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

ISBN 0-471-15045-2

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This geometry theorem states that (see figure a):

The exterior angle of a triangle is equal to the sum of the two interior opposite angles.

Example. A trapdoor AB covers a hole in the ground and is hinged at A (see figure b). The trapdoor is propped open at an angle of of 47Â° by a support which makes a right angle with the trapdoor. Find the angle x that the support makes with the ground.

Solution. Write

x = 47Â° + 90Â° Exterior angle of triangle = sum of interior opposite angles JC = 137Â°

The obtuse angle the support makes with the ground is 137Â°.

References: Exterior Angle of a Polygon, Obtuse Angle.

EXTRAPOLATION

This is a method of estimating the value of a function beyond the values that are already known. In order to extrapolate, we assume that the function will continue in the same pattern it has already followed. It is because we make this assumption that extrapolation, and interpolation, are not absolutely reliable unless one is certain that the pattern of the function will not change.

Example. The table shows the equivalent ages of dogs and men.

Age of dog in years 1 2 3 4 81215 20

Equivalent age of man in years 15 22 28 34 49 64 76 96

Estimate the equivalent age of a man when a dog is (a) 26 years old, (b) 10 years old.

Solution. To estimate these values, we will need a graph of the data, which is drawn in the figure. Interpolation is a method of estimating the value of a function between

196 EXTRAPOLATION

the values that are already known, so the age of a man equivalent to a dogâ€™s age of 10 years old is read off the graph. The table does not extend as far as a dog of 26 years, so the graph is extended, continuing the line that is drawn so far, which appears to have straightened. This process is called extrapolation. Since we do not know for certain that the relationship between the ages of dogs and men continues beyond the values given in the table, the result is not completely reliable.

The answer to (a) is found using extrapolation, and the answer to (b) using interpolation:

(a) 26 ->â– 120 years.

(b) 10 ->â– 56 years.

The ages of the men are estimated to be 120 and 56 years, respectively.

References: Estimation, Graphs.

F

FACE

Reference: Edge.

FACTOR

A positive integer that divides exactly into another positive integer, without a remainder, is a factor of that number. The integer 1 is a factor of all positive integers, and every positive integer is a factor of itself. For example, 8 is a factor of 24, because 8 divides exactly into 24, without a remainder. The integer 24 is a factor of 24, and 1 is also a factor of 24. A complete list of the eight factors of 24 is {1, 2, 3, 4, 6, 8, 12, 24}.

A prime factor is a factor that is a prime number. The prime factors of 24 are {2, 3}. The integer 1 is a factor of 24, but it is not a prime number.

A number that is a factor of two numbers is a common factor of those two numbers. For example, 3 is a factor of 6 and of 9, so 3 is a common factor of 6 and 9.

Example 1. Find all the common factors of the two numbers 30 and 36; what is their highest common factor?

Solution. The factors of 30 are {1, 2, 3, 5, 6, 10, 15, 30}, and the factors of 36 are {1, 2, 3, 4, 6, 9, 12, 18, 36}. The common factors of 30 and 36 are numbers that are highlighted in both lists, which are {1, 2, 3, 6}. The highest common factor (HCF) of 30 and 36 is the largest number in the list of common factors. The HCF of 30 and 36 is 6.

When numbers are factorized they are written as the products of their prime factors. This is also explained under the entry Factor Tree.

Example 2. Factorize 36.

197

198 FACTOR TREE

Solution. Write

36 = 4 x 9 Breaking down 36 into any two products

= 2x2x3x3 Factorizing 4 and 9 into prime numbers

= 22 x 32. Using indices

36 is factorized as 22 x 32.

A term often confused with factors is multiples. The multiples of a number are obtained by multiplying that number by 1, 2, 3, 4,... in turn.

Example 3. Find the first five multiples of 6.

Solution. The numbers 1, 2, 3, 4, 5 multiply the number 6 in turn as follows:

6 x 1 = 6, 6 x 2 = 12, 6 x 3 = 18, 6 x 4 = 24, 6 x 5 = 30.

Note that 6 is regarded as a multiple of 6

The first five multiples of 6 are {6, 12, 18, 24, 30}.

Example 4. Find the lowest common multiple of the two numbers 6 and 8.

Solution. The lowest common multiple (LCM) of 6 and 8 is the smallest number that is a multiple of both of them. Multiples of 6 = {6, 12, 18, 24, 30,...} and multiples of 8 = (8, 16,24, 32,...}.

The LCM of 6 and 8 is 24.

References: Factor tree, Integers, Product.

FACTOR TREE

The factor tree is used to factorize numbers, which means write them as the products of their prime factors.

Example. Write 840 as the product of primes.

FACTORIZE 199

Solution. Find any two numbers that multiply together to give 840, say 12 x 70. Split 12 into any two numbers which multiply together to give 12, say 3x4, and 70 into 7 x 10, an so on. All the prime factors 3, 2, 2, 2, 5, 7 that multiply to make 840 are highlighted in the figure. Arranging them in order of size gives

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