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

ISBN 0-471-15045-2

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Plato Socrates

X X-X {x-xf X X-X {X-XŸ

56 56-74= -18 (-18)2 = 324 62 62 - 74 = -12 (-12)2 = 144

59 -15 225 62 -12 144

62 -12 144 63 -11 121

63 -11 121 64 -10 100

66 -8 64 69 -5 25

69 -5 25 70 -4 16

70 -4 16 72 -2 4

75 1 1 76 2 4

78 4 16 77 3 9

80 6 36 77 3 9

83 9 81 79 5 25

84 10 100 81 7 49

85 11 121 85 11 121

87 13 169 86 12 144

93 19 361 Total = 1804 87 13 169 Total = 1084

/E(*-*)2 2

y 15 a-'\\ 15

a = /1804 V 15 /1084 a~i 15

a = 11.0 to 1 dp a = 8.5 to 1 dp

STANDARD FORM 425

These figures confirm that the weights of the players from Plato High School are more spread about the mean than the weights of the players from Socrates High School, because the greater the value of the standard deviation, the greater is the spread.

In the example, the number of items of data for each team was 15. In the following general formula for standard deviation there are n items of data:

References: Mean, Normal Distribution, Quartiles, Variance.

STANDARD FORM

Standard form is sometimes called scientific notation. It is a convenient way of writing very large or very small numbers in shortened form. This is especially useful in science, which often deals with very large and small numbers. In astronomy there is a unit of distance called the “light year,” which is approximately equal to 6,000,000,000,000 miles. When it is written in that form it takes up a lot of room and cannot easily be compared with other numbers of similar size. In standard form a number is written as a x 10n, where n is an integer and a is a number between 1 and 10, or, more precisely 1< a < 10. This makes the size of a clear: it can equal 1, but cannot equal 10.

Example 1. Write the number 6,000,000,000,000 in standard form.

Solution. Write

6,000,000,000,000 = 6 x 1,000,000,000,000 In this case a is 6

The number in standard form is 6 x 1012.

Example 2. The greatest distance of the Moon from the Earth is 405,470 km. Write this distance in standard form correct to three significant figures.

Solution. Write

6 x 1012

The index n is 12

405,470 = 4.05470 x 100,000

= 4.05 x 100,000 Writing the number to 3 sf

= 4.05 x 105

The number in standard form is 4.05 x 105.

Example 3. The length of a room is 4 meters to 1 sf. Write this length in standard form.

426 STAR POLYGONS

Solution. Write

4 = 4x1 = 4 x 10° Since 10° = 1 The length of the room, in standard form, is 4 x 10° meters.

Very small numbers can be written in standard form using negative indices, as illustrated with the following example.

Example 4. The length of the wavelength of ultraviolet light is 0.0000254 cm. Write this in standard form.

Solution. Write

0.0000254 = 2.54 a- 100,000 -a 100,000 is equivalent to xlO-6

= 2.54 x 10-5 The index is negative because we are dividing

The number in standard form is 2.54 x 10-5.

Example 5. Write the number 3.4 x 10-4 in decimal form.

Solution. Write

3.4 x 10-4 = 3.4 -T- 10,000 xlO-4 is equivalent to ^10,000

= 0.00034 Moving the decimal point 4 places to the left.

The number written in decimal form is 0.00034

We can easily compare the sizes of numbers that are in standard form, as illustrated by the following examples:

♦ 3.4 x 105 is greater than 8.5 x 104, because it has a greater index (5 > 4).

♦ 2.8 x 10s is less than 3.0 x 108, because 2.8 is less than 3.0 and the indices are the same.

♦ 5.9 x 10-6 is less than 2.8 x 10-5, because it has a smaller index (-6 < -5). References: Decimal, Indices.

STAR POLYGONS

These are star shapes formed by joining the vertices of regular polygons, as described below. Regular star polygons were well known to Pythagoras and his followers, who lived about 500 years BC. In fact, the pentagram was adopted as the badge of the ancient Pythagorean school.

STATIONARY POINT 427

Pentagon The first vertex is joined with a straight line to the third vertex, the second vertex is joined to the fourth vertex, the third vertex to the fifth vertex, and so on (figure a). Then the polygon is removed, and the result is a star pentagon, which is also known as a pentagram.

1

Hexagon The process is repeated for the hexagon.

Octagon There are two star octagons (figure b). In the first octagon the first vertex is drawn to the third vertex, the second to the fourth, the third to the fifth, and so on. In the second octagon the first vertex is joined to the fourth vertex, the second to the fifth, the third to the sixth, and so on.

1 1

(b)

References: Hexagon, Pentagon, Pentagram, Regular Polygon, Vertex.

STATIONARY POINT

In this entry, a stationary point on a curve is defined in terms of the gradient of the curve at that point. See the entry Gradient of a Curve to ensure you understand the term. A stationary point on a curve is the point where the gradient of the tangent to the curve is zero. In the figure a the points A, B, and C are stationary points, because the tangents at those points, which are drawn dashed, have zero gradients.

A

(a)

428 STEM AND LEAF GRAPH

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