IB Maths AI HLNumber ToolkitPaper 1 & 2exact values~6 min read
Accuracy & Estimation
An exact value carries a number’s full precision — a surd, a fraction, a constant like π. Estimation does the opposite: it rounds numbers to easy values for a quick, rough answer. Together they help you give precise answers and sense-check whether an answer is reasonable.
📘 What you need to know
An exact value represents a number fully and precisely — no rounding.
Exact values include fractions, surds (roots), logarithms and constants like π and e.
A decimal with trailing dots, e.g. √2 = 1.414…, is not exact — it never ends.
Estimation means rounding numbers to easy values before calculating, for a rough answer.
Use estimation to check a method — if the rough answer is far from the exact one, recheck.
If a calculation gives more than one answer, use logic to pick the valid one (e.g. a length cannot be negative).
Exact values
An exact value is the full, precise form of a number, written without rounding. Many numbers cannot be written exactly as a finite decimal — their decimal expansion runs on forever — so the exact form keeps the surd, fraction or constant instead.
Forms of an exact value
fractions 27 · surds √3 · logarithms ln 2 · constants π, e
a decimal that never terminates — 1.414… — is only an approximation
Each exact value on the left equals an unending decimal on the right. The exact form keeps full precision; the decimal is always an approximation.
Estimation
Estimation means rounding the numbers in a calculation to easy values — often 1 significant figure — before working it out. The result is rough, but quick enough to do without a calculator.
Its main job is checking: if a full calculation gives an answer wildly different from the estimate, the method has a mistake — a misplaced decimal point, or the wrong operation.
Sense-check habit: a quick estimate confirms the order of magnitude of an answer. If your method for a mean age gives 200, it is clearly wrong — estimation catches that instantly.
Choosing the correct answer
Some methods produce more than one mathematical answer — a quadratic gives two roots, a square root gives a positive and a negative value. Use the context to choose the valid one.
A length, a mass, a time or a count of objects cannot be negative, so a negative solution is rejected. Always read the situation before stating a final answer.
🧠Recipe — estimating a calculation
Round each number to an easy value — usually 1 significant figure.
Rewrite the calculation with those easy numbers.
Work it out by hand — it should now be simple enough without a calculator.
State the estimate, and compare it with the exact answer if you have one.
Recheck the method if the exact answer is a very different size from the estimate.
Worked examples
WE 1
Writing an exact value
A square has an area of 18 cm2. Write down the exact length of one of its sides.
side length = √areaside = √18 cmsimplify the surd: 18 = 9 × 2√18 = √9 × √2 = 3√2side = 3√2 cm (exact)leave it as a surd — 3√2 is exact; 4.24… would only be an approximation.
WE 2
Estimating a calculation
Estimate the value of (4.87 × 61.3) ÷ 2.95 by first rounding each number to 1 significant figure.
round each number to 1 s.f.4.87 ≈ 5, 61.3 ≈ 60, 2.95 ≈ 3calculate with the easy numbers(5 × 60) ÷ 3 = 300 ÷ 3estimate ≈ 100the exact value is 101.2… — the estimate is a good rough check.
WE 3
Estimation in context
A rectangular patio has an area of 90 m2, to the nearest m2. It is to be covered with square paving slabs of side 48 cm. Estimate the number of slabs needed.
each slab ≈ 50 cm × 50 cmslab area ≈ 2500 cm2patio area in cm290 m2 = 900 000 cm2900 000 ÷ 2500 = 9000 ÷ 25≈ 360 slabsround 48 to 50 first, and convert both areas to the same unit.
WE 4
Using estimation to spot an error
A student calculates 312 × 0.48 and writes the answer as 1497.6. Use estimation to show that this answer must be wrong.
round to easy values312 ≈ 300, 0.48 ≈ 0.5estimate = 300 × 0.5 = 150compare with the student’s answer1497.6 is about 10× too biganswer should be near 150 — 1497.6 is wrongthe true value is 149.76 — the student misplaced the decimal point.
WE 5
Choosing the correct answer
The length x cm of a rectangle satisfies x2 + 3x − 28 = 0. Solve the equation and state the length of the rectangle.
factorise the quadratic(x + 7)(x − 4) = 0x = −7 or x = 4use logic — a length cannot be negativelength = 4 cm (reject x = −7)both values solve the equation, but only the positive one fits the context.
WE 6
Full question: exact value, estimate, logic
A square field has an area of 250 m2. (a) Write down the exact length of one side. (b) Estimate this length to check it is sensible. (c) Solving for the side gives x = ±√250 — state which value is the side length and why.
(a) side = √250; simplify: 250 = 25 × 10√250 = √25 × √10 = 5√10 m(b) estimate — 250 is close to 256√250 ≈ √256 = 16, so side ≈ 16 m(c) a length cannot be negative(a) 5√10 m · (b) ≈ 16 m · (c) +5√10 m, since length > 05√10 = 15.8… — the estimate of 16 confirms the exact value is sensible.
💡 Top tips
Asked for an exact value? Leave surds, fractions and π as they are — never round.
Simplify surds for an exact answer: √18 = 3√2, √250 = 5√10.
Estimate by rounding to 1 significant figure — it makes mental arithmetic easy.
Use an estimate to check the size of a calculator answer — it catches decimal-point slips.
Always read the context: reject negative lengths, masses, times and counts.
âš Common mistakes
Rounding an exact value when the question asks for the exact form.
Leaving a surd unsimplified — √18 should be written as 3√2.
Treating an estimate as the final answer — it is only a rough check.
Forgetting to convert units when estimating — e.g. m2 and cm2.
Keeping a negative solution for a length, mass or other quantity that must be positive.
Next up: Solving Equations using a GDC — using your calculator to solve systems of linear equations and polynomial equations. The logic from this note carries over: the GDC gives the solutions, but you still decide which ones make sense in context.
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.
