Approximations and Errors

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.
In this chapter, we learn how to round numbers correctly, how to quantify the error introduced by an approximation, how measurement accuracy is linked to the smallest division on a scale, and how to determine bounds (an interval of possible true values).

We can round numbers in two main ways:

• Decimal places (dp): the number of digits kept after the decimal point.
• Significant figures (sf): 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 \(0,1,2,3,4\), round down; if it is \(5,6,7,8,9\), round up.

Consider the number \(N = 1204.5678\).

• Rounded to 2 decimal places (2 dp): \(1204.57\) (the 3rd decimal digit is \(7\), so we round up).
• Rounded to 3 significant figures (3 sf): \(1.20 \times 10^3\) (the next digit is \(4\), so we round down).
  (Writing \(1200\) is acceptable here, but \(1.20\times 10^3\) makes the ``3 sf'' explicit.)

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).

A common strategy is:

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

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

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 $$ \epsilon = r \times 100\pourcent = \left|\frac{V_{\text{approx}} - V_{\text{exact}}}{V_{\text{exact}}}\right|\times 100\pourcent. $$

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

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 \(\pm \dfrac{u}{2}\).

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.

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

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

When a value is rounded to the nearest unit \(u\), 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 \(2\) dp \(\Rightarrow u=0.01\).)

For a rounded value:

• The Lower Bound (LB) is the smallest possible value.
• The Upper Bound (UB) is the greatest possible value (usually not included), so we often write ``\(<\)'' for the upper bound.

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

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

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	Maximum Value	Minimum Value
\(A + B\)	\(A_{\max} + B_{\max}\)	\(A_{\min} + B_{\min}\)
\(A - B\)	\(A_{\max} - B_{\min}\)	\(A_{\min} - B_{\max}\)
\(A \times B\)	\(A_{\max} \times B_{\max}\)	\(A_{\min} \times B_{\min}\)
\(A \div B\)	\(A_{\max} \div B_{\min}\)	\(A_{\min} \div B_{\max}\)

A rectangle has length \(l = 10\) cm and width \(w = 5\) cm, both measured to the nearest cm. Calculate the maximum and minimum possible area.