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

ISBN 0-471-15045-2

**Download**(direct link)

**:**

**18**> 19 20 21 22 23 24 .. 126 >> Next

x 2

x

Solution. Since the length of the original sandpit is not known, let its length be x meters (see figure). The total length of the new sandpit is x + 2 meters and its width is x meters. Since the expression x + 2 is a single quantity, which is the length of the sandpit, it is enclosed in brackets and written as (x + 2). Write

Area of new sand pit = width x length

= x(x + 2)

This expression can be expanded to be x2 + 2x if it is desired to write the expression without brackets.

3. Square brackets [ ], which are used in conjunction with curved brackets in expressions where one set of curved brackets is insufficient to make clear what is intended. When simplifying expressions that contain a mixture of brackets, the rule is to work from the inside out.

62 BREADTH

Example. Simplify 4[x + (x — 2)2].

Solution. Write

4[x + (x — 2)2] = 4[x + x2 — 4x 4] Expanding inside brackets first

= 4[x2 — 3x + 4] Collecting terms, x — 4x = —3x

= 4x2 — 12.x + 16 Expanding square bracket last

4. Angle brackets { }. The numbers, or terms, between these brackets are a sequence. For example, the sequence of triangle numbers = (1, 3, 6, 10, 15,...}.

References: Braces, Collecting Terms, Expanding Brackets, Parentheses, Sequence, Triangle numbers.

BREADTH

Another, more common name for breadth is width. The breadth of a geometrical shape is the distance measured across the shape at its widest place in a direction at right angles to the shape’s length. The length of a shape is its longest measurement. The breadth and length of the ellipse are shown in figure a.

The units for measuring breadth are millimeters, centimeters, meters, and kilometers, and are the same as those for measuring length. The word breadth may be used in place of the word width in the formula for finding the area of a rectangle (see figure b).

References: Area, Ellipse, Length, Rectangle.

c

CALCULATE

To calculate means to use numbers to find the answer to a question. When the question is about an algebraic expression we use simplify, instead of calculate, as the instruction.

References: Algebra, Circle, Indices.

CALCULATOR

A direct algebraic logic, abbreviated DAL, scientific calculator is recommended to aid your study of the mathematics in this book. DAL means that you enter information into the calculator in the same order that you normally write it down. Different calculators have slightly different instructions, so it has been decided not to include in this book the calculator processes. Instead you are referred to the calculator manual.

CANCELING

When fractions are canceled down they are rewritten as equivalent fractions in their simplest form.

Example. Cancel down the fraction 15/72.

Solution. Find the greatest common factor of 15 and 72 that is a positive integer greater than 1. This factor is 3. Write each of the numbers 15 and 72 as a product of two factors using 3 as one of the factors. Then cancel out the threes, since 3-^3 = 1:

5x3

24 x 3

5 3

— x -

24 3

5

24

63

64 CANCELING

Example. Cancel down the fraction 1080/1701.

Solution. By inspection it is found that 3 and 9 are both factors of 1080 and of

Algebraic fractions can also be canceled down by writing the numerator and denominator as products of factors.

Example. Simplify 2a2b/6ab3.

Solution. Identify the common factor of 2a2b and 6ab3, which is 2ab. Write each term as the product of two factors, where one factor is 2ab. The steps of working are as follows:

Solution. Canceling cannot take place at present because the numerator and denominator are not yet written as the product of factors. The numerator is a quadratic expression and the denominator is the difference of two squares. Write

1701:

1080 40 9 3

~irr~ = — X - X -

1701 63 9 3

1701

3

40

63

Canceling the 9’s and 3’s

2a2b a x 2ab

6ab3 3b2 x 2ab

a 2 ab

a

3b2

Example. Simplify

x2 + a: — 12 x2 — 16

x2 + a: — 12 (x — 3)(x + 4)

x2 — 16 (x — 4)(x + 4)

Factoring

(* _3) (x + 4) (* - 4) X (x + 4)

(*-3)

(x-4)

Canceling the common factor (x + 4)

References: Algebraic Fractions, Difference of Two Squares, Fractions, Equivalent Fractions, Factor, Numerator, Product of Prime Factors, Quadratic Equations, Rational Expression.

CARRY 65

CAPACITY

Capacity refers to the volume of liquid that a container holds. The units of capacity are as follows:

♦ Milliliter (ml). One milliliter is equivalent to a volume measure of 1 cubic centimeter (cm3, or cc). It is roughly the capacity of a teaspoon.

♦ Liter (1). One liter is equal to 1000 milliliters. It is roughly the capacity of a large jug, or about 41 cups.

♦ Kiloliter (kl). One kiloliter is equal to 1000 liters. It is the capacity of a cube that has a volume of 1 cubic meter (1 m3).

References: Cube (geometry), Metric Units, Volume.

CAPITAL

This is a sum of money that a person has to start a company, or the money accumulated by a person that may be invested at a profit.

Reference: Principal.

CARRY

Three of the column headings for base 10 numbers are shown in figure a; the greatest digit that can be written in any column is 9. When adding numbers together one column at a time, it is possible to obtain a sum greater than 9, which is too large to be written in that column. When this happens, 10, or multiples of 10, are subtracted from the sum and carried to the next column of higher value, and the remnant is written in the column in which you are adding. This process is demonstrated in the following example.

**18**> 19 20 21 22 23 24 .. 126 >> Next