Mathematics for the Practical Man

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MATHEMATICS FOR

THE PRACTICAL MAN EXPLAINING SIMPLY AND QUICKLY ALL THE ELEMENTS OF

ALGEBRA, GEOMETRY, TRIGONOMETRY, LOGARITHMS, COORDINATE

GEOMETRY, CALCULUS WITH ANSWERS TO PROBLErtS BY

GEORGE HOWE,

M.E.

ILLUSTRATED

SEVENTH THOUSAND

D.

NEW YORK VAN NOSTRAND COMPANY 25

Park Place 1918 •0

Copyright,

191 1,

by

D.

VAN NOSTRAND COMPANY

D.

VAN NOSTRAND COMPANY

Copyright,

igig,

by

Stanbope Ipnes

G1LSON COMPANY BOSTON. U.S.A.

r. H.

Dedicated To

Mtavm Agrea, pf.S. PRESIDENT OF THE UNIVERSITY OF TENNESSEE "MY GOOD FRIEND AND GUIDE."

Cornell University Library

The

original of this

book

is in

the Cornell University Library.

There are no known copyright

restrictions in

the United States on the use of the

text.

http://www.archive.org/details/cu31924031266871

PREFACE In preparing this work the author has been prompted by many reasons, the most important of which are: The dearth of short but complete books covering the fundamentals of mathematics.

The tendency

of those elementary books

which " begin

at the beginning " to treat the subject in a popular rather scientific manner. Those who have had experience

than in a bodies of

men

in lecturing to large

know that they are comengineers who have had consider-

in night classes

posed partly of practical

able experience in the operation of machinery, but no scientific training

whatsoever; partly of

men who have de-

voted some time to study through correspondence schools

and similar methods of instruction; partly of men who have had a good education in some non-technical field of work but, feeling a distinct calling to the engineering profession,

have sought

special training

from night lecture

courses; partly of commercial engineering salesmen,

preparation has been non-technical and

who

whose

realize in this

fact a serious handicap whenever an important sale is to be negotiated and they are brought into competition with

the

skill

of trained engineers;

and

finally, of

young men become

leaving high schools and academies anxious to engineers but

purpose.

who

are unable to attend college for that

Therefore

it is

apparent that with this wide

PREFACE

iv

any must begin with studies which are quite familiar to a large number but which have been forgotten or perhaps never undertaken by a large number of others. And here lies the best hope of this textbook. It "begins at the beginning," assumes no mathematical knowledge beyond arithmetic on the part of the student, has endeavored to gather together in a concise and simple yet accurate and difference in the degree of preparation of its students

course of study

scientific

form those fundamental notions of mathematics

without which any studies in engineering are impossible, omitting the usual diffuseness of elementary works, and

making no pretense at elaborate demonstrations, believing that where there is much chaff the seed is easily lost. I have therefore made it the policy of this book that no technical difficulties will be waived, no obstacles circumscribed in the pursuit of any theory or any conception. Straightforward discussion has

been adopted;

where

an attempt has been made to very roots, and proceed no further until

obstacles have been met, strike at their

they have been thoroughly unearthed.

With this introduction, I beg to submit this modest attempt to the engineering world, being amply repaid if, even in a small way, it may advance the general knowledge of mathematics.

GEORGE HOWE. New

York, September, 1910.

TABLE OF CONTENTS Chapter I.

Page

Fundamentals of Algebra.

Addition and Subtrac-

tion II.

i

Fundamentals of Algebra.

Multiplication and Divi-

sion, I III.

-.

Fundamentals of Algebra.

7

Multiplication and Divi12

sion, II

IV.

Fundamentals of Algebra.

Factoring

21

Involution and Evolu-

V. Fundamentals of Algebra.

tion VI. VII.

25

Simple Equations

Fundamentals of Algebra.

Fundamentals

of

Algebra.

Simultaneous

29

Equa-

tions VIII.

Fundamentals of Algebra.

41

Quadratic Equations

DC. Fundamentals of Algebra.

Variation

X. Some Elements of Geometry

XII. Logarithms

XIV.

55

61

XI. Elementary Principles of Trigonometry

XIII. Elementary Principles of Coordinate

48

75

85

Geometry

Elementary Prlnclples of the Calculus

95

no

MATHEMATICS CHAPTER

I

FUNDAMENTALS OF ALGEBRA Addition and Subtraction

As an

introduction to this chapter on the fundamental

principles of algebra, I will say that essential to

an understanding

it

absolutely

is

of engineering that the

fundamental principles of algebra be thoroughly digested

and

redigested,

— in

mind and method Algebra if

is

short, literally

soaked into one's

of thought.

a very simple science

— extremely

looked at from a common-sense standpoint.

seen thus,

it

If

can be made most intricate and, in It is arithmetic simplified,

cut to arithmetic.

In arithmetic we would say,

hat costs 5 cents, 10 hats cost^ 50 cents.

cents,

say,

if

fact,

if

one

In algebra

one a costs 5 cents, then 10 a cost 50

a being used here to represent "hat."

we term

not

— a short

incomprehensible.

we would

simple

in algebra a symbol,

handled by means

of

to represent one thing;

and

all

such symbols, b,

a

is

what

quantities are

a

another symbol,

is

presumed

is

presumed

MATHEMATICS

2

to represent another thing, c another,

on

so

and

any number

for

is

objects.

therefore, of using

simplicity,

sent objects

of

d another, and

The

usefulness

symbols to repre-

Suppose a merchant in the

obvious.

furniture business to be taking stock.

He would

through his stock rooms and,

10 chairs, he

seeing

would actually write down "10 chairs"; 5 would actually write out "5 actually write this out,

Now,

and so on.

by the

he had at

by the

beds by the letter

letter b,

if

c,

he

he would

tables"; 4 beds,

the start agreed to represent chairs tables

tables,

go

letter a,

and so on,

he would have been saved the necessity of writing

down

the

names

have written 10

of these articles

a, 5

and 4

b,

Definition of a Symbol.

which

it is

When

each time, and could

c.

— A symbol

is

some

letter

by

agreed to represent some object or thing.

a problem

thing necessary

is

is

to

to be

worked in algebra, the

make a

first

choice of symbols, namely,

to assign certain letters to each of the different objects

concerned with the problem,

up a

code.

When

this

code

— in

is

be rigorously maintained; that

any problem or

other words, to get

once established is, if,

it

must

in the solution of

set of problems, it is

once stipulated

that a shall represent a chair, then wherever a appears it

means a

chair,

and wherever the word

be inserted an a must be placed be changed.

— the

chair

would

code must not

FUNDAMENTALS OF ALGEBRA Positivity

and Negativity.

3

— Now, in algebraic thought,

not only do we use symbols to represent various objects

and

things,

but we use the signs plus (+) or minus (—

before the symbols, to indicate

what we

call

the positivity

or negativity of the object.

Addition and Subtraction.

— Algebraically,

if,

and accounts, a merchant

ing over his stock

in go-

finds that

he has 4 tables in stock, and on glancing over his

books finds that he owes 3 the 4

tables

stock

in

tables,

he would represent

by such a form

as

+4 a,

a

representing table; the 3 tables which he owes he would represent

by —3

a,

the plus sign indicating that which

he has on hand and the minus sign that which he owes.

Grouping

the

quantities

+4 a

and

—3a

in other words, striking a balance, one

together,

would get +a,

which represents the one table which he owns over and above that which he owes.

The plus

sign, then, is

taken

to indicate all things on hand, all quantities greater

The minus

than zero.

sign

those things which are owed,

is

all

taken to indicate

Suppose the following to be the inventory tain

quantity

+4 a, — 2

&,

of

stock:

— 2 c, +5 c.

quantities together

and

all

things less than zero. of a cer-

+8 a, — 2 a, +6 b, —3 c, Now, on grouping

these

striking a balance, it will be

seen that there are 8 of those things which are repre-

sented

4

a,

by a on hand;

on hand;

2

likewise 4 more, represented

are owed, namely,

—2

a.

by

Therefore,

MATHEMATICS

4

+8 a, +4 a,

on grouping will

be the

Now,

result.

and

we have

senting the objects which

+6b

and —2

—3

c,

— 2 c,

+5

c

b,

+5

represents 5

represent that 5 tities neutralize

+ 8a —

2

together^ will

on hand, and

c's

reduces to

+46.

Grouping

give o,

—3c

because

and

—2c

Therefore,

strike a balance.

+ 66 — 3 c + 4a — + 10 a + 4

a

we have

b,

are owed; therefore, these quan-

c's

and

c

+10

together,

called

as a result

giving

and

—2a

collecting those terms repre-

2b

—

2 c

+ 5c

b.

This process of gathering together and simplifying a collection of

in

call

terms having different signs

algebra addition and subtraction.

is

what we

Nothing

is

more simple, and yet nothing should be more thoroughly understood before proceeding further.

It

is

obviously

impossible to add one table to one chair and thereby get two chairs, or one bojbk to one hat and get two

books; whereas to another

it is

perfectly possible to

book and

get'

add one book

two books, one chair to an-

other chair and thereby get two chairs.

Rule.

— Like symbols can be added and

subtracted,

and

8a; a

+b + b,

only like symbols.

a will

+ a will not give

give 2 a; 2

a or 2

this being the simplest

these

30 b,

+

but

50

will

will

give

simply give a

form in which the addition of

two terms can be expressed.

FUNDAMENTALS OF ALGEBRA Coefficients.

— In

+8 a

any term such as

sign indicates that the object zero, the 8 indicates the

is

and the a

coefficient,

ject,

whether

them on hand,

of

a chair or a book or a table that

it is

In the term

a.

greater than zero, the 6 indicates the

on hand, and the a in his pocket

their nature.

and he owes $50,

his

represented $1

by the

owned, or of objects

If

it is

owed by

still

letter a,

together,

which

— 50 a. On

Algebraic Expressions.

owe

an algebraic expression;

+ $a +

$30.

simply, for the 5 b

+ 3a + written.

11b

+

c,

he If

then the $20 in

+ 20 a,

the $50

grouping these terms

—30

a.

for instance,

+a-|-2&

$b

+

6b

+c

+a+

+ cisan is

and 6

n

b,

b

b is

algebraic

another

braic expression, but this last one can be written

one term, making

if

— An algebraic expression con-

two or more terms;

expression;

has $20

the same process as the settling of

is

accounts, the result would be

sists of

man

a

evident that

pocket would be represented by

that he

+6 a,

is

paid up as far as he could, he would

we had

we

number

sign indicates that the object

(+)

it

the

called

is

indicates the nature of the ob-

have represented by the symbol the plus

the plus

on hand or greater than

number

the numerical part of the term and

is

5

alge-

more

can be grouped together in

and the expression now becomes

which

is

as

simple as

it

can be

It is always advisable to

group together into

any

algebraic expression

the smallest

number

of terms

MATHEMATICS

6

wherever

it is

met

in a problem,

manipulation or handling of

When

there

is

To

sign

it.

no sign before the

expression the plus i

and thus simplify the

(+)

sign

is

term of an

first

intended.

,

subtract one quantity from another, change the

and then group the quantities into one term, as

just

+ 12 a

we

Thus: to subtract 4 a from

explained.

write

—

4a

-+•

i2j^wMclvsiniplifies into 4-8

a.

—

2

subtracting i"a from 4- 6 a

which equals

we would have

Again, a

+ 6 a,

+4 a. PROBLEMS

Simplify the following expressions 1. 2.

3.

4.

5.

6. 7.

4 ^ V- &B + 5b + 6c-8a~3d + a — 6 + c — 10 a — c + 2 10J + 3Z + 8& — 4 d — 6.z — 12 & + 5 a — 3d + 8z — ioa + 8b — $a — 6z + iob. 5« — 4 y + 3 — 2 &.+ 4 y + * + z + a — j + 6y. 6— 2 a + 5 c + 7 a — 10 6j— 8 c 4- 4 a— b + c. 3 — 2 » + a + 6 + ioy — 6 *— y— 7 a + 3 b + 2y. 4X — y + z + x + 15Z — 3X + 6y^7y+ 12Z. r r ioa

b.

7,

,z

&.

CHAPTER

II

FUNDAMENTALS OF ALGEBRA and Division

Multiplication

We

have seen how the use

of algebra simplifies the

operations of addition and subtraction, but in multiplication

and

division this simplification is far greater,

weapon

the great

to the student is

of

now

to

become If the

is

one foot, his result

one square

foot, the

being very different from the foot. multiply one chair the result? Is there

by one

What word

is

asked to multiply one foot by

student of arithmetic is

and

realized for the first time.

thought which algebra

Now, ask

How

table.

square foot hira to

can he express

can he use to signify the result?

any conception

in his

mind as

to the appear-

ance of the object which would be obtained by multiplying one chair simplified. b,

If

we

by one

which represents in

by a

to call it; to our

table.

its

b,

by

we obtain

a,

all

this is

and a chair by

the expression ab,

entirety the multiplication of a

We need

we simply

no word, no name by which

use the form ab, and that carries

mind the notion

multiplied

In algebra

represent a table

and we multiply a by

chair

table?

of the thing

by the thing which we 7

which we

call b.

And

call

a

thus the

MATHEMATICS

8

form to

is

any further thought being given

carried without

it.

Exponents.

— The

by

represented cut,

namely, a2

two

a's

multiplication of a

we have a further short

called

an exponent, indicates that

But

aa.

This

.

2,

by a may be

here

have been multiplied by each other; a

would give us a

3

signifies

been multiplied by

the

number

of times the

Now, suppose a were multiplied by a3 signifies

and a3 indicates that 3

The exposymbol has

;j

the result would

a's are multiplied together;

X

would likewise give us a10 a4 ,

X

a

a

Rule.

X

a

2

X

— The

representing

a

3

then

by each other simply

indicates that 5 a's are multiplied together,

4

a

that 2 a's are multiplied together,

multiplying these two expressions

4

X

itself.

2

be a5 , since a2

a

the 3 indicating that three a's have

,

been multiplied by one another; and so on. nent simply

X

a3

X

a7

a4 would give us a8,

would give us a13 and so on. ,

multiplication by each other of symbols

similar objects

is

accomplished by adding

their exponents.

Indentity of Symbols.

be

clearly seen that the

from

either

a or b

from a or b as

c

;

— Now,

in the foregoing it

combined symbol ab

different

ab must be handled as differently

would be handled;

an absolutely new symbol. from a as a square foot 2

is

must

is

in other words, it is

Likewise a2

from a linear

as different from a as one cubic foot

is

is

as different

foot,

and a3

is

from one square

FUNDAMENTALS OF ALGEBRA a2

foot,

a3

a distinct symbol,

is

9

a distinct symbol,

is

and can only be grouped together with other a3 example,

if

an algebraic expression such as

+ a + ab + a + 3 a —

a2

3

2

2

—

a

For

's.

were met

this ab,

to simplify it

we could group

+3 a2

+4 a2 the +a and the — 2a give — a; the — ab neutralize each other; there is

the

,

giving

+ab and

together the a2 and the

;

term with the symbol a3

only one

Therefore the

.

above expression simplified would be 4 a2

—

This

Above

is

as simple as

can be expressed.

it

things the most important

a

and

separate

distinct

8 .

all

never to group unlike

is

symbols together by addition and subtraction.

member fundamentally

+a

that a,

b, ab,

symbols,

a

2 ,

a

3 ,

Re-

a* are all

each representing a

separate and distinct thing.

X c. It gives us the term X b we get a2b. If we have ab X ab, we have 2 ab X 2 ab we get 4 a b

Suppose we have a If

abc.

we

we have a

get a2 b 2

6 a2W

X

X

b

2

3

c,

-

2 2

If

we

;

get 18 a2 b3c; and so on.

Whenever two

terms are multiplied by each other, the coefficients are multiplied together, and the similar parts of the symbols are multiplied together. •

Division.

— Just

down -to mean

2

as

when

divided

in

by

arithmetic

3, in

algebra

mean a

divided

by

b.

a

is

called a

b a denominator, and the expression

write

we writer o

3 to

we

7

numerator and is

called a frac-

MATHEMATICS

IO tion.

If

result is If

a

3

is

a3 5

a

,

is

multiplied

by a2 we have seen that the ,

obtained by adding the exponents 3 and

divided by a

by subtracting

2

2

the result

,

from

is

obtained

— would

equal a,

which

is a,

Therefore

3.

2.

a?b

2 the a in the denominator dividing into o in the nu-

merator a times, and the b in the denominator cancelDivision

ing the b in the numerator.

inverse of multiplication, which

is

is

a b

c

,

and

way

logical

of multiplication,

.

we obtain

may be

written

a b ,

and so

on.

C

— But there

a more

is

scientific

of explaining division as the inverse

and

This

fraction—. a2

—

simplify-

2

Or OCr

Negative Exponents.

On

patent.

A 2 3

ing such an expression as

then simply the

—

it is

may ;

thus

:

Suppose we have the

be written a

that

is,

-2 ,

or the term b 2

any term may be changed

from the numerator of a fraction to the denominator by simply changing the sign of

its

a5

2

exponent.

For example,

written o5

X

a

terms together, which

is

accomplished by adding their

-j

may be

Multiplying these two

.

exponents, would give us a3

,

the addition of 5 and —2.

It

therefore, to

made

make a

3

being the result of is

scarcely necessary,

separate l'aw for division

for multiplication,

when

it is

if

one

is

seen that division

simply changes the sign of the exponent.

This should

FUNDAMENTALS OF ALGEBRA

II

be carefully considered and thought over by the pupil,

Take such an

for it is of great importance.

as

.

abc

l

Suppose

all

the symbols in the denominator

are placed in the numerator, then simplifying, ab~

or,

—

z

which