IB Mathematics Applications & Interpretation Topic SL…

Why We Approximate

Real measurements are never perfectly exact, and a GDC often returns a long string of decimals, so we round to a sensible length to communicate a value clearly.
Rounding always throws a little information away, so this topic is also about measuring how far a rounded value can be from the truth.

Note

Unless a question says otherwise, AI answers are given exact or to three significant figures, so knowing how to round correctly protects easy marks on every paper.

Rounding: Decimal Places And Significant Figures

Decimal places (d.p.) count the digits after the decimal point, so you round by looking at the first digit you are about to drop.
Significant figures (s.f.) count from the first non-zero digit onwards, so they measure how many meaningful digits a number carries.
The rounding rule is the same for both: if the next digit is 5 or more, round up, otherwise round down.

Tip

Leading zeros are never significant (they only place the decimal point), but zeros between non-zero digits always are.

Example

$0.040\,72$ to 2 s.f. is $0.041$ (the first significant digit is the 4).
$0.040\,72$ to 2 d.p. is $0.04$ (two digits after the point).
$3567$ to 2 s.f. is $3600$ (keep the 3 and 6, and use zeros to hold place value).

Example question

Round $0.008\,254$ as stated:

(a) to 3 significant figures

(b) to 3 decimal places

Solution

(a) The significant digits are $8, 2, 5$; the next digit is 4, so $0.008\,25$.
(b) Three places after the point give $0.008$.

Upper And Lower Bounds

A rounded number does not name one exact value; it stands for a whole range of values that all round to it.
The range reaches half a rounding unit either side, so the bounds sit half a unit below and half a unit above the stated value.
Write the range with the lower bound included and the upper bound excluded, because a value exactly on the upper bound would round up to the next figure.

Example

A length given as $24$ cm to the nearest cm lies in $23.5 \le L < 24.5$.
A mass given as $4.5$ kg to the nearest $0.1$ kg lies in $4.45 \le m < 4.55$.

Example question

The length of a table is $1.8$ m, correct to 2 significant figures. Write down its lower and upper bounds.

Solution

2 s.f. here means rounded to the nearest $0.1$ m, so the range reaches $0.05$ m either side.
Lower bound $1.75$ m, upper bound $1.85$ m, so $1.75 \le L < 1.85$.

Percentage Error

When we replace an exact value with an approximation, percentage error measures how big that mistake is relative to the true value. $$ \varepsilon = \left| \frac{v_A - v_E}{v_E} \right| \times 100\% $$
$v_E$ is the exact (accepted) value and $v_A$ is the approximate value; the exact value always goes on the bottom.
The bars mean absolute value, so percentage error is never negative: it measures size, not direction.

Example question

The fraction $\tfrac{22}{7}$ is often used as an approximation for $\pi$. Find the percentage error, giving your answer to 3 significant figures.

Solution

Here $v_A = \tfrac{22}{7} = 3.142857\dots$ and $v_E = \pi = 3.141593\dots$
$\varepsilon = \left| \dfrac{3.142857 - 3.141593}{3.141593} \right| \times 100\%$.
$\varepsilon = 0.0402\%$ (3 s.f.).

Estimation As A Sanity Check

Rounding every number in a calculation to one significant figure gives a quick estimate you can compare with your GDC answer.
If the estimate and the exact answer are wildly different, you have probably mis-keyed something, so estimation is a fast way to catch errors.

Exam technique

The formula booklet (SL 1.6) gives the percentage error formula, so the marks are for substituting the right values, especially the exact value on the bottom.
Keep full precision in your GDC and round only the final answer, because rounding partway through introduces its own error.
Match the rounding the question asks for: "3 s.f." and "3 d.p." are not the same instruction.

Active recall

What is the difference between a significant figure and a decimal place?
Round $0.030\,49$ to 2 significant figures.
A distance is $350$ m to the nearest $10$ m. State its lower and upper bounds.
In the percentage error formula, which value goes in the denominator, and why?
Why should you avoid rounding in the middle of a multi-step calculation?

Why We Approximate

Real measurements are never perfectly exact, and a GDC often returns a long string of decimals, so we round to a sensible length to communicate a value clearly.
Rounding always throws a little information away, so this topic is also about measuring how far a rounded value can be from the truth.

Note

Unless a question says otherwise, AI answers are given exact or to three significant figures, so knowing how to round correctly protects easy marks on every paper.

Rounding: Decimal Places And Significant Figures

Decimal places (d.p.) count the digits after the decimal point, so you round by looking at the first digit you are about to drop.
Significant figures (s.f.) count from the first non-zero digit onwards, so they measure how many meaningful digits a number carries.
The rounding rule is the same for both: if the next digit is 5 or more, round up, otherwise round down.

Tip

Leading zeros are never significant (they only place the decimal point), but zeros between non-zero digits always are.

Example

$0.040\,72$ to 2 s.f. is $0.041$ (the first significant digit is the 4).
$0.040\,72$ to 2 d.p. is $0.04$ (two digits after the point).
$3567$ to 2 s.f. is $3600$ (keep the 3 and 6, and use zeros to hold place value).

Example question

Round $0.008\,254$ as stated:

(a) to 3 significant figures

(b) to 3 decimal places

Solution

(a) The significant digits are $8, 2, 5$; the next digit is 4, so $0.008\,25$.
(b) Three places after the point give $0.008$.

Upper And Lower Bounds

A rounded number does not name one exact value; it stands for a whole range of values that all round to it.
The range reaches half a rounding unit either side, so the bounds sit half a unit below and half a unit above the stated value.
Write the range with the lower bound included and the upper bound excluded, because a value exactly on the upper bound would round up to the next figure.

Example

A length given as $24$ cm to the nearest cm lies in $23.5 \le L < 24.5$.
A mass given as $4.5$ kg to the nearest $0.1$ kg lies in $4.45 \le m < 4.55$.

Example question

The length of a table is $1.8$ m, correct to 2 significant figures. Write down its lower and upper bounds.

Solution

2 s.f. here means rounded to the nearest $0.1$ m, so the range reaches $0.05$ m either side.
Lower bound $1.75$ m, upper bound $1.85$ m, so $1.75 \le L < 1.85$.

Percentage Error

When we replace an exact value with an approximation, percentage error measures how big that mistake is relative to the true value. $$ \varepsilon = \left| \frac{v_A - v_E}{v_E} \right| \times 100\% $$
$v_E$ is the exact (accepted) value and $v_A$ is the approximate value; the exact value always goes on the bottom.
The bars mean absolute value, so percentage error is never negative: it measures size, not direction.

Example question

The fraction $\tfrac{22}{7}$ is often used as an approximation for $\pi$. Find the percentage error, giving your answer to 3 significant figures.

Solution

Here $v_A = \tfrac{22}{7} = 3.142857\dots$ and $v_E = \pi = 3.141593\dots$
$\varepsilon = \left| \dfrac{3.142857 - 3.141593}{3.141593} \right| \times 100\%$.
$\varepsilon = 0.0402\%$ (3 s.f.).

Estimation As A Sanity Check

Rounding every number in a calculation to one significant figure gives a quick estimate you can compare with your GDC answer.
If the estimate and the exact answer are wildly different, you have probably mis-keyed something, so estimation is a fast way to catch errors.

Exam technique

The formula booklet (SL 1.6) gives the percentage error formula, so the marks are for substituting the right values, especially the exact value on the bottom.
Keep full precision in your GDC and round only the final answer, because rounding partway through introduces its own error.
Match the rounding the question asks for: "3 s.f." and "3 d.p." are not the same instruction.

Active recall

What is the difference between a significant figure and a decimal place?
Round $0.030\,49$ to 2 significant figures.
A distance is $350$ m to the nearest $10$ m. State its lower and upper bounds.
In the percentage error formula, which value goes in the denominator, and why?
Why should you avoid rounding in the middle of a multi-step calculation?

IB Mathematics Applications & Interpretation SL 1.6—approximating and Estimating Notes

IB Mathematics Applications & Interpretation SL 1.6—approximating and Estimating Notes

Why We Approximate

Rounding: Decimal Places And Significant Figures

Upper And Lower Bounds

Percentage Error

Estimation As A Sanity Check

Why We Approximate

Rounding: Decimal Places And Significant Figures

Upper And Lower Bounds

Percentage Error

Estimation As A Sanity Check

Number and Algebra

Functions

Geometry and Trigonometry

Statistics and Probability

Calculus

Number and Algebra

Functions

Geometry and Trigonometry

Statistics and Probability

Calculus

IB Mathematics Applications & Interpretation SL 1.6—approximating and Estimating Notes

IB Mathematics Applications & Interpretation SL 1.6—approximating and Estimating Notes

Why We Approximate

Rounding: Decimal Places And Significant Figures

Upper And Lower Bounds

Percentage Error

Estimation As A Sanity Check

Why We Approximate

Rounding: Decimal Places And Significant Figures

Upper And Lower Bounds

Percentage Error

Estimation As A Sanity Check