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The A to Z of mathematics a basic guide - Sidebotham T.H.

Sidebotham T.H. The A to Z of mathematics a basic guide - Wiley publishing , 2002. - 489 p.
ISBN 0-471-15045-2
Download (direct link): theatozofmath2002.pdf
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♦ David’s measurement of 50 km had been rounded to one significant figure.
♦ Jane’s measurement of 48 km had been rounded to two significant figures.
♦ William’s measurement of 47.4 km had been rounded to three significant figures.
Numbers are rounded in order to supply different people with the kind of information they require. For example, suppose a journalist was told that the attendance for the final of the 100 meters race at the Olympic Games was 95,287 people. In his report he would probably round the figure to the nearest thousand, because readers would not be interested in the exact figure. In his article he would write up the attendance as
95,000. The organizers of the games, who are interested in the receipts, would prefer a more exact figure of 95,290, which is rounded to the nearest 10 people.
Suppose the length of a small room is measured to be 3.472 meters. This measurement can be rounded to different degrees of accuracy:
3.47 m (2 dp) Showing two decimal places
3.5 m (1 dp) Showing one decimal place
3.0 m (2 sf) Showing two figures
The zero must be inserted, otherwise one figure would be showing:
3 m (1 sf) Showing one figure
6 ACCURACY
The method used in rounding numbers to a given number of decimal places is illustrated in the following examples.
Example 1. Suppose we are rounding the number 3.47 to 1 dp, which means that only one number is written after the decimal point. We round up to 3.5 and not down to 3.4, because 3.47 is closer to 3.5 than it is to 3.4 (see figure a).
3.4
—O—
3.47
3.5
(a)
Example 2. Round the number 3.423 to 2 dp. In this example we write the answer with two numbers written after the decimal point. The number 3.423 rounds down to 3.42, because it is closer to 3.42 than it is to 3.43 (see figure b).
—1----------------------------------------—О-----------------------------1—
3.42 3.423 3.43
(b)
Example 3. Write 3.45 to 1 dp. For a number like 3.45, which is exactly halfway between 3.4 and 3.5, we always round up to the higher value, which in this example means we round to 3.5 (see figure c).
l____
3.4
The examples above illustrate rounding numbers to a specific number of decimal places. Rounding numbers to significant figures works in a similar way. The difference is that figures are counted irrespective of the position of the decimal point. Sometimes zero is not included in the count of significant figures. This will be explained in some of the following examples.
Example 4. Round 20.8 to two significant figures. The number 20.8 rounds up to 21, because it is closer to 21 than it is to 20 (see figure d).
3.45
(c)
3.5
20
20.8
21
(d)
Example 5. Round 0.35 to one significant figure. The number 0.35 rounds up to
0.4, because 0.35 is exactly halfway between 0.3 and 0.4 (see figure e). Note that we write a zero to show there is no whole number, but this zero does not count as a significant figure.
ACRE 7
1-----------------------O------------------------1-----------------------
0.3 0.35 0.4
(e)
Example 6. Round 8.032 to 3 sf. The number 8.032 rounds down to 8.03, because it is closer to 8.03 than it is to 8.04 (see figure f). Note that zero counts as a significant figure when it acts as a placeholder between other figures.
1 O----------------------------------------------1---------------------------------------------
8.03 8.032 8.04
(f)
Example 7. Write 0.6049 to 2 sf. The number 0.6049 rounds down to 0.60, because it is closer to 0.60 than it is to 0.61 (see figure g). This example shows that when rounding to 2 sf we only examine the third figure, which is 4, and ignore the 9, which does not affect the second significant figure. Note also that the zero in the answer must be inserted to show the required number of two figures.
* O----------------------------------------------■---------------------------------------------
0.60 0.6049 0.61
(g)
Example 8. Round off 208 to 2 sf. The number 208 rounds up to 210, because it is closer to 210 than it is to 200 (see figure h). This is an interesting example, because we cannot write 21 as the answer. It has the required two figures, but 21 is not roughly the same size as 208. So a zero must be inserted to make sure the answer of 210 is approximately the same size as 208. Alternatively, the answer can be expressed in standard form as 2.1 x 102, which is rounded to 2 sf.
1-------------------------------------O-------1--
200 208 210
(h)
References: Approximation, Decimal, Limits of Accuracy, Standard Form.
ACRE
The acre is a unit of area for measuring the size of a piece of land, and is an imperial unit. One acre is 4840 square yards, or 43,560 square feet. Historically, the acre was defined to be the amount of land that a team of two oxen could plough in 1 day. One acre is approximately 0.40 hectare, or 1 hectare is about 2.47 acres.
For practical purposes, 1 hectare is roughly 2| acres, or 5 acres is approximately 2 hectares.
Reference: Hectare, Imperial System of Units, SI Units.
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