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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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♦ Hexomino. This is made of up six squares joined together. There are 35 types of hexominoes (see the entry Hexominoes).
POWERS 339
♦ Heptomino. This is made up of seven squares j oined together. There are 108 types of heptominoes, including one with a “hole,” shown in figure e.
(e)
References: Hexomino, Pentominoes.
POPULATION
Population is a term used in statistics and means the complete set of data being considered when choosing a sample. Before you can choose a sample you must be clear about what population you are going to choose your sample from. For example, if you were choosing a sample of students to answer a questionnaire on how they feel about having a representative on the Board of Trustees, the population will probably be all the students in your school. But if you were working for the Examination Board and were choosing a sample of students to obtain some general information on the examination results, your population would probably be all the students in the country. It would not be a good idea to have all the students in the world as the population, because they would not all have taken the same examination.
Reference: Bias, Sample.
POSITIVE
Reference: Negative.
POSITIVE INTEGERS
Reference: Negative Integers.
POUND
The pound is a unit of weight in the imperial system of units. The abbreviation is lb. Reference: Imperial System of Units.
POWERS
Reference: Exponent.
340 prism
PRIME FACTOR
References: Factor, Prime Number.
PRIME NUMBER
A positive integer is prime if it has exactly two factors, which are 1 and itself:
♦ 3 is a prime number because its only factors are 1 and 3.
♦ 2 is a prime number because its only factors are 1 and 2.
♦ 6 is not a prime number because its factors are 1, 2, 3, and 6. Since it has four
factors, it is not a prime number.
♦ 1 is not a prime number because it has only one factor, which is 1.
There are 25 positive prime numbers less than 100:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
The next prime number is 101.
Through history mathematicians have been interested in prime numbers and in obtaining a formula for them. This has never been achieved, but some interesting attempts are discussed in the entry Counterexample.
References: Counterexample, Eratosthenes’ sieve, Factor.
PRINCIPAL
Reference: Capital.
PRISM
A prism is a polyhedron (solid) which has a base and a top that are congruent and parallel, and cross sections which are parallel and identical to the base (see figure a).
The prisms that we will consider are upright prisms, which are usually called “right” prisms. The names of some prisms are given according to the shape of their cross section. Some well-known prisms are drawn in figure b. The cross section may be a regular polygon, as in the cube, or it may be a nonregular prism, as in the trapezium prism. Notice that the trapezium prism is the shape of a swimming pool that has a deep end and a shallow end.
prism 341
Upright
prism
t<7\
Oblique
prism
Examples of prisms in everyday life are pencils, coins, swimming pools, and boxes.
Volume of a Prism The distance between the cross-sectional ends of a prism is called the length. The volume is given by
Volume of prism = area of cross section x length
20 m
(c)
342 PROBABILITY OF AN EVENT
Example. The length of the swimming pool is 20 meters, the width is 8 meters, the deep end is 3 meters deep, and the shallow end is 2 meters deep (see figure c). Find the volume of the pool.
Solution. The cross section is the trapezium drawn shaded in figure c. Write
(a + b)
Area of a trapezium A = —-— x h Here a and b are the lengths of the
parallel sides and h is the distance between them
Write
(2 + 3)
Area of trapezium = x 20 Substituting a = 2, & = 3, /2 = 20 in the
area formula
= 2.5 x 20 = 50
The area of the trapezium is 50 square meters.
To find the volume, write
Volume of prism = area of cross section x length = area of trapezium x length = 50 x 8
= 400 cubic meters
The volume of the pool is 400 m3.
Note: The length of 8 meters in the volume formula is the distance between the parallel faces of the cross section, not the length of the pool.
References: Congruent Figures, Cross Section, Parallel, Trapezium.
PROBABILITY OF AN EVENT
Suppose the event we are going to consider is rolling a die once and obtaining a 3. The die could land in a total of six different ways. We say that the total number of outcomes of rolling the die is six, which means there are six ways it could land. The
The distance between the parallel sides is 8 meters
PROBABILITY OF AN EVENT 343
number of ways of obtaining the particular outcome of rolling a 3 is one. The formula for probability is
„ , number of ways the event can happen
Probability of a particular event =---------------------------------------------
total number of outcomes
We can apply this formula to our example of the probability of rolling a 3 with a die:
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