Algebra

If you want to compare more than two quantities, a first important step is to arrange them in tables. A higher degree of clarity is achieved by representing them graphically by means of all kinds of pictures. For example, the wealth of nations can be represented by differently large purses, the size of their armies by soldiers of different height, their mercantile fleets by varying size ships, etc. All these methods of representation involve different size distances, areas or bodies. Let us look at these possibilities in detail.

Give an estimate of the size ratios of the squares shown on the left hnd side. The second square is twice, the third three times as large as the first, but this is not readily seen.

If a square is to be twice as large as a given one, you cannot simply double the length of its sides, because it would yield 4 times its size. In the case of a cube, doubling its edges would yield a cube which is 8 times as large.

If you examine representations by areas and bodies, you will frequently discover, that the images do not at all correspond to the numbers to be compared. Some such images can be seen to be totally at fault. If you want to compare numbers, you should demand that, in particular, magnitudes of numbers are represented as closely as possible .

Since, as a rule, estimates tend to be unreliable, you supply your figures with numbers or scales which allow you to extract more accurate values. The simplest mode of measurement is then also that of lengths, so that one tends to prefer distances for representation of numbers .

In equations and

formulae only contain numbers. Graphical representations also present numbers. If you want to extract from a graph information, you must study its scales and identify the correct objects. In this graph of the students at a school, one student

1 mm.

Form	Number of students	In the drawing
6	45	the difference
7	51	in the length
8	38	of the largest
9	42	class from that
10	38	of the smallest
11	35	class is 16 mm

Task: Extract from this graph with 10⁶km²

1 mm the magnitudes of the continents.

This is a representation by rectangles rather than bars. Their height is unimportant, only their lengths yield information, whence, if a rectangle is twice as long as another, it represents twice the area. The rectangles only give the graph a more pleasant appearance. Bars could have been used.

Countries	Coal in 10⁶ to	%
USA	581	49.1	The production of coal
England	282	23.8	in 1923 shows amounts as
Germany	62	5.2	well as percentages
France	37.7	3.2	of the total. The scale
			of the rectangle is
World	1184	100	100% 10 cm

This figure shows that 5% of the 41·10⁶ inhabitants of Germany in 1871 lived in towns with more than 10⁵, 8 % in towns with more 20·10⁴ inhabitants, etc. In 1910, with a population of 65·10⁶, 11% of all towns had 2000 - 5000 inhabitants. More exact values can be interpolated from the graph.

country	Iron in 10⁶ to	% of world production	º

USA	31.6	47.6	171.4
Germany	7.8	11.7	42.1
France	7.6	11.4	41.0
England	7.4	11.1	39.96

World	66.4	100	360

Sectors can be used for the representation of percentages, which must be converted into corresponding angles 100%

360º, 1%

3.6 º. The size of the radius is unimportant.

1. The use of land in Japan and Germany is given by the table. Represent the details and percentages by strip diagrams.

Country	forest %	meadows 1%	fields 15%	other 16%
Japan	68	1	15	16
Germany	20	22	33	16

Until now, you have only placed lines, rectangles, etc., side by side at arbitrary distances in between. However, this distance can be employed, in order to represent a second quantity, which is somewhat related to the first. In the example above, we have graphed the hours along the horizontal axis at a certain scale and at the markers for the definite hours we have represented the temperature by vertical lines.

For example, if you want to know the temperature at 9 a.m., you would be inclined to link the end points of the verticals, as has been done by dashed lines, and read off for 9 a.m. the temperature 21 ºC. In doing so, you assume that the temperature has changed uniformly between 8 and 10 a.m. Of course, you do not know for certain that this is the case. In fact, the temperature could have varied in one way or another, as is shown by the temperature curve.

There exist so called thermographs - instruments for the recording of temperature - which draw continuous curves. From their charts, you can read off the temperature at any time of the day. This chart shows that at 7 a.m., the temperature was 19ºC, at 16 p.m. 28ºC.

Task: 1 kg of a substance costs 2 Bath, what is the cost of 2, 3, . . . , kg? Make a table and draw the graph!

If you enter in the graph points, which you compute for intermediate weights, you will see that they all lie on the line shown, because the growth rate of the price is linear.

Exercise: The distance, a car should have from a car in front varies with speed. The table lists suggested safety speeds.

1871
population	millions	%
over 10⁵	2	4.9
2·10^{4 -}10⁵	3.1	7.6
5000 - 2·10⁴	4.6	11.2
2000 - 5000	5.1	12.4
to 2000	26.2	63.9

m³ gas	Baht	If the growth rate is uniform,
1	0.17	you obtain a straight line.
2	0.34	Uniform growth serves as
3	0.51	a measure for all rates
4	0.68	of growth.

speed	distance	tasks
km/hr	m	1. Draw the graph of speed
60	25	against distance
80	33
100	42	.2. Read off it the distance for
120	50	speeds of 70 km/hr, 145 km/hr.
140	58
160	67
180	75

Time		temperature
6		14
8		18
10		25
12		23
14		25
16		29
18		28
20		22

kg Substance			Baht
1			2
2			4
3			6
4			8
5			10