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### Chapter 1. Integers

Abstract
Numbers were first used for counting, and the roman numerals, which were based on finger counting, were used in Britain up to the seventeenth century. They were replaced by Hindu-Arabic symbols similar to the familiar 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, used today, and these are called digits after the Latin word for ‘finger’. In most countries the denary system of counting in tens is used, but other systems are considered in Chapter 6, including binary arithmetic.
O. Perry, J. Perry

### Chapter 2. Common Fractions

Abstract
To most of us a fraction means ‘part of a whole’. For instance, if a fruit flan is divided into six ‘equal’ portions, each one is one-sixth part of the whole flan, 1 ÷ 6 or 1/6, but in fact the portions can never be exactly equal. In mathematics fractions are exact, and they have a much wider meaning.
O. Perry, J. Perry

### Chapter 3. Decimal Fractions

Abstract
In the denary system of powers of 10, fractions occupy columns to the right of the units column. They are called decimal fractions (or just decimals) and the fractional part of a number is separated from the integer part by a dot (.) called a decimal point.
O. Perry, J. Perry

### Chapter 4. Roots, Indices, Four-Figure Tables, Calculators

Abstract
In Chapter 3 we discussed exact and recurring decimal fractions but there are some decimal numbers that never end or repeat however many decimal places are calculated. These are called irrational numbers because they cannot be written as common fractions and they are corrected to a specified number of decimal places. An irrational number is a real number and on the number line it has a position between two rational numbers, although we cannot locate the position exactly.
O. Perry, J. Perry

### Chapter 5. Percentage, Ratio and Proportion

Abstract
Per cent (symbol %) means per hundred, and a percentage may be considered as a common fraction with 100 as denominator
$$3\%=\frac{3}{{100}},\quad \;72\%=\frac{{72}}{{100}}\,\,or\,\,\frac{{18}}{{25}},\quad \quad 2\tfrac{1}{2}\%=\frac{5}{{200}}\,\,or\,\,\frac{1}{{40}}$$
A percentage may also be expressed as a decimal fraction, so that
$$7\%=7\div 100=0.07,\quad \quad 2.5\%=0.025,\quad \quad 100\%=1.00$$
To change a percentage to a fraction, divide by one hundred.
O. Perry, J. Perry

### Chapter 6. Other Number Bases

Abstract
The denary system of counting used in Chapter 1 has ten digits and the columns are powers of ten. Any positive integer other than one could be used as the base of a counting system, for instance the octal system, base eight, uses the eight digits 1, 2, 3, 4, 5, 6, 7, 0 and the columns are powers of eight.
O. Perry, J. Perry

### Chapter 7. Algebraic Expressions

Abstract
The name algebra is derived from an Arabic word, and most of the elementary algebra was known by the fourth century A.D. In algebra the processes of arithmetic are described in a general but compact form by using letter symbols to represent many different numbers and the ability to manipulate algebraic expressions and to solve equations is important in all branches of science as well as in mathematics.
O. Perry, J. Perry

### Chapter 8. Algebraic Equations and Inequalities

Abstract
Placing an equals sign between two algebraic expressions implies that they have the same numerical value
$$2x=10,\quad \quad 3x-2=x+4,\quad \quad {{x}^{2}}-5x-6=0$$
are all equations in one unknown, x, which is called the variable. In order to maintain the equality, whatever change is made to the expression on one side of an equation must also be made on the other side.
O. Perry, J. Perry

### Chapter 9. Sets

Abstract
Sets are the basis of modern algebra, and set theory and notation are used in other branches of mathematics, particularly in probability and topology.
O. Perry, J. Perry

### Chapter 10. Matrices

Abstract
A set is a collection of elements in no particular order, but a matrix is a mathematical unit in which the elements are numbers arranged in rows and columns. The same numbers arranged in a different order would give a different matrix, and matrices are used to present numerical information in a compact form.
O. Perry, J. Perry

### Chapter 11. Introduction to Geometry

Abstract
The need to measure distances and directions precisely for erecting buildings and monuments stimulated the study of geometry in Greece and Egypt in the sixth century B.C. In the third century B.C. Euclid enlarged the knowledge of the subject and presented it as a continuous logical development in his books, which remained the main source of geometrical studies for the succeeding 2000 years.
O. Perry, J. Perry

### Chapter 12. Geometrical Constructions

Abstract
A locus in plane geometry is a line or curve joining all points in the plane which satisfy the given conditions. For example, a circle is the locus of all the points in a plane which are equidistant from a fixed point in the plane; the fixed point is the centre, the fixed distance is the radius and the locus is the circumference of the circle. An arc is part of the circumference.
O. Perry, J. Perry

### Chapter 13. Perimeter, Area and Volume

Abstract
The perimeter of a polygon is the total length of its sides, so that a regular polygon with n sides of length l cm has perimeter nl cm. There is no general formula for the perimeter of irregular polygons but when the dimensions are known the perimeter is easily calculated.
O. Perry, J. Perry

### Chapter 14. Mappings and Functions, Variation

Abstract
This chapter contains three topics, which although closely related, could be studied separately. Mappings and mapping diagrams (Sections 14.1 and 14.2) and the composition of functions (Section 14.3) are required by only a few Examining Boards. Function notation and the ability to evaluate functions for given values of the variable (Section 14.4) are used later in the book, while a knowledge of variation (Section 14.5) is essential in engineering and science, and is required by most Boards.
O. Perry, J. Perry

### Chapter 15. Cartesian Graphs of Functions

Abstract
An algebraic equation in the form y = f(x), such as y = x2 − 2x + 3, is called a Cartesian equation (after the French mathematician Descartes), and a mapping diagram referred to perpendicular axes is the Cartesian graph of the function f(x). Since the value of y is calculated from a given value of x, x is the independent variable and y is the dependent variable.
O. Perry, J. Perry

### Chapter 16. Applications of Graphs

Abstract
The solution of many problems in industry and commerce depends on the manner in which changes in variable quantities such as profit, value of stock and cost of labour and transport affect each other. When these variables can be written algebraically as a number of linear inequalities, computer programs are devised to find the best (optimum) conditions, and this process is called linear programming.
O. Perry, J. Perry

### Chapter 17. Differential Calculus

Abstract
The invention of the calculus was one of the greatest mathematical achievements of the seventeenth century and many famous mathematicians contributed, including Kepler, Fermat, Newton and Leibnitz, but the notation finally adopted was that of Leibnitz.
O. Perry, J. Perry

### Chapter 18. Integral Calculus

Abstract
It was Leibnitz who first used the elongated ‘s’ (for ‘summa’) as the symbol for an integral. If y is a function of x
$$\int {y\;dx}$$
means ‘the integral of y with respect to x’.
O. Perry, J. Perry

### Chapter 19. Statistics

Abstract
The subject of statistics includes the collection and processing of large amounts of numerical information (data). Vital statistics, for instance, are not the measurements of a beauty queen but the numbers of births, marriages and deaths in the population of a country. Social statistics are the information collected on behalf of the government, such as the number of people employed in various industries, and these are published in Statistical Abstracts by HMSO. Mathematical statistics is concerned with calculations based on samples from a population, and includes such quantities as, for example, the average weekly income of employees and the expected life of electrical batteries.
O. Perry, J. Perry

### Chapter 20. Probability

Abstract
A possible result of an experiment or game is called an outcome and each combination of outcomes is an event. For instance, if an experiment consists of throwing two dice, the possible outcomes are all the ordered pairs such as (2,3) (6,6) (5,1) and there are 36 altogether. One event, A, might be showing the same number on both dice and event B might be a total score of 9.
O. Perry, J. Perry

### Chapter 21. Trigonometry

Abstract
In geometry the angle between two lines is measured in degrees, but in trigonometry the size of an acute angle is described by the ratio of two of the sides of a right angled triangle containing the angle. The main advantage of using trigonometrical ratios is that distances can be calculated which can not be measured directly, in astronomy, surveying, navigation and travel problems. Another advantage is that angles can be introduced into equations without any change of units because ratios are numbers, and this is important in mathematics and physics.
O. Perry, J. Perry

### Chapter 22. Vectors

Abstract
A north-east wind blowing steadily at a speed of 30 km/h has magnitude (30 km/h) and direction (bearing 225°) and the velocity of the wind is a vector. Displacement, acceleration and force are vector quantities, and we are studying the theory of vectors here because the mathematics of one type of vector can be applied to every other vector.
O. Perry, J. Perry

### Chapter 23. Transformations of the Cartesian Plane

Abstract
Transformation problems can be solved by graphical methods and by algebraic methods using matrices, but you may not be given a choice in a particular question.
O. Perry, J. Perry

### Chapter 24. Further Geometry

Abstract
Traditional or Euclidean geometry (named after Euclid) comprises a few basic axioms which are so obviously true that no proof is necessary, and a sequence of theorems, the proof of each theorem following from those earlier in the sequence. Formal proofs of theorems are not given here, since they are no longer required in the syllabus of the major Examining Boards, but a number of the important theorems are stated, with examples to show how they are applied to problems in geometry.
O. Perry, J. Perry

### Chapter 25. Further Geometrical Constructions

Abstract
In this chapter the following constructions are used, as described in Chapter 12.
O. Perry, J. Perry