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

ISBN 0-471-15045-2

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1. The only digits that can be used are 0 and 1.

2. 1 + 1 = 10,1 + 1 + 1 = 11, 1 + 1 + 1 + 1 = 100, etc.

3. 10 - 1 = 1,11 - 1 = 10, 111 - 1 = 110, 110 - 1 = 101, 100 - 1 = 11.

4. Borrowing is sometimes necessary when numbers are subtracted. When we borrow from the next column we borrow a 2, or a 4, or an 8, or a 16,..., depending on which column we borrow from.

5. 1x1 = 1,1x0 = 0. Also, when multiplying a number by 10, we simply add a zero to the number, and when multiplying by 100, we add two zeroes, and so on. For example, 101 x 10 = 1010.

The first 20 counting numbers in binary form are as follows:

1 1 6 110 11 1011 16 10000

2 10 7 111 12 1100 17 10001

3 11 8 1000 13 1101 18 10010

4 100 9 1001 14 1110 19 10011

5 101 10 1010 15 1111 20 10100

Binary numbers are the basis of digital computing. Electronic computers depend on the flow of electric current. Since every binary number can be written using only two digits, 0 and 1, they can be represented by off and on in an electrical circuit. A row of light bulbs can be set up to represent binary numbers. The number illustrated in the figure is 10010.

':M sm

Reference: Decimal System, Number Bases.

BINOMIAL

A binomial is an algebraic expression that contains two terms. Some binomials can be factored using common factors or the difference of two squares. Examples of binomials are x2 — 3x, 2xy + 4, — x3y4 + 7, and 4xy5 — la.

References: Coefficient, Common Factor, Difference of Two Squares, Powers.

BOX AND WHISKER GRAPH 59

BISECT

To bisect a geometrical figure is to divide it into two congruent or equal parts. The geometrical constructions, using ruler and compasses, for perpendicular bisection of a line segment is described under the entry Perpendicular Bisector, and the bisection of an angle is described under the entry Angle Bisector. In both cases the resulting construction line is a mirror line.

References: Angle Bisector, Congruent Figures, Mediator, Mirror Line, Perpendicular Bisector.

BOX AND WHISKER GRAPH

This graph is sometimes called a box plot. It is used in statistics and displays the spread of data about the median. The items involved in box and whisker graphs are listed as follows:

♦ Greatest value

♦ Upper quartile, UQ

♦ Median, M

♦ Lower quartile, LQ

♦ Least value

♦ Range

♦ Interquartile range

The appearance of the graph is illustrated in figure a.

• *

Least LQ M UQ Greatest

(a)

The shaded region is the box, and the solid lines to each side of the box are the whiskers. It is a good idea to draw small dots at the ends of the whiskers to clearly indicate where they start and end. The dashed lines are not part of the graph and the width of the box is not important, but the length of the box is a measure of the interquartile range. The total length from end to end of the whiskers is the range of the data. This graph is very informative and is a method of comparing one set of data with another. A drawback to this presentation of data is that it gives no information about frequencies and mode, which the column graph provides. The graph can also be drawn in a vertical position.

60 BOX PLOT

Example. The marks out of 10 in a mathematics test taken by 30 students are recorded in the frequency table.

(a) Calculate the median and upper and lower quartiles.

(b) Draw the box and whisker graph for this frequency distribution.

Mark out of 10 (x) 01 23456789 10

Frequency (f) 0024368511 0

Solution. To understand how to calculate the median, upper quartile, and lower quartile, search for the methods that are explained under their particular entries.

(a) The number of marks is 30 and there are two middle ones, the 15th, which is 5, and the 16th, which is 6. The median is the mean value of 5 and 6:

A/f A' 5 + 6 Median =-----------

2

= 5.5

The lower quartile is the eighth mark counted from the bottom of the list:

LQ =4

The upper quartile is the eighth mark counted from the top of the list:

UQ = 6

(b) The top mark is 9 and the bottom mark is 2. The box and whisker graph can now be drawn as in figure b.

Results of Math Test

0123456789 10 Marks out of 10 (b)

References: Box Plots, Column Graph, Lower Quartile, Median of a Set of Data, Range, Spread, Statistics, Upper Quartile.

BOX PLOT

This another name for a box and whisker graph.

BRACKETS 61

BRACES

These are a type of brackets; the symbol for braces is { }. They are used to indicate a set, and the elements of the set are written inside the braces. For example, the set of winter sports listed at Luke’s Community Center is {baseball, basketball, hockey, soccer}.

Reference: Brackets.

BRACKETS

Brackets, which are used to enclose something, is a general name for any of the following:

1. Braces { }, which are used in set theory to enclose a set. For an example see the entry Braces.

2. Curved brackets (), sometimes called parentheses, which are used in algebra to enclose terms that represent a single quantity or expression.

Example. Darren has a square sandpit and wishes to extend it in one direction by 2 meters. Write down an expression for the total area of the new sandpit.

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