Introduction
In mathematics, approximations are used as an estimation or a rough calculation of a quantity or value. Errors in mathematics refer to the difference between the approximation of a value and the actual value. Understanding approximations and errors is crucial as it helps in making informed decisions and judgments based on available data.
Significant Figures
Significant figures are the digits in a number that carry meaning contributing to its precision. When approximating a number, it is important to consider the significant figures to maintain accuracy. For example, if we have the number $123.45$, there are five significant figures.
Example: Round $456.789$ to three significant figures.
Solution: The third significant figure is $6$, which is greater than $5$, so we round the number up to $456.79$.
Absolute Error
Absolute error is the magnitude of the difference between the approximate value and the true value. It is always positive and is used to measure how accurate an approximation is.
Example: If the true value of a length is $15.6$ cm, and it was approximated as $15$ cm, find the absolute error.
Solution: Absolute error = $|$True value - Approximated value$|$
Absolute error = $|$15.6 - 15$|$ = $|$0.6$|$ = $0.6$ cm.
Relative Error
Relative error is the ratio of the absolute error to the true value. It gives a relative measure of accuracy compared to the size of the actual value.
Example: If the true value of a mass is $50$ kg, and it was measured as $48$ kg, find the relative error.
Solution: Relative error = $\dfrac{\text{Absolute error}}{\text{True value}}$
Relative error = $\dfrac{50 - 48}{50}$ = $\dfrac{2}{50}$ = $0.04$ or $4%$.
Percentage Error
Percentage error is the relative error expressed as a percentage. It is useful in comparing the accuracy of different measurements or approximations.
Example: If the true value of a speed is $60$ m/s, and it was calculated as $55$ m/s, find the percentage error.
Solution: Percentage error = $\left(\dfrac{\text{Absolute error}}{\text{True value}}\right) \times 100%$
Percentage error = $\left(\dfrac{60 - 55}{60}\right) \times 100%$ = $\left(\dfrac{5}{60}\right) \times 100%$ = $8.33%$.
Bounds of Error
Bounds of error are used to determine the range within which the true value lies. Upper and lower bounds can be calculated based on the given approximation and the error involved.
Example: If a measurement is $23.5$ cm with an absolute error of $0.2$ cm, find the upper and lower bounds of the measurement.
Solution: Upper bound = Approximated value + Absolute error
Upper bound = $23.5 + 0.2$ = $23.7$ cm
Lower bound = Approximated value - Absolute error
Lower bound = $23.5 - 0.2$ = $23.3$ cm
Common Mistakes
- Forgetting to consider significant figures when rounding numbers can lead to inaccurate approximations.
- Confusing absolute error with relative error can result in incorrect calculations.
- Misinterpreting upper and lower bounds can lead to incorrect conclusions about the range of possible values.
Key Points
- Significant figures are important in maintaining precision when approximating numbers.
- Absolute error measures the accuracy of an approximation.
- Relative error and percentage error provide a relative measure of accuracy.
- Bounds of error help determine the range within which the true value lies.
Practice Questions
- Round $3456.789$ to four significant figures.
Answer:
The fourth significant figure is $6$, which is greater than $5$, so we round the number up to $3456.8$.
- If the true value of a time interval is $7.2$ s and it was measured as $7.8$ s, calculate the absolute error.
Answer:
Absolute error = $|$True value - Approximated value$|$
Absolute error = $|$7.2 - 7.8$|$ = $|$-0.6$|$ = $0.6$ s.
- Find the relative error when the true value of a temperature is $25$ °C and it was recorded as $26$ °C.
Answer:
Relative error = $\dfrac{\text{Absolute error}}{\text{True value}}$
Relative error = $\dfrac{26 - 25}{25}$ = $\dfrac{1}{25}$ = $0.04$ or $4%$.
- Calculate the percentage error if the true value of a distance is $100$ m and it was approximated as $95$ m.
Answer:
Percentage error = $\left(\dfrac{\text{Absolute error}}{\text{True value}}\right) \times 100%$
Percentage error = $\left(\dfrac{100 - 95}{100}\right) \times 100%$ = $\left(\dfrac{5}{100}\right) \times 100%$ = $5%$.
- A length is measured as $18.7$ cm with an absolute error of $0.3$ cm. Determine the upper and lower bounds of the measurement.
Answer:
Upper bound = Approximated value + Absolute error
Upper bound = $18.7 + 0.3$ = $19.0$ cm
Lower bound = Approximated value - Absolute error
Lower bound = $18.7 - 0.3$ = $18.4$ cm