Approximation, error and estimation

IB Mathematics: Applications and Interpretation HL· Unit 1: Number and algebra, Topic 2· 25 min read

1. Significant Figures and Rounding★★☆☆☆⏱ 8 min

📘 Definition

Significant figures

abbreviated s.f.

Digits that carry meaningful information about the precision of a measurement. Leading zeros are never significant; trailing zeros are significant only after a decimal point.

Example:

0.0042 has 2 s.f., 2400 has 2 s.f., 2400.0 has 5 s.f.

Rounding to a specified number of significant figures is required for almost all calculations in IB exams, as it communicates the precision of your result. Follow these consistent rules to avoid mistakes:

Start counting from the first non-zero digit on the left
Count the required number of significant digits
Round the last digit up if the next digit is 5 or greater, otherwise leave it unchanged

📐 Worked Example

Round 0.024571 to 3 significant figures

1
The first non-zero digit is 2 (1st s.f.), so we count 3 digits: 2 (1st), 4 (2nd), 5 (3rd). The next digit after 5 is 7.
2
Since 7 ≥ 5, round the 3rd digit up from 5 to 6.
3
Final rounded value: 0.0246

2. Absolute, Relative and Percentage Error★★★☆☆⏱ 10 min

📘 Definition

Error measures

Quantify the discrepancy between a true (exact) value and an approximate or measured value. Three standard measures are used in IB AI HL:

Errors arise from rounding, measurement device precision limits, and estimation. Each error type communicates different information about the size of the discrepancy.

\text{Absolute error: } |\Delta x| = |x_{\text{true}} - x_{\text{approx}}|

\text{Relative error: } \frac{|\Delta x|}{|x_{\text{true}}|}, \quad x_{\text{true}} \neq 0

\text{Percentage error: } \frac{|\Delta x|}{|x_{\text{true}}|} \times 100\%

📐 Worked Example

A runner's race time is measured as 42.3 seconds. The actual official time is 42.7 seconds. Calculate the absolute, relative and percentage error of the measurement.

1
Identify values: true time $x_{t r u e} = 42.7$ s, approximate time $x_{a pp r o x} = 42.3$ s
2
Calculate absolute error:
3
$|\Delta x| = |42.7 - 42.3| = 0.4 \text{ s}$
4
Calculate relative error:
5
$\text{Relative error} = \frac{0.4}{42.7} \approx 0.0094$
6
Calculate percentage error:
7
$\text{Percentage error} = 0.0094 \times 100\% \approx 0.94\%$

3. Estimation of Calculations★★★☆☆⏱ 7 min

Estimation is used to quickly check if a final calculated answer is reasonable, or to get an approximate result when an exact value is not needed. The standard technique is to round every value in the calculation to 1 or 2 significant figures before computing.

📐 Worked Example

Estimate the value of $\frac{10.2 \times 58.7}{1.98}$

1
Round each value to 1 significant figure:
2
$10.2 \approx 10, \quad 58.7 \approx 60, \quad 1.98 \approx 2$
3
Substitute and simplify:
4
$\frac{10 \times 60}{2} = 300$
5
The exact value is ~ 302.6, so the estimate is accurate for a quick check.

✓ Quick check

Test your understanding of maximum error

A mass is given as 1400 g to 2 significant figures. What is the maximum possible absolute error?
- 0.5 g
- 5 g
- 50 g
- 100 g
Reveal answer
50 g —
1400 g to 2 s.f. means the true value lies between 1350 g and 1450 g, so the maximum error is $1450 - 1400 = 50$ g.

4. Common Pitfalls

Wrong move:

Counting leading zeros as significant figures

Why:

Leading zeros only mark the position of the decimal point, they do not contribute to precision. Overcounting leads to incorrect rounding

Correct move:

Always start counting significant figures from the first non-zero digit from the left

Wrong move:

Using the approximate value instead of the true value in the denominator for percentage error

Why:

The IB syllabus defines percentage error relative to the true value, so this will always give an incorrect result

Correct move:

Always divide the absolute error by the true value when calculating relative or percentage error

Wrong move:

Writing 1500 to indicate 4 significant figures, with no additional notation

Why:

Trailing zeros without a decimal point are ambiguous, and examiners will assume they are placeholders

Correct move:

Write it as $1.500 \times 1 0^{3}$ to clearly show 4 significant figures

Wrong move:

Forgetting to include units for absolute error

Why:

Absolute error is a measure of the difference in the original quantity, so it shares the original units. Examiners often penalize missing units

Correct move:

Add the correct units to your absolute error answer, leave relative error unitless and add a % symbol for percentage error

5. Quick Reference Cheatsheet

Term	Formula	Key Note
Absolute Error	$∣ x_{t r u e} - x_{a pp r o x} ∣$	Same units as original quantity
Relative Error	$\frac{∣ x _{t r u e} - x _{a pp r o x} ∣}{∣ x _{t r u e} ∣}$	Unitless proportional error
Percentage Error	$\frac{∣ x _{t r u e} - x _{a pp r o x} ∣}{∣ x _{t r u e} ∣} \times 100%$	Add % symbol to final answer
Max Measurement Error (precision $u$ )	$\frac{u}{2}$	Use when true value is unknown
Estimation Rule	Round all values to 1 s.f.	Quick check for calculation reasonableness

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2021 · 1
Percentage error calculation
2022 · 1
Significant figure rounding
2023 · 2
Estimation of calculation

Going deeper

syllabusIB DP AI HL Official SyllabusFull Unit 1 syllabus reference

What's Next

Approximation and error analysis is a foundational skill that appears across every topic in IB AI HL, from descriptive statistics to differential equations to real-world modelling. Examiners enforce strict significant figure rules, so mastering this topic helps you avoid losing easy marks on every exam question. The focus on error also aligns with the applications focus of AI HL, as all real-world data has inherent uncertainty. Your next topic will build on this foundation to work with large and small numbers in scientific notation.