Approximations
$\pi$ has infinitely many digits, but in the laboratory or the workshop nobody uses them all. A digital scale, a stopwatch, even the price shown on the supermarket scales: they are all approximate numbers. Learning to approximate well, and to know how big your error is, is just as important as calculating.
Truncating and rounding
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Truncating: cutting a number off straight after a given digit, without looking at what comes next.
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Rounding: keeping the closest value. Look at the first digit you are going to remove: if it is 5 or more, add 1 to the last digit you keep; if it is less than 5, leave it as it is.
Round the number $\pi \approx 3,14159265...$ to the nearest tenth, hundredth and thousandth.
Round the number $\pi \approx 3,14159265...$ to the nearest tenth, hundredth and thousandth.
Truncate the irrational number $\sqrt{7} \approx 2,645751...$ to two decimal places and calculate the absolute error made with this approximation.
Truncate the irrational number $\sqrt{7} \approx 2,645751...$ to two decimal places and calculate the absolute error made with this approximation.
Significant figures
Significant figures are the digits that genuinely contribute to the precision of a measurement.
State the number of significant figures in the following experimental measurements: $0,0045 \text{ kg}$, $30,020 \text{ s}$, $1500 \text{ m}$ (assuming it is exact to…
State the number of significant figures in the following experimental measurements: $0,0045 \text{ kg}$, $30,020 \text{ s}$, $1500 \text{ m}$ (assuming it is exact to the nearest unit) and $2,0 \cdot 10^5 \text{ A}$.
Express the number $45678,9$ to three significant figures, using standard form.
Express the number $45678,9$ to three significant figures, using standard form.
Rounding to $n$ significant figures
Approximate the fraction $\frac{17}{7}$ to three decimal places by overestimate and underestimate.
Approximate the fraction $\frac{17}{7}$ to three decimal places by overestimate and underestimate.
The universal gravitational constant has an approximate value of $G \approx 6,674 \cdot 10^{-11} \text{ N}\cdot\text{m}^2/\text{kg}^2$. Round this value so that it has…
The universal gravitational constant has an approximate value of $G \approx 6,674 \cdot 10^{-11} \text{ N}\cdot\text{m}^2/\text{kg}^2$. Round this value so that it has only three significant figures and find the bound on the relative error of this new approximation.
If you only know the approximation: bounding the true value
When a measurement is given already rounded or with a margin of error, the true value lies within an interval, even though you do not know the exact number.
When an unknown number is rounded to two decimal places, $4,38$ is obtained. State the interval containing the exact value of the number and determine the bound on the…
When an unknown number is rounded to two decimal places, $4,38$ is obtained. State the interval containing the exact value of the number and determine the bound on the absolute error.
The road distance between two cities is published as $340 \text{ km}$, stated to have two significant figures. Determine the interval of values containing the actual…
The road distance between two cities is published as $340 \text{ km}$, stated to have two significant figures. Determine the interval of values containing the actual distance and calculate the bound on the absolute error and the maximum possible relative error bound.
A circular town square is to be fenced off. The surveyor measures the radius and obtains $R = 45,2 \text{ m}$ (a value rounded to one decimal place). Calculate the…
A circular town square is to be fenced off. The surveyor measures the radius and obtains $R = 45,2 \text{ m}$ (a value rounded to one decimal place). Calculate the nominal perimeter of the square using $\pi \approx 3,1416$ and find the maximum absolute error (error bound) that can be made in calculating this perimeter due to the uncertainty in the radius measurement.
Back to the start: If the supermarket scales show $1{,}24$ kg, the absolute error bound is half a hundredth ($0{,}005$ kg or $5$ g): the real weight of your shopping is guaranteed to lie within the interval $[1{,}235; 1{,}245)$ kg.