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

ISBN 0-471-15045-2

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Out of your sample of 1000 people say there are 40 people who like the new flavor. There are 100 million people in the country, say, which is 100,000 groups of 1000 people. You would expect 100,000 x 40 = 4,000,000 people to like your new Chocko bar.

Example 3. In this example suppose you are using your calculator to work out the value of the expression

2.1 + 3.8 5.9 - 3.2

Using the steps 2.1 + 3.8 ^ 5.9 — 3.2, the answer from the calculator is —0.456, to 3 dp. If you have a suspicion that the answer is incorrect, then estimating the answer using approximate values for the numbers can check it.

The top line is approximated as 2 + 4 = 6. The bottom line is approximated as 6 — 3 = 3. The answer is approximately 6 ^3 = 2. This confirms that the answer —0.456 could not be correct. The calculator answer isincorrect because brackets need inserting in the calculator steps to make (2.1 + 3.8) ^ (5.9 — 3.2) = 2.185, to 3 dp.

Example 4. The following example demonstrates how the number of grains of sand on a beach can be estimated. The number of grains of sand on a large beach is finite and not infinite. The number may be estimated in the following way. Count the number of grains of sand in 1 cubic centimeter (cm3) and say it is 100. There are 1,000,000 cm3 in 1 cubic meter (m3) and therefore 1,000,000 x 100 = 108 grains of sand in 1 m3. Suppose we measured the length of the beach to be L = 2 km, which is 2000 m. The width of the beach is estimated to be IT = 50 m and its depth is estimated to be D = 10 m. Write

Volume of beach = length x width x depth = 2000 x 50 x 10

= 1,000,000 m3 = 106 m3

180 EVALUATE

In each cubic meter there are 108 grains of sand. Therefore

Number of grains of sand on this beach = 108 x 106

= 1014

1014 grains of sand is a finite number.

References: Error, Brackets, Exponent, Sample.

EULER’S FORMULA

This is a formula discovered by Leonhard Euler (1707-1783) that relates the number of vertices V, edges E, and faces F of any simple closed three-dimensional polyhedron. Euler’s formula is

F+V = E + 2

An application of this formula is given in the entry Edge.

There is a similar formula relating the number of nodes N, arcs A, and regions R of a network that is drawn in one plane:

N+R=A+2

References: Arc, Edge, Face, Networks, Node, Polyhedron, Regions, Vertex.

EVALUATE

To evaluate an expression means to find the numerical value, which is the number value, of the expression.

Example. The equation of a parabola is y = x2 + 2. Evaluate y when x = 5.

Solution. Write

y = 52 T 2 Substituting x = 5 in the expression x2 + 2

y = 25 + 2 Squaring 5

y = 27

By finding a numerical value for x2 + 2 we have evaluated the expression.

References: Formula, Numerical Value, Parabola.

EVEN FUNCTION 181

EVEN

A number is even if it is divisible by 2 with zero remainder. For example, 6 is divisible by 2 with zero remainder, so 6 is an even number. Similarly, —4 is an even number, because it is divisible by 2 with zero remainder. The set of even numbers is

{...,-6, -4,-2, 0,2,4,6,...}

Even numbers are multiples of 2, and form an infinite set. The formula for even numbers is E = 2n, and the set of even numbers is obtained by substituting each of the integer numbers I = {..., —3, —2, — 1, 0,1, 2, 3,...} in turn for n.

A number is odd if it is one more or one less than an even number. The set of odd numbers is

{• • •> —5, -3,-1,1,3,5,...}

Odd numbers form an infinite set. The formula for odd numbers is O = 2n + 1, and the set of odd numbers is obtained by substituting each of the integer numbers I = {..., —3, —2, —1, 0, 1, 2,3,...} in turn for n.

In everyday life, the most frequently used even and odd numbers are those that are greater than zero. The results of adding and multiplying even and odd numbers are given in the two tables below.

+ Even Odd X Even Odd

Even Even Odd Even Even Even

Odd Odd Even Odd Even Odd

References: Divisible, Multiple, Quotient, Remainder.

EVEN FUNCTION

A function can be even, odd, or neither even nor odd. The simplest way to recognize whether a function is even or odd is to look at its graph. The graph of an even function is symmetrical about the y-axis. The two examples drawn in figure a are y = x2 + 2 and y = cos x, and both functions are even because their graphs are symmetrical about the y-axis.

182 EVEN FUNCTION

In a similar way, odd functions can be recognized from their graphs, because an odd function has half-turn rotational symmetry about the origin. In other words, the graph has rotational symmetry of order two, and the center of rotational symmetry is the origin. The two curves drawn in figure b are y = x3 and y = sinx, and both these functions are odd, because their curves have half-tum rotational symmetry about the origin.

In addition to the symmetry properties of the graphs of odd and even functions there is an algebraic property of each:

If f(—x) = fix) the function is even

If f(—x) = — fix) the function is odd

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