Approximations and errors
A GPS gives your position with a margin of error of a few metres; a precision scale rounds to grams; Avogadro's number is used with only a few figures. In science and engineering you almost never work with the exact value, but with an approximation and a measure of how far you can trust it.
Truncating and rounding
Approximate the decimal number $4,8275$ to two decimal places (hundredths) using truncation and rounding.
Approximate the decimal number $4,8275$ to two decimal places (hundredths) using truncation and rounding.
Approximate the irrational number $\sqrt{7}$ to three decimal places (thousandths), stating clearly which is the lower estimate (approximation by defect) and which is…
Approximate the irrational number $\sqrt{7}$ to three decimal places (thousandths), stating clearly which is the lower estimate (approximation by defect) and which is the upper estimate (approximation by excess).
Absolute error and relative error
Calculate the absolute error made when using the approximation $0,33$ for the exact value of the fraction $\frac{1}{3}$. Express the result as a fraction.
Calculate the absolute error made when using the approximation $0,33$ for the exact value of the fraction $\frac{1}{3}$. Express the result as a fraction.
Calculate the relative error made when approximating $\frac{1}{3}$ by $0,33$. Express the result as a percentage.
Calculate the relative error made when approximating $\frac{1}{3}$ by $0,33$. Express the result as a percentage.
During an athletics competition, the head judge measures the length of the track and records $399,5$ m. If the official and exact length of the track should be $400$ m,…
During an athletics competition, the head judge measures the length of the track and records $399,5$ m. If the official and exact length of the track should be $400$ m, calculate the absolute error and the percentage relative error of the judge's measurement. Do you consider this measurement acceptable if the regulations allow a maximum relative error of $0,2\%$?
Error bound when there is no exact value
Often you do not know $V_r$, only the precision of the instrument or of the rounding. In that case you cannot calculate $E_a$ exactly, but you can bound it: you can guarantee that it is at most a certain amount.
State the absolute error bound assumed when asserting that the weight of an object, rounded to three decimal places (thousandths), is $12,458$ kg.
State the absolute error bound assumed when asserting that the weight of an object, rounded to three decimal places (thousandths), is $12,458$ kg.
A kitchen scale gives a reading of $250$ g with an absolute error bound of $5$ g. Write in the form of a mathematical interval the range of values in which the actual…
A kitchen scale gives a reading of $250$ g with an absolute error bound of $5$ g. Write in the form of a mathematical interval the range of values in which the actual weight of the object lies.
Propagation of error in operations
When you calculate with approximate values (a sum of measurements, an area, a volume), the error accumulates. Knowing how much, in a reasoned way, is essential in any experimental science.
An architect measures the dimensions of a rectangular room, obtaining a length of $4,5$ m and a width of $3,2$ m. He knows that his tape measure has an absolute error…
An architect measures the dimensions of a rectangular room, obtaining a length of $4,5$ m and a width of $3,2$ m. He knows that his tape measure has an absolute error bound of $0,1$ m for each measurement. Calculate the measured area and determine between which maximum and minimum values the actual area of the room lies.
In the manufacturing of a cylindrical part for a high-precision engine, the radius is required to be $r = 2,50$ cm and the height $h = 8,00$ cm. If the machinery has a…
In the manufacturing of a cylindrical part for a high-precision engine, the radius is required to be $r = 2,50$ cm and the height $h = 8,00$ cm. If the machinery has a relative error bound of $1\%$ for any linear measurement, prove algebraically whether the maximum relative error in calculating the volume of the cylinder ($V = \pi r^2 h$) will be $1\%$, $2\%$ or $3\%$. Assume that the relative errors are small enough to sum their linear approximations.