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

ISBN 0-471-15045-2

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A regular octagon has eight sides of equal length, and each of its eight angles is of equal size, which is 135° (see figure b). The regular octagon is made up of eight isosceles triangles, and each angle at the center of the octagon is 45°.

306

ODD FUNCTION 307

(b)

Tiling Patterns The regular octagon and the square form a tiling pattern (tessellation) as shown in figure c.

References: Isosceles Triangle, Regular Polygon, Tessellations, Triangle.

OCTAHEDRON

This is a solid shape (polyhedron) that has eight faces; the prefix octa means eight. All the eight faces of a regular octahedron, which is one of the Platonic Solids, are congruent equilateral triangles. In the figure one can see that the octahedron is made up of two square-based pyramids stuck together. The net of a regular octahedron is made up of eight equilateral triangles.

References: Congruent Figures, Equilateral Triangle, Net, Platonic Solids, Polyhedron.

ODD FUNCTION

Reference: Even Function.

308 ODDS

ODD NUMBERS

Reference: Even.

ODDS

In order to understand odds it is important first to read the entry Probability of an Event. The odds of an event happening are defined to be the ratio of the probability of the event happening to the probability of the event not happening.

Suppose the probability of an event happening is P. Then the probability of it not happening is I-P. Therefore

The odds of rolling a 6 can be referred to as “odds on rolling a 6” and is equal to the ratio of 1:5, or 1 to 5. Similarly, the odds against rolling a 6 is equal to the reverse ratio of 5:1, or 5 to 1.

Example 2. If a playing card is drawn from a pack of 52 cards, what are the odds against it being a black king?

Solution. Write

2

Probability of drawing a black king = — There are 2 black kings out of 52

Odds of an event happening = y—— Using the definition Example 1. If a die is rolled, what are the odds of it being a 6?

Solution. Write

Probability of rolling a six is P = |

Then

Odds of rolling a six = ^ 6 1 Using -——

5

Odds of drawing a black king =

P

Using -j—p

1

25

The odds of drawing a black king are 25 to 1 against.

OPPOSITE INTEGERS

309

When a coin is tossed, the odds of it being a head are 1 to 1 against, or 1 to 1 for. In this case we say the odds are even!

References: Canceling, Probability of an Event.

OGIVE

An ogive is the name of the graph of a cumulative frequency distribution.

Reference: Cumulative Frequency Graph.

ONE-MANY CORRESPONDENCE

Reference: Correspondence.

ONE-ONE CORRESPONDENCE

Reference: Correspondence.

OPERATIONS

An operation is a method of combining numbers. The four basic operations in arithmetic for combining numbers are listed below with their symbol in brackets:

♦ Addition (+)

♦ Subtraction (—)

♦ Multiplication (x)

♦ Division (e~)

See the entry B EDM AS for the order of operations.

Reference: BEDMAS.

OPPOSITE INTEGERS

One integer is the opposite of another integer when they are positioned on a number line on opposite sides of the origin and at equal distances from the origin. The “opposite of an integer” is a term used to explain the process of subtracting integers, which is done under the entry Integers. For example, the opposite of —2 is +2, or just 2. The opposite of 4 is —4.

References: Integers, Number Line.

310 ORDER OF ROTATIONAL SYMMETRY

OPPOSITE TRANSFORMATION

This is the same as an indirect transformation. References: Congruent Figures, Transformation Geometry.

ORDER OF A NODE

References: Königsberg Bridge Problem, Networks.

ORDER OF ROTATIONAL SYMMETRY

Rotational symmetry is the symmetry of rotation, or turning, about a fixed point. The order of rotational symmetry is the number of small turns of rotation a shape makes in completing a full turn. The following explanation brings out the meaning of order of rotational symmetry.

Suppose you take grandpa’s walking stick and rotate it about its end through four turns of 90° to obtain the complete figure shown here in figure a. This complete figure has four positions of the stick, and each time the stick has been rotated about the same point, through the same angle of 90°. We say the complete figure has rotational symmetry of order four. The end of the stick, about which it is rotated, is called the center of rotation.

It should be pointed out that after the stick has been rotated twice, giving three sticks, the complete figure does not appear to have rotational symmetry (see figure b), because the angle between each pair of sticks is not the same size.

On the other hand, the figure made up of two sticks, with 180° between the sticks, has rotational symmetry of order two. This is true in whatever position it is drawn, provided the final figure is obtained by rotation about the same point. Each of the three pairs of walking sticks has rotational symmetry of order two (see figure c).

Note these figures do not have any axes (or lines) of symmetry. A figure can have both rotational and axes of symmetry, as explained below.

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