International Baccalaureate · DP 1: Applications and Interpretation SL

Approximations and Error

⌚ ~90 min ▢ 9 blocks ✓ 42 exercises ➣ Prerequisites : Surds and Exponents, Measurement

I Rounding and Estimation

Recappreviously studied — click to reread

On 23 September 1999, NASA lost the Mars Climate Orbiter: the probe, part of a $\dollar 327$ million mission with its twin lander, burned up in the Martian atmosphere instead of entering orbit. The investigation found the cause. One piece of software reported the thruster impulses in pound-force seconds, while the navigation system read those numbers as newton seconds — and $1$ pound-force second $\approx 4.45$ newton seconds. Every reported value was therefore $4.45$ times too small.

The navigation system used the value $1$ where the true value was $4.45$. Compute the percentage error $\left|\frac{1 - 4.45}{4.45}\right| \times 100\pourcent$ committed on every single thruster firing.
Tiny errors are often harmless. Was this one tiny? What simple habit — checking the order of magnitude of a result — could have caught the problem months before the loss of the probe?

Approximating well is exactly that habit: rounding consciously, quantifying the error of an approximation, and giving the interval in which an unknown exact value lies.

In science and mathematics, exact values are often impossible to obtain, either because of measurement limitations or because some numbers (like $\pi$ or $\sqrt{2}$) cannot be written exactly as terminating decimals. Therefore, we often work with approximations.
Approximating well takes four skills: rounding numbers correctly, quantifying the error introduced by an approximation, linking measurement accuracy to the smallest division on a scale, and giving the interval in which the true value lies.

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Definition — Rounding Rules

We can round numbers in two main ways:

Decimal places (d.p.): the number of digits kept after the decimal point.
Significant figures (s.f.): the number of digits kept starting from the first non-zero digit.

Rounding rule (to the nearest): look at the first digit that will be removed. If it is $0$, $1$, $2$, $3$ or $4$, keep the last kept digit unchanged (round down); if it is $5$, $6$, $7$, $8$ or $9$, increase the last kept digit by $1$ (round up).

Example

Consider the number $N = 1204.5678$.

Rounded to 2 decimal places (2 d.p.): $1204.57$ (the 3rd decimal digit is $7$, so we round up).
Rounded to 3 significant figures (3 s.f.): $1.20 \times 10^3$ (the next digit is $4$, so we round down).
(Writing $1200$ is acceptable here, but $1.20\times 10^3$ makes the “3 s.f.” explicit.)

Definition — Estimation

Estimation means finding a value that is close enough to the correct answer using simpler calculations. It is useful for checking whether calculator results are reasonable (order of magnitude, size, and plausibility).

Method — Estimation Strategy

A common strategy is:

Round each number to 1 significant figure.
Perform the calculation with the rounded numbers.
Interpret the result as an approximate check (not an exact answer).

Example

A restaurant meal costs $\dollar$9.90 per person. Estimate the total cost for $5$ people.

Answer

To estimate, round to 1 significant figure:$$9.90 \approx 10 \quad \text{and} \quad 5 \approx 5.$$ So the estimated total cost is:$$10 \times 5 = 50.$$ (Exact total: $9.90 \times 5 = 49.50$, so the estimate is close.)

Skills to practice

Rounding Numbers
Estimating Values

II Absolute and Percentage Error

Recappreviously studied — click to reread

Definition — Absolute, Relative and Percentage Error

Let $V_{\text{exact}}$ be the true value and $V_{\text{approx}}$ be the approximated value.

The signed error is $e = V_{\text{approx}} - V_{\text{exact}}$.
The absolute error is $|e| = |V_{\text{approx}} - V_{\text{exact}}|$.
The relative error (when $V_{\text{exact}}\neq 0$) is $$ r = \left|\frac{V_{\text{approx}} - V_{\text{exact}}}{V_{\text{exact}}}\right|. $$
The percentage error is $$ \epsilon = r \times 100\pourcent = \left|\frac{V_{\text{approx}} - V_{\text{exact}}}{V_{\text{exact}}}\right|\times 100\pourcent. $$

Example

The estimated length of a table is $150$ cm, while the actual length is $145$ cm. Calculate the absolute error and the percentage error.

Answer

Absolute error: $|150 - 145| = 5$ cm.
Percentage error: $$\left|\frac{150 - 145}{145}\right|\times 100\pourcent \approx 3.45\pourcent.$$

Skills to practice

Calculating Absolute and Percentage Errors

III Measurement Accuracy

Recappreviously studied — click to reread

When we take measurements, we usually read a scale (a ruler, a thermometer, a balance, $\dots$). Often, the true value lies between two marks. We record the closest mark, so the recorded value is an approximation.
If the smallest division on the scale is $u$, then the measurement can be wrong by up to half of this division, i.e. by $\pm \dfrac{u}{2}$.
The same holds for any value displayed or rounded to a unit $u$ (a digital display, a value rounded to the nearest multiple of $u$): the true value lies within $\pm \dfrac{u}{2}$ of it. A digital scale shows $2.48$ kg, to the nearest $0.01$ kg: the true mass lies between $2.475$ kg and $2.485$ kg.

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Definition — Accuracy Rule

If the smallest division on the scale is $u$, then:$$ \text{accuracy} = \pm \frac{u}{2}. $$ This is sometimes called the reading error or the absolute uncertainty. It is the largest possible reading error: it assumes the reading is taken to the nearest mark, and it does not include other sources of error (instrument, method).

Example

A ruler has markings every $1$ cm (so $u=1$ cm). A pencil is measured to be $14$ cm.

Determine the accuracy of the measurement.
State the interval within which the true length lies.

Answer

The accuracy is $\pm \dfrac{1}{2}\times 1\text{ cm}=\pm 0.5\text{ cm}$.
The true length $L$ is in the interval $[13.5,\;14.5)$ cm, i.e. $13.5 \le L < 14.5$ cm.

Note: If the ruler had millimetre marks ($u=1$ mm $=0.1$ cm), the accuracy would be $\pm 0.5$ mm $=\pm 0.05$ cm.

Skills to practice

Understanding Approximation
Determining Accuracy and Ranges

IV Upper and Lower Bounds

As in Measurement Accuracy, a value rounded to the nearest unit $u$ is within $\dfrac{u}{2}$ of the exact value. Written as an interval, the exact value lies in:$$ \text{Rounded Value} - \frac{u}{2} \le x < \text{Rounded Value} + \frac{u}{2}. $$ (Examples: nearest cm $\Rightarrow u=1$; nearest $0.1$ $\Rightarrow u=0.1$; rounded to $2$ d.p. $\Rightarrow u=0.01$.)

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Definition — Lower and Upper Bounds

For a rounded value:

The Lower Bound (LB) is the smallest possible value.
The Upper Bound (UB) is the value that the possible values never reach: every possible value is strictly below it, so we write “$<$” for the upper bound. The upper bound itself is not a possible value, but it is the number used in calculations.

Example

The length of a book is given as $L = 12$ cm, rounded to the nearest cm ($u=1$).

Calculate the lower bound and the upper bound of the length.
Write the inequality representing the possible values for the true length.

Answer

Bounds:
- Lower bound: $12 - 0.5 = 11.5$ cm.
- Upper bound: $12 + 0.5 = 12.5$ cm.
Inequality: Therefore: $11.5 \le L < 12.5$.

Method — Calculating Maximum and Minimum Values

When performing calculations with rounded values (area, perimeter, speed, $\dots$), errors propagate.
Let $A$ and $B$ be two positive quantities with bounds $A\in[A_{\min},A_{\max})$ and $B\in[B_{\min},B_{\max})$ (with $B_{\min}>0$ if dividing by $B$).

Operation	Upper bound	Lower bound
$A + B$	$A_{\max} + B_{\max}$	$A_{\min} + B_{\min}$
$A - B$	$A_{\max} - B_{\min}$	$A_{\min} - B_{\max}$
$A \times B$	$A_{\max} \times B_{\max}$	$A_{\min} \times B_{\min}$
$A \div B$	$A_{\max} \div B_{\min}$	$A_{\min} \div B_{\max}$

Example

A rectangle has length $l = 10$ cm and width $w = 5$ cm, both measured to the nearest cm. Calculate the upper and lower bounds of the area.

Answer

Step 1: Determine the bounds.
Since the measurements are to the nearest 1 cm, the accuracy is $\pm 0.5$ cm.
- Length $l$: Lower bound $= 9.5$, Upper bound $= 10.5$.
- Width $w$: Lower bound $= 4.5$, Upper bound $= 5.5$.
Step 2: Calculate the areas.
- Upper bound of the area (using the upper bounds of the sides): $$ A_{\max} = 10.5 \times 5.5 = 57.75 \text{ cm}^2 $$
- Lower bound of the area (using the lower bounds of the sides): $$ A_{\min} = 9.5 \times 4.5 = 42.75 \text{ cm}^2 $$

Method — Finding the Maximum Percentage Error

A value calculated from rounded measurements (an area, a volume, a speed, $\dots$) is an approximation $V_{\text{approx}}$: the exact value $V_{\text{exact}}$ is unknown, but it lies between the lower bound and the upper bound of the calculation; this lower bound must be strictly positive, since step 3 divides by it. The maximum percentage error of the approximation is the largest value that its percentage error can take or approach.

Calculate the approximation $V_{\text{approx}}$ with the rounded measurements.
Calculate the lower bound and the upper bound of the quantity, with the rules for the bounds of a sum, a difference, a product or a quotient.
Calculate the percentage error twice, replacing $V_{\text{exact}}$ by the lower bound, then by the upper bound: $$\epsilon = \left|\frac{V_{\text{approx}} - V_{\text{exact}}}{V_{\text{exact}}}\right| \times 100\pourcent.$$
Keep the larger of the two values: it is the maximum percentage error. Keep the bounds unrounded, and round only this final percentage.

For positive quantities, the percentage error grows as $V_{\text{exact}}$ moves away from $V_{\text{approx}}$, on either side: it is therefore largest at one of the two bounds — often the lower bound, since dividing by a smaller $V_{\text{exact}}$ gives a larger quotient. A bound of the quantity may be impossible to reach (when it comes from the upper bound of a measurement); as for any bound, it is still the number used in calculations, and the percentage error found there is then a limit that the error approaches without reaching.

Example

The radius of a circular table top is measured as $45$ cm, to the nearest centimetre.

Calculate the area of the table top given by this measurement.
Calculate the lower bound and the upper bound of the area.
Find the maximum percentage error in the area calculated in question 1.

Answer

With the measured radius $r = 45$ cm: $$\begin{aligned}[t] A_{\text{approx}} &= \pi \times 45^2 && \text{(area of a circle } \pi r^2\text{)}\\ &= 2025\pi && \text{(square)}\\ &\approx 6361.7 \text{ cm}^2. \end{aligned}$$
To the nearest centimetre, $44.5 \le r < 45.5$. The area $\pi r^2$ grows with $r$: $$\begin{aligned}[t] A_{\min} &= \pi \times 44.5^2 && \text{(lower bound of } r\text{)}\\ &= 1980.25\pi\\ &\approx 6221.1 \text{ cm}^2,\\ A_{\max} &= \pi \times 45.5^2 && \text{(upper bound of } r\text{)}\\ &= 2070.25\pi\\ &\approx 6503.9 \text{ cm}^2. \end{aligned}$$ The rounded values are for reading only: question 3 uses the exact forms $1980.25\pi$ and $2070.25\pi$.
The percentage error is calculated with $V_{\text{exact}}$ replaced by each bound, keeping $\pi$ exact (it cancels): $$\begin{aligned}[t] \epsilon_1 &= \frac{2025\pi - 1980.25\pi}{1980.25\pi} \times 100\pourcent && \text{(at } A_{\min}\text{)}\\ &= \frac{44.75}{1980.25} \times 100\pourcent && \text{(simplify by } \pi\text{)}\\ &\approx 2.26\pourcent,\\ \epsilon_2 &= \frac{2070.25\pi - 2025\pi}{2070.25\pi} \times 100\pourcent && \text{(at } A_{\max}\text{)}\\ &= \frac{45.25}{2070.25} \times 100\pourcent && \text{(simplify by } \pi\text{)}\\ &\approx 2.19\pourcent. \end{aligned}$$ The larger value is at the lower bound: the maximum percentage error in the area is about $\mathbf{2.26\pourcent}$.

Skills to practice

Determining and Calculating with Bounds
Finding the Maximum Percentage Error

V Going further

Skills to practice

Reasoning about Errors
Solving Problems with Errors and Bounds
Quiz

Operation	Upper bound	Lower bound
\(A + B\)	\(A_{\max} + B_{\max}\)	\(A_{\min} + B_{\min}\)
\(A - B\)	\(A_{\max} - B_{\min}\)	\(A_{\min} - B_{\max}\)
\(A \times B\)	\(A_{\max} \times B_{\max}\)	\(A_{\min} \times B_{\min}\)
\(A \div B\)	\(A_{\max} \div B_{\min}\)	\(A_{\min} \div B_{\max}\)