.
Look at the next digit: 5 or more rounds up, 4 or less rounds down.
Keep the place value — replace dropped whole-number digits with zeros so the number stays the right size.
Check the context — if the situation demands it, round up (or down) regardless of the digit.
Worked examples
WE 1
Rounding to significant figures
A grain of sand has a mass of 0.0046382 g. Write this mass correct to 3 significant figures.
count 3 s.f. from the first non-zero digit0.004 6 382 → keep 4, 6, 3deciding digit is 88 ≥ 5, so 3 rounds up to 4≈ 0.00464 g (3 s.f.)leading zeros are not significant — counting starts at the 4.
WE 2
Rounding to decimal places
A wooden plank measures 7.38561 m. Write this length correct to 2 decimal places.
keep 2 digits after the decimal point7.38561 → keep 3, 8deciding digit is 55 ≥ 5, so 8 rounds up to 9≈ 7.39 m (2 d.p.)a deciding digit of exactly 5 still rounds up.
WE 3
Significant figures in a large number
A concert was attended by 48 627 people. Write this attendance correct to 2 significant figures.
count 2 s.f. from the first digit4 8627 → keep 4, 8deciding digit is 66 ≥ 5, so 8 rounds up to 9replace the dropped digits with zeros≈ 49 000 (2 s.f.)keep the place value — the zeros hold the number at the right size.
WE 4
Rounding a calculated result
A circle has radius 6.4 cm. Find its area, giving your answer correct to 3 significant figures.
area = πr2A = π × 6.42 = π × 40.96A = 128.679… cm2round to 3 s.f. — deciding digit 6A ≈ 129 cm2 (3 s.f.)keep the full value through the working, then round only the final answer.
WE 5
Rounding up in context
310 students are going on a trip. Each coach seats 48 students. How many coaches are needed?
divide the students by the seats31048 = 6.458…6 coaches seat only 288 — not enoughround up so every student has a seat7 coachesthe context forces a round up — 6.458 does not round to 6 here.
WE 6
Full question: exact value then rounding
A square photo frame has sides of 8 cm. (a) Find the exact length of the diagonal. (b) Give the diagonal correct to 2 decimal places. (c) Give the diagonal correct to 3 significant figures.
(a) diagonal by Pythagorasd = √(82 + 82) = √128 = 8√2 cm(b) 8√2 = 11.31370… — deciding digit 3round down ⇒ 11.31 cm(c) 3 s.f. of 11.31370… — deciding digit 1round down ⇒ 11.3 cm(a) 8√2 cm · (b) 11.31 cm · (c) 11.3 cmleave (a) as a surd — that is the exact value; round only when asked.
💡 Top tips
No accuracy stated? Use 3 s.f. by default — but a stated accuracy always wins.
Significant figures start at the first non-zero digit; decimal places start at the decimal point — don’t mix them up.
The rounding rule is the same for both: 5 or more rounds up.
Rounding a large number to s.f.? Fill with zeros so it keeps the right place value.
Carry 4 s.f. or more (or exact values) through working; round once, at the end.
âš Common mistakes
Counting leading zeros as significant — in 0.0046, the first s.f. is the 4.
Confusing s.f. with d.p. — 0.0408 is 3 s.f. but 4 d.p.
Dropping the zeros from a rounded large number — 48 627 to 2 s.f. is 49 000, not 49.
Rounding too early — rounding mid-calculation can throw off the final 3 s.f.
Ignoring the context — 6.458 coaches must round up to 7, not down to 6.
Next up: Upper & Lower Bounds — the smallest and largest values a rounded number could really have. Approximation works the other way round there: instead of rounding a number, you ask what range an already-rounded number came from.
Need help with AI HL Number Toolkit?
Get 1-on-1 help from an IB examiner who knows exactly what Paper 1 & 2 are looking for.
