Approximation and errors
A video with 3 894 215 views appears in the app as "3,9 M". Your scales say 58,4 kg even though your real weight has more decimal places. And nobody pays 2,3456 € at the petrol station. We use approximate numbers all the time. The important question is: how much are we getting wrong?
Truncating and rounding
There are two ways of keeping fewer digits:
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Truncating: simply cutting off. $\pi = 3{,}14159\ldots$ truncated to the hundredths is $3{,}14$.
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Rounding: keeping the nearest value. Look at the first digit you 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.
Move the slider and compare truncating with rounding. Notice that the rounding error is less than or equal to half a unit of the place you are rounding to. For example, going from 2,35 to 2,4, the error is exactly 0,05.
The irrational number $\sqrt{10} = 3,16227766...$ appears on your calculator screen. Copy the following table into your exercise book and complete it by carrying out the…
The irrational number $\sqrt{10} = 3,16227766...$ appears on your calculator screen.
Copy the following table into your exercise book and complete it by carrying out the indicated approximations for that number:
| Type of approximation | To the nearest unit | To the nearest tenth | To the nearest hundredth | To the nearest thousandth |
|---|---|---|---|---|
| By truncation | ||||
| By rounding |
You have gone to the supermarket to buy ingredients for dinner. The actual prices of the products are: Pasta: 1.85 € Tomato sauce: 2.15 € Grated cheese: 3.48 € Soft…
You have gone to the supermarket to buy ingredients for dinner. The actual prices of the products are:
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Pasta: 1.85 €
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Tomato sauce: 2.15 €
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Grated cheese: 3.48 €
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Soft drink: 1.25 €
- a) Make a quick estimate of the total cost by rounding each price to the nearest unit before adding.
- b) Calculate the exact sum of the prices.
- c) Calculate the difference (absolute error) between your estimate and the actual price. Do you think it was a good estimate to know if you had 10 € in your pocket?
Significant figures
The significant figures are the digits that give information about precision. When we give 3 894 215 as "3,9 million" we use 2 significant figures.
A famous streamer has uploaded a video that has exactly 3 894 215 views. a) Write down this quantity rounded to the nearest million. b) Write down the quantity rounded…
A famous streamer has uploaded a video that has exactly 3 894 215 views.
- a) Write down this quantity rounded to the nearest million.
- b) Write down the quantity rounded to the nearest hundred thousand.
- c) How many significant figures have you used in part b?
In science and technology, it is crucial to know how precisely we are working. Round the following numbers to three significant figures: a) The number of views of a…
In science and technology, it is crucial to know how precisely we are working. Round the following numbers to three significant figures:
- a) The number of views of a viral video: $14\,592\,301$.
- b) The mass of a dust particle: $0,00040961$ grams.
- c) The number $\pi$: $3,141592...$
- d) The capacity of a hard drive in bytes: $1\,024\,576\,000$.
Afterwards, express each result using standard form.
Absolute error and relative error
Two measurements: you are off by 1 cm when measuring your desk and 1 cm when measuring the thickness of a coin. The error is the same… or is it? On the desk it is nothing; on the coin it is a disaster. That is why there are two types of error.
Error bound
Often you do not know the exact value, but you do know the precision of the instrument. If an ideal set of scales rounds to 100 grams and reads 58,4 kg, the value before rounding lies between 58,35 and 58,45 kg. This bound only describes the rounding: a real instrument may have other measurement errors. The absolute error is at most half a unit of the last digit: $E_a \le 0{,}05$ kg. That is an error bound.
When measuring the length of your study desk, the actual factory value is 120 cm. You use a tape measure and get a measurement of 119 cm. a) Calculate the absolute error…
When measuring the length of your study desk, the actual factory value is 120 cm. You use a tape measure and get a measurement of 119 cm.
- a) Calculate the absolute error ($E_a$) made in your measurement. Remember that the absolute error is the positive difference between the actual value and the approximate value: $E_a = |V_{real} - V_{aprox}|$.
- b) If your classmate measures the school corridor, which actually measures 5 000 cm, and gets 4 999 cm, their absolute error is 1 cm, the same as yours. Does this mean both measurements have the same precision? Reason your answer (you can use the concept of relative error to support your answer).
Calculate the relative error made in the following approximations and state which of them is the worst approximation (the one with the largest relative error): a)…
Calculate the relative error made in the following approximations and state which of them is the worst approximation (the one with the largest relative error):
- a) Approximate the fraction $\frac{1}{3}$ as $0,33$.
- b) Approximate the number $\sqrt{2}$ as $1,41$.
- c) Approximate a budget of $150.000$ € as $140.000$ €.
Digital bathroom scales usually have a precision of $100$ grams ($0,1$ kg). This means that if the scales read $54,2$ kg, the actual weight could be slightly above or…
Digital bathroom scales usually have a precision of $100$ grams ($0,1$ kg). This means that if the scales read $54,2$ kg, the actual weight could be slightly above or slightly below, but rounded to that decimal place.
- a) Write down the interval of possible actual values for a measured weight of $54,2$ kg.
- b) What is the absolute error bound for these scales?
Back to the start: For the video with 3 894 215 views shown as 3,9 M (3 900 000), the absolute error is only 5 785 views and the relative error is $E_r = \frac{5\,785}{3\,894\,215} \approx 0{,}0015 = 0{,}15\,\%$: a negligible error that makes the data far easier to read.