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

ISBN 0-471-15045-2

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(a) (b)

For the transformations of rotation and translation the object and the image are directly congruent, so they are called direct transformations. For reflection the object and the image are oppositely congruent, so reflection is an opposite, indirect, transformation. Reflection may be called an indirect transformation. When the object and the image are congruent shapes the transformation is called an isometric. This means that rotation, translation, and reflection are isometrics. Enlargement is not an isometric, since the object and the image are not congruent, because they are not the same size.

References: Circle Geometry Theorems, Geometry Theorems, Direct Transformation, Enlargement, Image, Indirect Transformation, I some try, Object, Pentominoes, Reflection, Rotation, Translation.

CONIC SECTIONS

Conic sections are a set of curves {circle, parabola, ellipse, hyperbola} that are formed when a plane surface cuts a cone. Each of these curves is discussed in turn.

108 CONIC SECTIONS

Circle. This conic section is formed when the plane surface is parallel to the base of the cone (see figure a).

Parabola. This conic section is formed when the plane surface is parallel to one of the generators of the cone (see figure b).

Ellipse. This conic section is formed when the plane surface is not parallel to one of the generators or the base of the cone (see figure c).

Hyperbola. This conic section is formed when the plane surface is perpendicular to the bases of two cones (see figure d). Two cones are needed, because the hyperbola has two branches.

CONSTANTS 109

CONJUGATE ANGLES

Angles are conjugate if their sum is 360°.

References: Angles at a Point, Complementary Angles.

CONSECUTIVE NUMBERS OR TERMS

These are numbers or terms that are next to each other in a sequence. For example, the consecutive counting numbers are (1,2, 3,4, 5,...}. The consecutive odd numbers, used, for example, for numbering houses, on the same side of the street, are (1, 3,5,7, 9,...}. The formula for these odd numbers is (2n — 1), where n may be replaced one after the other by successive counting numbers. The consecutive even numbers, for houses on the other side of the street, are (2,4, 6, 8,10,...}, and the formula for these even numbers is (2n), where n may be replaced one after the other by successive counting numbers.

References: Formula, Sequence.

CONSTANT OF PROPORTIONALITY

Reference: Proportion.

CONSTANTS

The symbols we use to represent numbers whose values are fixed are called constants. These symbols may be the actual numbers, or they may be letters of the alphabet which represent numbers. Constants occur in formulas, and they also can be part of algebraic terms and expressions. A constant is used when the value of a number has not been

110 CONSTRUCTIONS

given, but may be given at a later time. It is a value that is fixed and is unchanging for a particular calculation.

In algebra, the convention is that lower case letters near the end of the alphabet, like x, y, z, and also t and u, are used for variables. Other letters of the alphabet, like a, b, c, and k, are used for constants. Upper case letters, in particular, like C and K, are often used for constants. On the other hand, a variable is a quantity which can be assigned any of a set of values. A variable is used to represent any number, and its value(s) may be found by solving an equation. Furthermore, a variable may be replaced by a range of values if we wish to draw a graph of a relation, or a variable may be replaced by one value at a time in a formula.

In trigonometry we often use letters of the Greek alphabet, especially a, fi, and 6, to stand for angles which are variables. The Greek letter it is used as a constant for the ratio of a circle’s circumference to its diameter. A constant can stand for a fixed physical quantity, such as c for the speed of light or g for the acceleration due to gravity.

An example that illustrates constants and variables is the formula y = mx + c for the equation of a straight-line graph. For a particular straight line, say y = 2x + 3, the gradient m = 2 and the y intercept c = 3 are constants, but for these values for the constants, the variables x and y can take an infinite number of different values. For another straight line the constants m and c can take different values, which in turn generate another set of values for the variables x and y.

Another example is the formula A = ttR2 for finding the area of a circle, in which it is a constant in all the calculations. The variable R, for the radius of the circle, can take an infinite number of different values and A will take the same number of corresponding values. In the algebraic term 3xy, 3 is the constant and x and y are variables.

References: Algebra, Area, Circle, Formula.

CONSTRUCTIONS

The constructions, done using ruler and compasses only, explained under this entry are as follows:

♦ Angles of 60°, 30°, 90°, 45°

♦ Regular hexagon

♦ Rectangle

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