Dictionary

Real Numbers

All real numbers are either rational numbers or irrational numbers.

Real numbers
Rational numbers	Irrational numbers

A rational number can be written as a vulgar or common fraction i.e. as an integer divided by another integer (not zero) e.g. ½, -7/45, 21/6, these are all rational numbers.

An irrational number is one that is not rational. It cannot be written as a vulgar fraction e.g. pi - you cannot find an exact value for pi.

Square and Square Root

The Square of any number is the number multiplied by itself i.e. n² = n x n.

The square root of any positive number multiplied by itself gives the number i.e. sqrt n x sqrt n = n

Infinity

Without any limit or end. E.g. when an irrational number is written as a decimal the decimal is infinite i.e. it goes on forever.

Imaginary Numbers

The imaginary number i = sqrt -1 (the square root of -1). The imaginary unit is denoted i. Although there are two possible square roots of any number, the square roots of a negative number cannot be distinguished until one of the two is defined as the imaginary unit, at which point +i and -i can be identified. Either choice is possible.

Negative Numbers

A number classified into a set called a negative number -1, -2, -3, -4, -5.

This set of numbers is infinite i.e. goes on forever. Negative numbers are also called negative integers.

Algebra

In algebra letters are used to stand for numbers. For example:

a means 1 x a or 1a

-a means -1 x a or -1a

ab means a x b

a² means a x a

and so on..

The basic laws for arithmetic are also true in algebra.

An algebraic expression is a calculation written using letters. Therefore, it is a collection of letters and symbols combined by at least one of the operations +, -, x, ¸ . For example:

x and ¸ signs are not usually written in algebraic expressions but ab is taken to mean a x b and a/b is taken to mean a ¸ b

Complex numbers

Complex numbers are of the form x + iy, where x and y are real numbers and i is an imaginary unit equal to the square root of -1. When a single letter is used to denote a complex number, it is written z = x + iy or (x,y).

Trigonometry

Trigonometry is used to solve problems about unknown sides and angles in right-angles triangles. To solve these problems we use both pythagoras' theorem and the trigonometric ratios.

Pythagoras' theorem is about right-angled triangles. Where one of its angles is always a right-angle. The side opposite the right angle is called the hypotenuse. It is always the longest side in the triangle.

Pythagoras' theroem states: In a right angled triangle, the square on the hypotenuse equals the sum of the squares on the other two sides. For triangle ABC, Pythagoras' theorem states

c² = a² + b²

The sides of the triangle are labelled a, b and c. Side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. You can use Pythagoras' theorem to find any side of a right-angled triangle.

You can use the trigonometric ratios sine (sin), cosine (cos) and tangent (tan) to find the angle in a right angled triangle.

Geometry

A branch of mathematics dealing with points, lines, curves and surfaces.

Co-ordinates

Co-ordinates are an ordered pair of numbers. They give the position of a point using a grid, axes and an origin. The co-ordinates of any point P are: (x, y). The first number is always 'across'. The 'across' axis is the x-axis. So the first number is called the x-co ordinate.

The second number is always 'up or down.' The 'up and down' axis is the y-axis. So the second number is called the y-co ordinate.

Simultaneous Equations

To solve two simultaneous linear equations you find the values of x and y which make both equations true simultaneously. If you draw two straight lines on the same axes. When the two lines cross, the point at which the two lines cross or intersect is on both lines. Its co ordinates satisfy both equations simultaneously and give the solution of the equations.

Differential Calculus

The portion of Calculus dealing with derivatives.

Calculus

Calculus or real analysis is the branch of mathematics studying the rate of change of quantities (interpreted as the slopes of curves) and the length, area and volume of objects. Calculus is sometimes divided into differential and integral calculus, concerned with derivatives;

and integrals;

Probability

The outcome of an experiment or action is a single result of the experiment or action. For example, one outcome of tossing a coin is getting a head.

The set of all possible outcomes of an experiment or action is called the outcome set or sample space. This set is usually defined S. For example, the outcome set for rolling a die is

S = { 1, 2, 3, 4, 5, 6}

The number of members in any set X is written as n(X). For example, when rolling a die, the outcome set S has 6 possible outcomes.

So n(S) = 6

The event of an experiment or action is any subset of the outcome set S. For example, when a die is rolled, one event E is getting an odd number.

E = {1,3,5}

This set contains 3 outcomes. So n(E) = 3

When the outcomes are equally likely, the probability that an event E will occur is given by:

A short way to write the 'the probability of event E' is P(E). So the formula for probability can be written