Approximation & Error (AI SL)
Every measurement and every rounded number carries some error, and this topic is about controlling and quantifying it. You'll practise rounding to decimal places and significant figures, working out the upper and lower bounds a rounded value could have come from, and measuring how far an estimate strays from the exact answer with percentage error. It's calculator-permitted throughout, but the GDC only does the arithmetic - you still need to know which rule to apply.
What the syllabus says
This topic maps onto one point in the official IB Applications & Interpretation syllabus, covering four related skills.
| Code | Syllabus content |
|---|---|
| SL1.6 | Approximation: decimal places and significant figures. Students should be able to choose an appropriate degree of accuracy based on given data. |
| SL1.6 | Upper and lower bounds of rounded numbers - e.g. if \(x = 4.1\) to one decimal place, then \(4.05 \le x < 4.15\). |
| SL1.6 | Percentage errors, including finding the maximum percentage error caused by a measurement error - for example, the percentage error in the area of a circle if the radius was measured as \(2.5\) cm to one decimal place. |
| SL1.6 | Estimation: recognising whether an answer is reasonable in the context of the given data. |
SL1.6 sits in the Number & Algebra unit and underpins how you're expected to present and check every numerical answer across the AI syllabus.
Key terms
Four words worth knowing cold before you touch the formulas below - each with a worked example showing exactly what it means.
What is rounding?
Rounding replaces a number with a nearby one that has fewer digits, using the next digit to decide whether to round up or down. If that next digit is 5 or more, the last kept digit increases by one; otherwise it stays the same.
e.g. \(3.678\) rounded to 2 s.f.: the next digit is \(7 \ (\ge 5)\), so it rounds up to \(3.7\).
What are significant figures?
Significant figures (s.f.) count a number's meaningful digits, starting from the first non-zero digit. Leading zeros before that point are never significant, but zeros between or after significant digits usually are.
e.g. \(0.003080\) has 4 significant figures: \(3, 0, 8, 0\) - the three leading zeros don't count.
What is percentage error?
Percentage error measures how far an approximate value is from the exact value, as a percentage of the exact value. It's always given as a positive number, using the modulus of the difference.
e.g. a length measured as \(48\) cm when the true length is \(50\) cm has percentage error \(\left|\dfrac{48-50}{50}\right|\times100\% = 4\%\).
What are upper and lower bounds?
When a value has been rounded, its upper and lower bounds are the largest and smallest numbers that would have rounded to it. The gap on each side is half the place value you rounded to.
e.g. a length of \(24\) cm to the nearest cm has bounds \(24 \pm 0.5\), so \(23.5 \text{ cm} \le x < 24.5\text{ cm}\).
Key formulas
There's really only one formula to memorise here - the rest of the topic is about applying rounding conventions correctly. The tables below cover both.
Formula reference
The percentage error formula is given in the IB formula booklet; working out bounds and estimates is a technique rather than a formula, so nothing else needs to be memorised.
| Formula | Used for | Booklet? |
|---|---|---|
| \(\varepsilon = \left|\dfrac{v_A - v_E}{v_E}\right|\times100\%\) | Percentage error | ✓ Yes |
| lower bound \(= x - \tfrac12 \times\) (place value) | Bounds of a rounded number | Not in the formula booklet - prior knowledge |
| upper bound \(= x + \tfrac12 \times\) (place value) | Bounds of a rounded number | Not in the formula booklet - prior knowledge |
Degrees of accuracy
Decimal places and significant figures are the two standard ways IB questions ask you to state a level of accuracy.
Decimal places (d.p.)
Count digits after the decimal point, then look one further digit to decide whether to round up.
\(8.297\) to 1 d.p. \(= 8.3\) - the 2nd decimal, 9, rounds the 1st decimal up.
Not in the formula booklet - prior knowledgeSignificant figures (s.f.)
Start counting from the first non-zero digit, in either direction from the decimal point.
\(48\,372\) to 2 s.f. \(= 48\,000\) - only the 4 and 8 are kept as meaningful digits.
Not in the formula booklet - prior knowledgeChoosing accuracy
The precision you round to should match the precision of the data you were given - never invent extra accuracy.
If a question gives data to 3 s.f., an answer stated to 6 d.p. is over-precise and can lose marks.
Not in the formula booklet - prior knowledgeBounds and error
These skills quantify what rounding costs you - how uncertain a rounded value really is, and how far an estimate can drift from the truth.
Bounds
\[x - \tfrac12(\text{unit}) \le x < x + \tfrac12(\text{unit})\]
A mass of \(3.6\) kg to 1 d.p. lies in \(3.55 \le m < 3.65\).
Not in the formula booklet - prior knowledgePercentage error
\[\varepsilon = \left|\dfrac{v_A - v_E}{v_E}\right|\times100\%\]
Always divide by the exact value \(v_E\), and keep the answer positive.
✓ In the formula bookletEstimation
Round each number to 1 s.f. before combining, to get a quick sanity-check value.
\(\dfrac{198\times5.1}{0.49}\approx\dfrac{200\times5}{0.5}=2000\).
Not in the formula booklet - prior knowledgeWorked examples
Two full exam-style questions, marked exactly like the real thing. Try each one yourself before checking the worked solution.
Consider \(3.678\).
(a) Round it to 2 s.f.
(b) Round it to 3 s.f.
(c) Find the difference between the two results.
Worked solution
(a) Round \(3.678\) to 2 s.f.: figures \(3,6\); the next digit \(7\ (\ge5)\) rounds up \(\Rightarrow 3.7.\) A1
(b) Round \(3.678\) to 3 s.f.: figures \(3,6,7\); the next digit \(8\ (\ge5)\) rounds up \(\Rightarrow 3.68.\) A1
(c) \(3.7-3.68=0.02.\) A1
Keeping more significant figures gives a value closer to \(3.678\), so the 2 s.f. answer carries the larger rounding error. R1
Consider \(\dfrac{612\times0.48}{31}.\)
(a) Estimate by rounding each number to 1 s.f.
(b) Find the exact value to 3 s.f.
(c) Find the percentage error of the estimate.
Worked solution
(a) \(612\approx600,\ 0.48\approx0.5,\ 31\approx30.\) M1
Estimate \(=\dfrac{600\times0.5}{30}=\dfrac{300}{30}=10.\) A1
(b) Exact value: \(\dfrac{612\times0.48}{31}\approx9.48\) (3 s.f.). A1
(c) Using the exact value as \(v_E\): \(\left|\dfrac{10-9.48}{9.48}\right|\times100\%.\) M1
\(\varepsilon\approx5.49\%.\) A1
Common mistakes
The four slip-ups that account for most of the marks lost on this topic - worth reading before you start practising, not just after you get one wrong.
- Mixing up decimal places and significant figures. Decimal places count from the decimal point; significant figures count from the first non-zero digit. \(0.004567\) is \(0.00\) to 2 d.p. but \(0.0046\) to 2 s.f.
- Rounding intermediate values before the final step. Rounding too early compounds error through a calculation. Keep full calculator precision throughout, and round only the final answer.
- Counting leading zeros as significant figures. In \(0.003080\), the three zeros before the \(3\) are not significant - only \(3,0,8,0\) count, giving 4 s.f., not 7.
- Dividing by the approximate value instead of the exact value in percentage error. The formula is \(\left|\dfrac{v_A-v_E}{v_E}\right|\times100\%\) - the denominator must always be the exact (true) value, and the modulus keeps the result positive.
Using your GDC
Every step below is a real button sequence, not a vague "use your calculator" hint - covering the TI-84 Plus, TI-Nspire, and Casio fx-9860/fx-CG50. Pick your model to filter down to just the steps that apply to you.
For very large or very small numbers - avoids typing long strings of zeros and prevents rounding errors, useful when a rounded answer comes out like \(4.77\times10^5\).
- Scientific notation means \(a\times10^n\), e.g. \(3.2\times10^8\) or \(4.5\times10^{-3}\).
- Use 2nd → , (EE) to enter the ×10 part: type 3.2 2nd , 8 to enter \(3.2\times10^8\). Do NOT type ×10^ separately.TI-84
- Use the EE key (or type ×10^ from the keyboard template) to enter scientific notation. Or just type 3.2×10^8 using the ^ key.Nspire
- Use the ×10ˣ key (EXP key) - type 3.2 then EXP then 8. Do NOT type ×10^ manually.Casio
- To display answers in scientific notation: on TI-84 press MODE and choose SCI; on Casio set the display mode in SET UP.
Tip: A common mistake is typing ×10^ instead of using the EE/EXP key - this gives ×10×... (multiplication, then a power) rather than proper scientific notation.
The most common arithmetic errors on the GDC come from missing brackets - especially awkward in a percentage error calculation with a fraction and a subtraction stacked together.
- Use brackets whenever you have a fraction, a negative, or a compound expression in an exponent.
- Fraction: type (3+5)÷(2−1) not 3+5÷2−1 - the calculator respects order of operations, so division binds tightly.
- Power: type (2x+1)^3 not 2x+1^3 - without brackets only the 1 is cubed.
- Powers: use the ^ key. Square root: 2nd → √ then close the bracket.TI-84
- Powers: use the ^ key or the exponent template. Roots: use the √ template from the maths palette (ctrl+B or the template key).Nspire
- Powers: use the ^ key (x□ key). Square root: SHIFT → √. Fractions: use the fraction template (SHIFT → ÷) for clean stacked fractions.Casio
- Always check the answer makes sense - a wildly large or small result usually means a missing bracket.
Tip: When in doubt, add extra brackets. (3+5)÷(2) and 3+5÷2 give different answers - the first is almost always what you mean, and it's exactly the trap a percentage error calculation sets.
See the full GDC guide for more calculator models and topics.
Ready to practise properly?
Approximation & error questions, marked instantly like the real exam.
Quick answers
The questions students on this topic ask most often.
What's the difference between decimal places and significant figures?
Decimal places count digits after the decimal point; significant figures count meaningful digits starting from the first non-zero digit. So \(0.004567\) is \(0.00\) to 2 d.p. but \(0.0046\) to 2 s.f.
How do I find the percentage error?
Use \(\varepsilon = \left|\dfrac{v_A - v_E}{v_E}\right|\times100\%\), where \(v_A\) is the approximate value and \(v_E\) is the exact value. Always divide by the exact value, keep the result positive with the modulus, and use unrounded values in the calculation itself.
What are upper and lower bounds?
If a value has been rounded, its bounds are the range of numbers that would round to it. For a length given as 24 cm to the nearest cm, the true value lies between the lower bound 23.5 cm and the upper bound 24.5 cm.
Can I use my GDC for this topic?
Yes - every question on this topic is calculator-permitted. Your GDC handles the arithmetic; the skill being tested is knowing which rounding rule, bound, or error formula to apply and reading the calculator's output correctly, including scientific notation. See the GDC guide for model-specific instructions.
Sub-topics
Approximation & Error broken down into its individual skills, each with its own focused page.
Related topics
More Number & Algebra topics from the same AI SL syllabus unit, in case you want to keep going.