APPROXIMATE NUMBERS. ERRORS 5

Observe a) 34 m has 2 significant figures. b) 0.0863 hm3 has 3 significant figures. c) It is possible that 53 000 L only has 2 significant figures if the zeros are just used to designate the number. In this case, it would be better to say 53 thousands of litres.

Observe

Measurement: 34 m a)

Absolute error< 05 m b) :.

Measurement0 0863 hm

Absolute error< 0 00005 hm In other words, bs.error< 50 m a

Approximations and errors

For practical applications, we generally work with approximate numbers. Let’s review some of the concepts and procedures for using them.

We call the figures used to express an approximate number significant figures. We only need to use those we are sure of, and in such a way that they are relevant for the information we want to transmit.

For example, if the capacity of a swimming pool is 718 900 L, it might be more sensible to say it was 719 m3, using only three significant figures. However, if the measurement was not very accurate, or we did not want to give such an exact measurement, it might be better to say 720 m3, or even 72 tens of m3

The absolute error of an approximate measurement is the difference between the real value and the approximate value.

Absolute error = |Real value – Approximate value|

Measurement: thousand Ae error c) L L 53 500 bsolut <

Observe

The relative errors of the previous measurements are: a) R.e. < . 34 05 < 0.015 = 1.5 % b) R.e. < 0.0863 0 00005 < 0.0006 = 0.06 % c) R.e. < 53 000 500 < 0.0095 < 0.01 = 1 %

Problems solved

1 Use a reasonable number of significant figures to express the following quantities: a) Visitors to an art gallery in one year: 183 594. b) People attending a demonstration: 234 590. c) Number of bacteria in 1 dm3 of a preparation: 302 593 847. a) It is quite possible that this number is accurate, as the visitors to a museum pay for a ticket, which is recorded and counted. Let’s suppose that this number, 183 594, is the number of tickets sold. b) It would be impossible to count the demonstrators this accurately. Even if this figure has not been ‘inflated’ or ‘played down’ for political reasons, it is impossible to get so close to the actual figure. For example, it would be reasonable to say ‘more than two hundred thousand’, or even ‘between 200 000 and 250 000’. c) One, or at most two, figures is the most accurate approximation possible with this type of quantity: 3 hundred millions of bacteria, or 30 ten millions of bacteria. a) If we say the number of visitors is 180 thousand (or, even better, 18 ten thousands), we have an absolute error of 183 594 – 180 000 = 3 594 people. We can be this accurate because we know the exact amount. However, whoever we give this information to (18 ten thousands) will need to understand that there may be an error of up to 5 units from the first unused figure: 5 000 people. To summarise:

The real value is generally unknown. Therefore, the absolute error is also unknown. The most important thing is to be able to put limits on it: the absolute error is less than… We get the limit of the absolute error from the last significant figure used.

In the previous example (capacity of the swimming pool: 719 m3), the last significant figure (9) gives units of m3. The absolute error is less than half a cubic metre (error < 0.5 m3).

The relative error is the quotient of the absolute error and the real value. It is therefore lower the more significant figures we use. The relative error is usually also expressed as a percentage (%).

In the example, the relative error is less than: 719 05 < 0.0007 = 0.07 %.

Nevertheless, the figure can be simplified: ‘almost two hundred thousand’ or ‘more than one hundred eighty thousand’ are valid estimations.

2 Give a limit of absolute error and of relative error for each of the estimations of the amounts in the previous exercise.

Estimation: 180 thousand

Absolute error < 5 000

Relative error < 180 000 5 000 < 0.028 < 0.03 8 R.e. < 0.03 = 3 % b) Estimation: 200000

Absolute error < 50000

Relative error < 200 000 50 000 = 0.25 = 25 % c) Estimation: 3 hundred millions = 300 million

Absolute error < 0.5 ten millions = 5 million

Relative error < 300 5 < 0.017 < 0.02 8 R.e. < 0.02 = 2 % a) Saying that a swimming pool holds 147 253 892 thousand drops of water is correct if the measurements are very accurate. b) Saying that a swimming pool holds 147 253 892 thousand drops of water is not reasonable, as it is impossible to measure it this accurately. It would be much more sensible to say that the swimming pool holds 15 ten thousands of millions of drops. c) If we correctly estimate that the number of drops of water that a swimming pool holds is 15 ten thousands of millions, the absolute error is less than half of ten thousand million drops; in other words, absolute error < 5 000 000 000 drops. d) If the relative error for a certain measurement is less than 0.019, we can say that it is less than 19 %. e) If the relative error for a certain measurement is less than 0.019, we can say that it is less than 2 %. f ) The calculator tells us that π = 3.14159265. If we take π = 3.14, we can say that the absolute error is less than 0.00159266, but it is more reasonable to say that the absolute error < 0.0016 or even that the absolute error < 0.002.

Think and practise anayaeducacion.es Practise calculating errors.

1 True or false? Explain your answers.

2 Explain why it is not reasonable to say that there are 11 892 583 grains of rice in a sack.

3 Express it in a suitable way and give a limit of absolute error and relative error for the expression.

4 Give a limit of absolute error and relative error for when you approximate π as 3.1416.