IB Maths AI SL Topic 1 — Number & Algebra Paper 1 & 2 Default: 3 s.f. ~7 min read

Approximation

Approximation just means rounding: cutting a long number down to a sensible length. You’ll meet two kinds — significant figures (s.f.) and decimal places (d.p.). The rule is the same for both: look at the next digit, and if it’s 5 or more, round up. The tricky bits are (1) knowing which digits are “significant”, (2) knowing when context tells you to round UP regardless of the digit, and (3) keeping enough digits in your working so the final answer is accurate.

📘 What you need to know

The default in IB Maths is 3 s.f. — unless the question specifies something else (like “to 2 d.p.” or “to the nearest whole number”), give your final answer to 3 significant figures.
Significant figures count from the first NON-ZERO digit. Leading zeros are NOT significant; every digit after the first non-zero one IS. e.g. in 0.005408 the sig figs are 5, 4, 0, 8.
Decimal places count from the decimal point. e.g. 17.4658 to 1 d.p. = 17.5; to 2 d.p. = 17.47.
The “decider” digit: look at the digit just AFTER the one you’re keeping. If ≥ 5, round up the last kept digit; if < 5, leave it.
Keep more s.f. in working than in the final answer: if the answer needs 3 s.f., carry at least 4 s.f. (or use exact values) through the calculation. Otherwise rounding errors compound.
Currency conventions: USD, EUR, GBP use 2 d.p. (the “cents”); JPY, KRW, IDR use whole numbers. Use the convention for whichever currency the question mentions.
Round UP in “container” contexts: how many buses for 142 students, how many boxes for 87 items, how many minutes to charge a battery — you can’t have 0.7 of a bus or box, and rounding DOWN leaves things undone. Round UP regardless of the decider digit.
Round DOWN when something is a maximum or limit: how many full boxes can you make from 100 items if each holds 12 — 100/12 = 8.3, but you can only make 8 FULL boxes.

Significant figures vs decimal places

Significant figures (s.f.)

count from FIRST non-zero digit

Works for any size of number. Leading zeros don’t count; everything after the first non-zero digit DOES. e.g. 0.00 5 4 0 8 has 4 s.f. shown in bold.

Decimal places (d.p.)

count from the DECIMAL POINT

Counts digits AFTER the decimal. e.g. 17.4658 — the 4 is 1 d.p., the 6 is 2 d.p., the 5 is 3 d.p., the 8 is 4 d.p.

Why does this distinction matter? Because 0.00541 (3 s.f.) and 0.005 (3 d.p.) are very different numbers! For tiny numbers, s.f. preserves more accuracy. For numbers near 1 it barely matters. Always read the question carefully to see which the examiner wants.

The “decider digit” rule — same for s.f. and d.p.

The rule Look at the digit JUST AFTER the one you’re keeping. If it’s 5, 6, 7, 8, or 9 → round UP. If it’s 0, 1, 2, 3, or 4 → round DOWN (keep as is).

The kept digits (teal) decide how many s.f. you have. The decider (red) tells you whether to round up. Digits AFTER the decider (grey) don’t matter for the decision — but for whole numbers, they become placeholder zeros to keep place value.

🧭 Recipe — rounding any number

Identify your target: how many s.f. or d.p. do you need? (Default is 3 s.f. if not stated.)
Find the last kept digit: for s.f., count from the first non-zero digit; for d.p., count from the decimal point.
Look at the decider: the digit just AFTER the last kept digit. Is it ≥ 5 or < 5?
Round: if decider ≥ 5, add 1 to the last kept digit. If decider < 5, leave the last kept digit alone.
Keep place value: for whole numbers, replace digits after the kept ones with zeros (e.g., 4738 → 4700). For decimals, just drop the digits after.

Worked examples

WE 1

Significant figures — small decimal with leading zeros

Round 0.005408 to 3 significant figures.

Step 1 — find the first non-zero digit 0.005408 — the 5 is the 1st sig fig (leading zeros DON’T count as sig figs) Step 2 — count out 3 sig figs 5 (1st), 4 (2nd), 0 (3rd) ← last kept digit Step 3 — look at the decider (next digit) next digit = 8 8 ≥ 5 → round UP the last kept digit (0 → 1) 0.005408 ≈ 0.00541 (3 s.f.) key insight: the zeros BEFORE the 5 are placeholders for size — they don’t count. But the 0 in the middle of “540” IS significant. Once you start counting from the first non-zero digit, every digit after counts.

WE 2

Decimal places

Round 17.4658 to 1 decimal place.

Step 1 — find the digit at the 1st decimal place 17.4658 — the 4 is at the 1st d.p. Step 2 — look at the decider (2nd d.p.) next digit = 6 6 ≥ 5 → round UP (4 → 5) Step 3 — drop digits after the 1st d.p. 17.4658 ≈ 17.5 (1 d.p.) don’t be fooled by what comes after the decider. The 5 and 8 in 17.4658 don’t affect the decision — only the FIRST digit after the cut matters. So 17.4632, 17.4699, and 17.4658 all round to 17.5 at 1 d.p.

WE 3

Significant figures — large whole number

Round 4738 to 2 significant figures.

Step 1 — first non-zero digit is 4 (1st sig fig) 4738 Step 2 — count out 2 sig figs 4 (1st), 7 (2nd) ← last kept digit Step 3 — look at the decider next digit = 3 3 < 5 → keep 7 as is Step 4 — fill remaining places with zeros (whole number — must keep place value) 47 _ _ → 47 0 0 4738 ≈ 4700 (2 s.f.) you MUST keep the zeros at the end — otherwise “47” would be a much smaller number! The zeros aren’t significant themselves; they’re just holding place to show the number is in the thousands.

WE 4

Real-world — round UP regardless

A school is taking 142 students on a trip. Each minibus holds 9 students. How many minibuses are needed?

Step 1 — do the division 142 ÷ 9 = 15.777… Step 2 — interpret the context 15.77 minibuses isn’t possible — you can’t hire 0.77 of a bus if you book only 15 buses, that’s 15 × 9 = 135 seats 142 − 135 = 7 students LEFT BEHIND ✗ Step 3 — round UP to cover everyone need 16 buses (16 × 9 = 144 seats — fits 142 with 2 to spare) 16 minibuses needed the maths gave 15.77 — normal rounding would give 16 anyway (since .77 ≥ .5). But what if it had been 15.1? Normal rounding gives 15 — yet the context demands 16. ALWAYS think about what your number means in real life.

WE 5

Currency — using the right convention

A bill comes to $87.456. What amount does the customer pay?

Step 1 — which currency? Use the right convention USD uses 2 d.p. (cents) Step 2 — round to 2 d.p. $87.456 — keep up to 2nd d.p. (the 5) decider = 6, and 6 ≥ 5 → round UP 5 → 6 customer pays $87.46 currency conventions matter! For ¥ (Japanese yen), the same bill would round to a whole number — ¥87. For £ or € you’d use 2 d.p. like USD. Always check what currency you’re working in before deciding how many d.p.

WE 6

Multi-step — keep extra accuracy in working

Find the area of a circle with radius 4.7 cm. Give your answer to 3 significant figures.

Step 1 — write the formula and substitute A = π r² A = π × (4.7)² Step 2 — keep at least 4 s.f. (or exact) in working A = π × 22.09 A = 69.3978… cm² Step 3 — round the FINAL answer to 3 s.f. first 3 sig figs: 6, 9, 3 decider = 9 → 9 ≥ 5 → round UP (3 → 4) A ≈ 69.4 cm² (3 s.f.) NEVER round mid-calculation. If you’d used π ≈ 3.14 (only 3 s.f.) you’d get A = 3.14 × 22.09 = 69.36… → 69.4 — same answer this time, but for trickier problems early rounding gives wrong final answers. Use the π button on your GDC, not “3.14”.

💡 Top tips

Default to 3 s.f.: if a question doesn’t specify, give 3 s.f. on your final answer. Examiners expect this — it’s the IB house style.
Keep 4+ s.f. in working: the safest habit is to leave numbers as exact values (like 2√3 or π/4) until the very end, then round once.
Use the π and √ buttons, not decimal approximations: typing “3.14” or “1.41” instead of using the GDC’s exact symbols loses accuracy and risks a wrong final answer.
Ask “does my answer make sense?” after rounding. If you got 69.4 cm² for a circle with radius 4.7 cm, that’s roughly half a square 9.4 × 9.4 — sanity check passes. If you got 0.694 you’d know something went wrong.
Re-read the question for context clues: words like “people”, “buses”, “boxes”, “tickets” usually mean integer answers; words like “cost in dollars” usually mean 2 d.p. The number alone doesn’t tell you — the units do.

⚠ Common mistakes

Counting leading zeros as sig figs: in 0.005408, the three leading zeros don’t count. 0.005408 has 4 s.f., not 7. Start counting only from the FIRST non-zero digit.
Looking at ALL the digits after the decider: only the FIRST digit after the cut matters. 17.4658 to 1 d.p. is 17.5 — you don’t add up or “average” the 5 and 8 with the 6; you only look at the 6.
Forgetting place-value zeros for whole numbers: rounding 4738 to 2 s.f. gives 4700, NOT 47. The trailing zeros are essential to keep the number the right size.
Rounding too early in multi-step problems: if you round π to 3.14 at the start, your final answer might be off in the 3rd s.f. Use the π button until the last step.
Rounding DOWN in “container” problems: 142 students ÷ 9 per bus = 15.77 → 16 buses (round UP). Even though normal rounding gives 16 here, for cases like 152/9 = 16.88 you might mistakenly think “round up to 17” matches normal rounding — but 153/9 = 17.0 would give 18 by context. Always think about what the number REPRESENTS.

Up next: Upper & Lower Bounds. Now that you can round, the reverse question becomes important: if a number is given as 14.3 (1 d.p.), what’s the smallest and largest the TRUE value could be? Useful for error analysis and for figuring out the worst-case bounds of any quantity built from rounded measurements.

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