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

ISBN 0-471-15045-2

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Reference: Binary Numbers.

COMPACT FORM OF NUMBERS

Reference: Compact Form of Decimals.

COMPARATIVE COSTS

In a supermarket the same product may be marketed differently and shoppers are keen to get the best buy, or in other words, the best value for their money. In order to find the best buy, the costs are compared as in the following example.

COMPASS POINTS 89

Example. A 10-kg bag of Gala apples costs $15.70 and a 3-kg pack of the same apples is on special at $4.65. Which is the better value for the money?

Solution. We compare the two prices by calculating the cost per kilogram in each case.

♦ 10-kg bag:

Cost per kg = $15.70 10

= $1.57 per kg

♦ 3-kg pack:

Cost per kg = $4.65 3

= $1.55 per kg

The 3-kg pack of apples is the better buy, because it costs less per kilogram.

Reference: Rate.

COMPASS POINTS

The main 16 points of the compass are shown in figure a. There is a 22.5° angle between each of the 16 points. The bearings are listed here:

N is due north NE is northeast E is due east SE is southeast

NNE is north of northeast ENE is east of northeast ESE is east of southeast SSE is south of southeast

N

(a)

90 COMPLEMENTARY ADDITION

S is due south SSW is south of southwest

SW is southwest WSW is west of southwest

W is due west WNW is west of northwest

NW is northwest NNW is north of northwest

Compass bearings can also be written as a mixture of compass points and angles, by expressing the direction as three instructions, in the following way. The compass bearing of W 30°S is shown in figure b, and its direction is explained by following three instructions:

1. W: Face west.

2. 30°: Turn through an angle of 30°.

3. S: Turn away from west toward the south.

Reference: Bearings.

COMPLEMENT OF AN ANGLE

Reference: Complementary Angles.

COMPLEMENTARY ADDITION

This is a method of using addition to subtract two numbers. Before electronic cash registers were installed in shops, the shopkeeper would frequently use a form of complementary addition when giving change to a customer. Suppose John bought a pen for $17 and offered the shopkeeper a $50 bill in payment. The shopkeeper starts counting out the change, and his starting point is the cost of the pen, which is $17. He gives John $3, which brings the amount up to $20. Then he gives him $30, to bring the amount up to $50.

Complementary addition is used to subtract two numbers in the following example.

W

<*

W30°S

(b)

COMPLEMENTARY ANGLES 91

COMPLEMENTARY ANGLES

Two angles are complementary if they add together to make a 90° angle. Figure a shows a drop-leaf table with the leaf initially in a horizontal position. Then it swings about the edge of the table through angle a, and then through angle b until the leaf is vertical. The two angles a and b are complementary angles, because they add together to make 90°. This means a + b = 90°. We say that angle a is the complement of angle b, and vice versa.

Similarly, two angles are supplementary angles if their sum is 180°. Suppose there is a trapdoor in the floor of a barn, and the hinged lid is free to open (figure b). It swings about the hinge through the angles a and b until the door is fully open. The two angles a and b are supplementary angles, because they add together to make 180°. This means that a + b = 180°. We say that angle a is the supplement of angle b and vice versa.

Two angles are conjugate angles if their sum is 360°. The weighing scale in the drug store has a circular dial, and weighs up to 150 kg (see figure c). When Nathan gets on the scales the pointer turns round the scale through an angle a. If it turns through a further angle b, the pointer will be back at the top of the scale. The two angles a and b are conjugate angles, because they add together to make 360°. This means that a + b = 360°. We say that angle a is the conjugate of angle b, and vice versa.

(a)

(b)

(c)

References: Adjacent Angles, Angles at a Point, Cointerior Angles.

92 COMPLEMENTARY EVENTS

COMPLEMENTARY EVENTS

Complementary events occur in considerations of probability and may be confused with two other similar terms used in probability, called mutually exclusive events and independent events. These three terms are defined in this entry, using examples to make clear their differences. The methods of calculating the probabilities of complementary events, mutually exclusive events, and independent events are explained under the entry Probability of an Event.

Amanda is playing a game which involves rolling a die, which has 6 faces numbered 1-6. Suppose there are three events in which we are interested. One event is rolling a 6, and let this be called 5. Another is rolling an even number, and let this be called E. Finally, let O be the event of rolling an odd number.

Two events that cannot happen at the same time are called mutually exclusive events. In our example, with one roll of the die you cannot roll an even number and an odd number at the same time, so E and O are mutually exclusive events. Also, 5 and O are mutually exclusive, because one die cannot show a 6 and an odd number at the same time. But, if you roll a 6, you have achieved event 5 and event E at the same time, so 5 and E are not mutually exclusive.

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