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

ISBN 0-471-15045-2

**Download**(direct link)

**:**

**57**> 58 59 60 61 62 63 .. 126 >> Next

16x2 — 25 = (4x)2 — 52 In the formula a is 4x and b is 5

= (4x — 5)(4x + 5) Substituting a = 4x and b = 5 into

(a — b)(a + b)

Some factorization involves combinations of two types of factors, one of which is a common factor, and is done first.

Example 3. Factorize 2x2 — 4x — 6.

Solution. Write

2x2 — 4x — 6 = 2(x2 — 2x — 3)

= 2(x - 3)(x + 1)

Example 4. Factorize ttR2 — nr2.

Solution. Write

ttR2 - nr2 = n(R2 - r2)

= tt(R — r)(R + r)

The common factor is 2 Factorizing the quadratic x2 — 2x — 3

The common factor is tt

Using the difference of two squares

FORMULA 203

Note. It is not possible to factorize all expressions, but when it is possible the process is to proceed as in the examples above. For example, x2 + 1 and x2 + x — 1 are expressions that do not factorize.

References: Brackets, Expanding Brackets.

FIBONACCI SEQUENCE

The Fibonacci sequence of numbers is one of the most famous sequences in mathematics. It was named after the Italian Fibonacci (1180-1250), who was also known as Leonardo of Pisa. The sequence is the set of numbers {1, 1, 2, 3, 5, 8, 13, 21, 34,...}. The next term is found by adding together the two previous terms. The next term after 34 is 55, because 55 = 21 + 34.

Fibonacci’s sequence of numbers originated with the following problem that he was solving. “How many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair which becomes productive from the second month on?”

References: Golden Ratio, Sequence.

FINITE DECIMALS

Reference: Decimal.

FIXED POINTS

This is another name for invariant points.

Reference: Invariant Points.

FOOT

Reference: Imperial System of Units.

FORMULA

The plural of formula is formulae or formulas. A formula is an equation between two or more quantities which shows how changes in one quantity will affect the other(s). The formula A = ttR2 is for finding the area of a circle that has a radius of R. This formula represents a relationship between two quantities A and R. As R changes, there are corresponding changes in A.

204 FRACTIONS

The formula is A = ^BH is for finding the area of a triangle that has a base of length B and a perpendicular height of H. This formula represents a relationship between three quantities A, B, and H. Changes in B and H produce a corresponding change in A.

FOUR-COLOR PROBLEM

Suppose you wish to color the different regions on a map, or any similar figure which has many regions. If regions with a common edge are to have different colors, you will never need more than four colors to color the whole map.

Y >>.

-Lb y(

V G \ FU ! ]

/yV J

The map shown in the figure has been colored using four colors, red (R), yellow (Y), blue (B), and green (G). The outside region also has to be colored. Regions that meet at a point may have the same color, but those that meet on an edge must be colored differently.

FOUR RULES

The four rules are adding, subtracting, multiplying, and dividing. If you need to leam the four rules of decimals, you will need to know how to add, subtract, multiply, and divide decimals.

FRACTIONS

Only numerical fractions will be studied under this entry. For information about algebraic fractions see the entries Algebraic Fractions and Canceling.

A fraction is a number that is made up of the ratio of two integers. If n and d are two integers then -d is a fraction, provided that d ^ 0, otherwise the fraction is undefined.

Fractions can be positive or negative. Examples of fractions are 3 (because

it can be written as |), y, —5, and 2| (because it can be written as y). The top number in a fraction is called the numerator and the bottom number is called the denominator. If the numerator is larger than the denominator, we say that the fraction is improper. For example, y is an improper fraction. If a fraction is joined to a whole number by addition we say it is a mixed number. For example, 2| is a mixed

FRACTIONS 205

number. A proper fraction is one with the denominator larger than the numerator, and an example is |. Other names for proper fractions are common fractions and simple fractions. In practical terms a proper fraction is part of a whole, as illustrated in these examples:

♦ Nathan scored 16 out of 20 in a math test, and his score can be written as the fraction 16/20. In turn this fraction can be written as the percentage 80%.

♦ It takes Nathan and Jacob 5 hours to paint a fence for their dad. Nathan works for 3 hours and Jacob works for 2 hours, and their dad gives them $10. If they divide the $10 into 5 parts, then Nathan should get 3 parts of the money and Jacob 2 parts. Using fractions, we write that Nathan gets | of $10 = $6 and Jacob gets | of $10 = $4.

Example 1. A chessboard is made up of 64 squares; shade | of it.

(a)

Solution. The board is divided up into 4 equal strips, and 3 of them are shaded (see figure a). This means | of the board is shaded. Alternatively, 48 small squares are also shaded out of a total of 64 small squares, so || of the board is shaded. The two fractions | and || are the same size, and they are called equivalent fractions.

**57**> 58 59 60 61 62 63 .. 126 >> Next