Mathematics for the General Course in Engineering

140 Pages·1963·5.57 MB·English

by  John C.Moore and Robert Robinson (Auth.)

Mathematics for the
General Course in
Engineering
VOLUME I
JOHN MOORE, M.A. (Cantab.), B.Sc. (Lond.)
Senior Lecturer in Mathematics at the Oxford College of Technology
PERGAMON PRESS
OXFORD · LONDON · NEW YORK PARIS
1963
PERGAMON PRESS LTD.
Headington Hill Hall, Oxford
4 & 5 Fitzroy Square, London W.l.
PERGAMON PRESS INC.
122 East 55th Street, New York 22, N.Y.
GAUTHIER-VILLARS ED.
55 Quai des Grands-Augustins, Paris 6
PERGAMON PRESS G.m.b.H.
Kaiserstrasse 75, Frankfurt am Main.
Copyright © 1963
PERGAMON PRESS LTD.
Library of Congress Card No. 62-18999
Set in 10 on 12 pt Times New Roman and Printed in Great Britain by
PAGE BROS. (NORWICH) LTD.
Note to the Student
THIS book is the first of two volumes which have been specially
written to cover the syllabus in Mathematics for the G. 1 and G.2
years of the new General Course in Engineering. The present
volume covers G.I. It is natural for certain parts of the book to
stand out from the main thread of explanation and these have
been printed in red. They are particularly important and the
student who makes the effort and commits them to memory will
have formed a sound basis for the work of the G.2 year. Also in
red in the text are 31 unworked examples. These form a compre-
hensive revision course and the student is recommended to work
them all through towards the end of the G.l year. Answers are
provided.
ν
CHAPTER ONE
Arithmetic
FRACTIONS
Suppose we take the number 1 and break it into smaller pieces.
These smaller pieces are called PROPER FRACTIONS or
simply FRACTIONS.
When 1 is divided into 2 equal parts each part is called a HALF
1
and is written ^.
ι
DIAGRAM 1.
Further fractions are illustrated below.
~ I "ι .
% ι ·* ι ι
•'4
*% ι • ι ;^ ι ι ;%
DIAGRAM 2.
1
2 GENERAL COURSE IN ENGINEERING; MATHEMATICS
Names of the Fundamental Fractions
11 1 11
2 3 4 5 6
HALF THIRD QUARTER FIFTH SIXTH
The above list can be extended indefinitely and we shall call these
fractions the FUNDAMENTAL fractions. Any other fraction
can be expressed in terms of them.
2 1
For example ^ (TWO THIRDS) means 2 χ ^
3 1
^ (THREE QUARTERS) means 3 χ ^
3
When we meet a fraction such as ^ the number at the top has a
special name. It is called the NUMERATOR. The number at
the bottom also has a special name. It is called the DENOMIN-
ATOR.
3< Numerator
A, Denominator
An important principle
If we take a fraction and MULTIPLY both the numerator and the
denominator by the same number then the new fraction is exactly
equal to the old.
If we take a fraction and DIVIDE both the numerator and the
denominator by the same number then the new fraction is exactly
equal to the old.
Addition of Fractions
It is easy to add fractions whose DENOMINATORS are equal.
ARITHMETIC 3
If we are called upon to add fractions whose DENOMINATORS are
not equal then we use the first part of the principle printed above
in red.
Ex. Calculate
Think of the smallest number into which 4, 6 and 8 will divide
exactly. It is 24.
Using the first part of the principle each fraction can be ex-
pressed with a 24 in the DENOMINATOR.
So
Method of setting out
The previous example is usually set out as follows.
4 GENERAL COURSE IN ENGINEERING; MATHEMATICS
If we write this means
Note that
A fraction such as in which the NUMERATOR is larger than the
DENOMINATOR is called an IMPROPER FRACTION.
Ex. Express as an IMPROPER FRACTION using a quick method.
Quick method
Multiply the 1 by the 4 and then add in the 3. Put this number
over 4.
Ex. Express as an IMPROPER FRACTION using a quick method,
Ex. Calculate
ARITHMETIC 5
Subtraction of Fractions
Ex. Calculate
Multiplication of Fractions
5 2
Ex. Calculate η of ^ using a diagram.
•A§ ^ ^
:\>f ill
|| m f| Iff;
H
Κ ψ
f
γ.Ύ///Λ
DIAGRAM 3.
If the total area represents 1 then the portion enclosed by the
thick black lines represents ^
5 2
^ of ^ will be the shaded area.
6 GENERAL COURSE IN ENGINEERING; MATHEMATICS
This shaded area is made up of 10 small rectangles and each
rectangle represents (there are 21 rectangles altogether).
So the shaded area is
We usually use the multiplication symbol (x) and write
Quick method
The student will notice that there is a quick way of working out
We simply say 5 X 2 = 10 and 7 X 3 = 21.
The answer is
Division of Fractions
Ex. Calculate using a diagram.
of the total area in Diagram 3 comes to 15 small rectangles.
of the total area in Diagram 3 comes to 14 small rectangles.
So 15 small rectangles 14 small rectangles
ARITHMETIC 7
Quick method
2
We can obtain the correct answer by turning the ^ upside-down
and multiplying.
In some examples where fractions are multiplied together it is
necessary to CANCEL.
Ex. Calculate
12
When we CANCEL we are dividing both the numerator and the
denominator by the same number. This does not alter the value
of the fraction.
9
Ex. Express ^ as a fraction in its LOWEST TERMS.