Approximations and Errors
Your phone says the Metrobús will arrive in 4 minutes, but it arrives in 5. The GPS shows your position with a margin of a few meters, and a store scale rounds to the nearest gram. 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. How do you measure that trust? We find out on this page.
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).
The size of a computer file is $4567.89$ megabytes (MB). When copying it to a hard drive, the operating system truncates the value to one decimal place, whereas a…
The size of a computer file is $4567.89$ megabytes (MB). When copying it to a hard drive, the operating system truncates the value to one decimal place, whereas a compression program rounds it to one decimal place. Calculate the difference between the absolute error made by the operating system and that made by the compression program.
Absolute error and relative error
To compare two measurements, convert the relative errors to the same form: $0.015 = 1.5\%$ versus $1.2\%$; the smaller percentage is the more precise.
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.
A digital thermometer displays a temperature of $38.2^\circ \text{C}$. If a high-precision instrument shows that the actual temperature is $38.45^\circ \text{C}$,…
A digital thermometer displays a temperature of $38.2^\circ \text{C}$. If a high-precision instrument shows that the actual temperature is $38.45^\circ \text{C}$, determine the absolute error and the relative error of the digital thermometer reading.
Two different measurements have been carried out. In the first, a relative error of $0.015$ was made. In the second, the relative error was $1.2\%$. Determine which of…
Two different measurements have been carried out. In the first, a relative error of $0.015$ was made. In the second, the relative error was $1.2\%$. Determine which of the two measurements is more precise and justify your answer.
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 $0.2\%$?
In a chemistry laboratory, a student uses the value $6.02 \times 10^{23}$ for Avogadro's number in their calculations. If the most accurate currently accepted value is…
In a chemistry laboratory, a student uses the value $6.02 \times 10^{23}$ for Avogadro's number in their calculations. If the most accurate currently accepted value is $6.02214 \times 10^{23}$, determine the absolute error in standard form and the percentage relative error made by the student.
Errors with fractions and scientific notation
When the exact value is a fraction, it is best to compute the error with fractions: that way you don't carry along rounding.
The average distance from the Earth to the Sun is approximately $149\,597\,870$ km. A high school textbook states that the distance is $1.5 \times 10^8$ km. Calculate…
The average distance from the Earth to the Sun is approximately $149\,597\,870$ km. A high school textbook states that the distance is $1.5 \times 10^8$ km. Calculate the percentage relative error introduced by the textbook when using this approximation in scientific notation.
Two carpenters cut wooden battens that must measure exactly $\frac{5}{7}$ of a meter. The first carpenter cuts the batten to $0.71$ m and the second cuts it to $0.72$ m.…
Two carpenters cut wooden battens that must measure exactly $\frac{5}{7}$ of a meter. The first carpenter cuts the batten to $0.71$ m and the second cuts it to $0.72$ m. Without using decimals, working with fractions, determine which of the two has made a smaller absolute error.
When preparing the budget for a construction project, the cost of materials is estimated to be $308\,000$ pesos. Upon completion of the project, the invoices sum to an…
When preparing the budget for a construction project, the cost of materials is estimated to be $308\,000$ pesos. Upon completion of the project, the invoices sum to an actual total of $316\,500$ pesos. Calculate the absolute and relative errors of the initial estimate. If the construction company covers losses whenever the relative error exceeds $3\%$, who will pay the difference?
Error bound when there is no exact value
Often you don't know $V_r$, only the precision of the instrument or of the rounding. You can't compute $E_a$, but you can bound it: you guarantee that it is at most a certain amount.
Bound on the relative error when rounding
If a positive number $x$ is rounded to the units and you get $A$, the true value satisfies $A-0.5\le x< A+0.5$ and the absolute error is $E_a\le 0.5$. Since $E_r=E_a/x$, the relative error is larger when $x$ is as small as possible, that is, when $x=A-0.5$.
If you are given the endpoints of an interval, the approximate value is the midpoint and the bound is half the length. For endpoints $14.35$ and $14.65$: $V_a=14.5$, $E_a=0.15$ and the maximum relative error is $0.15/14.5\approx1.03\%$.
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.
To calculate the density of a metal block, its mass is measured as $m = 12.4$ kg and its volume as $V = 0.004$ m$^3$. If both measurements have been rounded to the last…
To calculate the density of a metal block, its mass is measured as $m = 12.4$ kg and its volume as $V = 0.004$ m$^3$. If both measurements have been rounded to the last decimal place shown, determine the absolute error bound for the mass and for the volume.
The following diagram shows the confidence interval of a measurement in a physics experiment, where the central point represents the approximate value $V_a$ and the…
The following diagram shows the confidence interval of a measurement in a physics experiment, where the central point represents the approximate value $V_a$ and the endpoints represent the limits given by the absolute error bound $E_a$. If the endpoints of the interval are $14.35$ and $14.65$, determine the approximate value $V_a$, the absolute error bound $E_a$ and the maximum possible relative error.
It is known that when approximating a number $x$ by $3.14$, the absolute error made is less than $0.005$. On the other hand, when approximating the same number $x$ by…
It is known that when approximating a number $x$ by $3.14$, the absolute error made is less than $0.005$. On the other hand, when approximating the same number $x$ by $3.146$, the absolute error is less than $0.002$. Express both conditions as intervals, calculate their intersection and determine the range of possible values for the exact number $x$.
Prove mathematically that if a positive number $x$ is rounded to the nearest whole number obtaining an approximation $A$, the relative error $E_r$ always satisfies the…
Prove mathematically that if a positive number $x$ is rounded to the nearest whole number obtaining an approximation $A$, the relative error $E_r$ always satisfies the inequality $E_r \le \frac{0.5}{A - 0.5}$. Apply this formula to find the upper bound for the relative error when rounding to $A = 5$.
Error propagation in operations
When you compute with approximate values (a sum of measurements, an area, a volume), the error accumulates.
Back to the beginning: confidence in an approximation is measured with the relative error and with a bound that guarantees the maximum possible 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…
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.
The speed of propagation of a signal in a fiber optic network is calculated using the formula $v = \frac{d}{t}$. A distance of $d = 1200 \pm 5$ m and a time of…
The speed of propagation of a signal in a fiber optic network is calculated using the formula $v = \frac{d}{t}$. A distance of $d = 1200 \pm 5$ m and a time of $t = 0.006 \pm 0.0001$ s are measured. Calculate the maximum and minimum possible values for the speed $v$, and deduce from them the overall absolute error bound of the speed measurement.
When calculating compound interest on an initial capital of $200\,000$ pesos at $5\%$ per annum for $10$ years, an accountant decides to truncate the annual growth rate…
When calculating compound interest on an initial capital of $200\,000$ pesos at $5\%$ per annum for $10$ years, an accountant decides to truncate the annual growth rate from $1.05$ to $1.0$ in his preliminary calculations. Calculate the absolute error and the relative error made in the final capital calculation. What conclusion can you draw about the propagation of errors in operations with large exponents?
Problems with equations and inequalities
If the relative error is known, the true value is obtained with an absolute value equation.
A sports tracking app estimates the distance of a cycle route. We know that the relative error of its estimate was exactly $4\%$. If the estimated value displayed by the…
A sports tracking app estimates the distance of a cycle route. We know that the relative error of its estimate was exactly $4\%$. If the estimated value displayed by the app is $26.00$ km, set up and solve the necessary equations to find the two possible exact real values that the distance could have.
A student must calculate the value of $y = \sqrt{x}$. They know that the exact value of $x$ belongs to the interval $[15.8; 16.2]$. If they decide to use $x = 16$ to…
A student must calculate the value of $y = \sqrt{x}$. They know that the exact value of $x$ belongs to the interval $[15.8; 16.2]$. If they decide to use $x = 16$ to simplify, obtaining $y = 4$, determine the upper bound of the absolute error being made in the variable $y$.