Working With Arithmetic

410 Pages·1967·21.5 MB·English

by  D. M. Neal B.Sc.

WORKING WITH ARITHMETIC
Working With
ARITHMETIC
by
D. M. NEAL, B.sc.
Lecturer in Mathematics, Mid-Essex Technical College
Formerly Assistant Master, Owen's School
and
J.
D. NEW, B.Sc. (Eng.), A.C.G.I.
Formerly Assistant Master, Owen's School
Macmillan Education
MACMILLAN AND CLEAVER LTD
Little Essex Street, London W C 2
ISBN 978-1-349-00045-6 ISBN 978-1-349-00043-2 (eBook)
DOI 10.1007/978-1-349-00043-2
© 1967
D. M. NEAL
D . .J. NEW
+
Crown Bvo x 401 pages
82 line illustrations
Preface
It is the aim of this book to be a concise and yet complete
course in basic arithmetic, meeting the requirements of stu
dents studying for the General Certificate of Education and
Certificate of Secondary Education and similar ' ordinary
level ' examinations.
It can be divided into three parts, concerning respectively:
(i) the fundamental operations
(ii) the use of mathematical tables
(iii) applications.
The first eleven chapters give the student an introduction
to the fundamental operations of arithmetic. The authors
have attempted, by introducing such ideas as, ' The origin of
number ', ' What is a fraction?', ' Why decimals?', ' What do
we mean by measurement?', 'What is area?', 'What is
volume?', to give the student a sound understanding of these
basic ideas. At the same time, being guided by their class
room experiences, they recognise the importance of securing
accuracy. Thus it will be found that each chapter also in
cludes many worked examples and sufficient exercises for the
student, so that he may become thoroughly proficient in the
use of, and manipulation of, such basic principles as fractions,
decimals, ratio, proportion, and percentage.
In Chapter II the floating point representation is intro
duced. This is used in both Chapters II and 12 to give the
student a simple introduction to the working of logarithm
tables and all other tables likely to be needed in a course in
basic arithmetic. By introducing logarithms as the ' area
under the curve ', their properties are easily deduced with the
vi PREFACE
aid of the trapezoidal rule. Both chapters contain many
exercises designed to give a complete working knowledge
of mathematical tables.
The final chapters are devoted to applications of arithmetic
in everyday life. The numerous worked examples, and pro
blems set in each chapter, are of the type that the student
is likely to encounter in G.C.E., C.S.E., or other 'ordinary
level ' examinations.
The authors wish to express their gratitude to the Univer
sity of London Examination Board and the Associated Exam
ining Board for their permission to make free use of past
examination questions.
Spring 1967 D. M. Neal
D.J. New
Contents
Preface v
Chapter 1. Integral Numbers
Number Patterns. Factors. Prime Numbers and
Prime Factors. Multiples. Highest Common Factor.
Lowest Common Multiple. Use ofFactors. Tests for
Divisibility. Factorisation of Large Numbers. To
Find the H.C.F. Application of Factors to L.C.M.
Powers of Numbers. Index Notation. Roots of
Numbers. Roots by the Use of Prime Factors.
Chapter 2. Fractions 16
Definition of Fractions. Mixed Numbers. Reduction
of Fractions. Magnitude of Fractions. Addition
and Subtraction. Multiplication. Division. Combined
Operations. Problems in Fractions.
Chapter 3. Decimals 35
Ancient Numerals. Position-Value System of Count-
ing. Working with Decimals. Addition and Subtrac-
tion. Multiplication and Division by powers of ten.
Multiplication and Division by other numbers. Con-
version of Fractions to Decimals. Recurring Decimals.
Degrees of Accuracy. Corrected Approximations.
Decimalisation of Compound Quantities. Square
Roots of Decimals by Factors.
Chapter 4. Averages and Unitary Methods 59
Averages. Direct Variation. Inverse Variation.
Chapter 5. The Common Systems of Measurement 70
First Ideas of Measurement. British System. Metric
System. Table of Standard Units. Decimals and the
Metric System. Calculations in the British and Metric
Systems. Conversion of Units of Measurement.
viii CONTENTS
Chapter 6. Measurement of Area 89
Area by Weighing. Unit of Measurement. Fitting
Squares into Rectangles. Land Measure. Calcula~
tion of Areas by Formulae: Square, Rectangle, Tri~
angle, Parallelogram, Trapezium. Perimeter of Recti~
linear Plane Figure. Problems involving use of Area.
Pythagoras Theorem.
Chapter 7. Manipulation of Compound Quantities. . 124
Working with Mixed Units. Fractional Method.
Simple Practice. Application of Simple Practice. Sub~
rraction Method. Compound Practice.
Chapter B. Volumes 131
Volume and Capacity. Volume of Rectangular Ob~
ect. Volume of Prisms.
Chapter 9. Ratio and Proportion 142
Comparison. Ratio in the Form 1 : m and n : I. Ratio
Expressed in Whole Numbers. Direct Proportion.
Inverse Proportion. Ratio of more than Two Quan~
tities. Proportional Parts. Applications of Ratio:
( 1) Scale Drawings (2) Income Tax (3) Rates and
Rateable Value (4) Bankruptcy.
Chapter 10. Percentage 164
Meaning of Percentage. Fractions as Percentages.
Percentage in Daily Life: (I) Hire Purchase (2) In~
surance (3) Discount. Percentage Change. Profit and
Loss. Simple Interest. Use of Formula. Compound
Interest.
Chapter 11. Graphs 191
Plotting. Travel Graphs. Intersecting Graphs. Other
Straight Line Graphs. Trapezoidal Rule.
Revision Exercises Chapters 1 to 11 225
CONTENTS IX
Chapter 12. Use of Tables 234
Squares of Numbers from 1 to 10. Square Roots of
Numbers from 1 to 100. Square Root Tables. Re
ciprocals of Numbers from l to 10. Floating Point
Numbers. Operations with Floating Point Numbers:
Multiplication, Division, Addition, Subtraction.
Squares of Other Numbers. Square Roots of Other
Numbers. Reciprocals of Other Numbers. Exam
ples on Combined Tables.
Chapter 13. Logarithms 256
Purpose. Evaluation of Logarithms from a Graph.
Fundamental Properties of Logarithms. Logarithm of
any Number. The Bar Notation. Basic Rules for
Multiplication, Division, Powers and Roots. Use of
Four-Figure Tables. To Find the Logarithm of a
Number. To Find a Number given the Logarithm.
Multiplication. Division. Powers. Roots. Opera-
tions involving Negative Characteristics. Examples
on the use of Logarithms.
Chapter 14. The Circle 278
Diameter and Circumference. Experimental Deter
mination of Ratio between Diameter ami. Circum
ference. The Number Length of an Arc. Area of
7T.
the Circle. Area of the Annulus. Area of the Sector.
Surface Area of Cylinder. Surface Area of Cone.
Chapter 15. Rates and Percentages 298
Problems involving the Principles of Unitary Method,
Rate, Proportion, Compound Units, Mixtures, and
Percentage.
Chapter 16. Further Volumes 307
Cylinder. Sphere. Cone. Specific Gravity and Archi
medes' Principle.
Chapter 17 . Shares and Stocks 317
Formation of a Company. Dividend. Buying and
Selling of Shares. Stocks. Buying and Selling of
Stocks and Shares. _ Other Types of Shares.
X CONTENTS
Chapter 18. Miscellaneous Examples and Exercises . . 333
Answers 347
Tables 381
Square Roots 382
Reciprocals 390
Squares 396
Logarithms 398
Antilogarithms 400