CHAPTER I. 
THE GENERAL NUMBER. 

A. THE SYSTEM OF NUMBERS. 

Addition and Subtraction. 

. I. From the operation of counting and measuring arose the 
art of figuring, arithmetic, algebra, and finally, more or less, 
the entire structure of mathematics. 

During the development of the human race throughout the 
ages, which is repeated by every child during the first years 
of life, the first conceptions of numerical values were vague 
and crude: many and few, big and Httle, large and small. 
Later the ability to count, that is, the knowledge of numbers, 
developed, and last of all the ability of measuring, and even 
up to-day, measuring is to a considerable extent done by count- 
ing: steps, knots, etc. 

From counting arose the simplest arithmetical operation — 
addition. Thus we may count a bunch of horses: 

1, 2, 3, 4, 5, 

and then count a second bunch of horses, 

1, 2, 3; 

now put the second bunch together with the first one, into one 
bunch, and count them. That is, after counting the horses 


2 ENGINEERING MATHEMATICS. 

of the first bunch, we continue to count those of the second 
bunch, thus : 

1, 2, 3, 4, 5-6, 7, 8; 

which gives addition, 

5 + 3 = 8; 
or, in general, 

a-\-h = c. 

We may take away again the second bunch of horses, that 
means, we count the entire bunch of horses, and then count 
off those we take away thus: 

1, 2, 3, 4, 5, 6, 7, 8-7, 6, 5; 

which gives subtraction, 

8-3 = 5; 
or, in general, 

c—h = a. 

The reverse of putting a group of things together with 
another group is to take a group away, therefore subtraction 
is the reverse of addition, 

2. Immediately we notice an essential difference between 
addition and subtraction, which may be illustrated by the 
following examples : 

Addition: 5 horses +3 horses gives 8 horses, 
Subtraction: 5 horses —3 horses gives 2 horses, 
Addition: 5 horses +7 horses gives 12 horses, 
Subtraction: 5 horses— 7 horses is impossible. 

From the above it follows that we can always add, but we 
cannot always subtract; subtraction is not always possible; 
it is not, when the number of things which we desire to sub- 
tract is greater than the number of things from which we 
desire to subtract. 

The same relation obtains in measuring; we may measure 
a distance from a starting point A (Fig. 1), for instance in steps, 
and then measure a second distance, and get the total distance 
from the starting point by addition: 5 steps, from A to B, 


THE GENERAL NUMBER. 3 

then 3 steps, from B to C, gives the distance from A to C, as 
8 steps. 

5 steps +3 steps =8 steps; 


12 3 4 5 6 
(^ 1 1 1 1 CD I 


B 
Fig. 1. Addition. 

or, we may step off a distance, and then step back, that is, 
subtract. another distance, for instance (Fig. 2), 

5 steps— 3 steps = 2 steps; 

that is, going 5 steps, from A to B, and then 3 steps back, 
from B to C, brings us to C, 2 steps away from A. 


12 3 4 5 
H S 1 1 ®- 


C B 

Fig. 2. Subtraction. 

Trying the case of subtraction which was impossible, in the 
example with the horses, 5 steps -7 steps = ? We go from the 
starting point. A, 5 steps, to J5, and then step back 7 steps; 
here we find that sometimes we can do it, sometimes we cannot 
do it; if back of the starting point A is a stone wall, we cannot 
step back 7 steps. If A is a chalk mark in the road, we may 
step back beyond it, and come to C in Fig. 3. In the latter case, 


■> 


21O12345 
(t) I (|) I 1 1 h 


C A B 

Fig. 3, Subtraction, Negative Result. 

at C we are again 2 steps distant from the starting point, just 
as in Fig. 2. That is, 

5-3 = 2 (Fig. 2), 

5-7 = 2 (Fig. 3). 

In the case where we can subtract 7 from 5, we get the same 
distance from the starting point as when we subtract 3 from 5, 


4 ENGINEERING MATHEMATICS. 

but the distance AC in Fig. 3, while the same, 2 steps, as 
in Fig. 2, is different in character, the one is toward the left, 
the other toward the right. That means, we have two kinds 
of distance units, those to the right and those to the left, and 
have to find some way to distinguish them. The distance 2 
in Fig. 3 is toward the left of the starting point A, that is, 
in that direction, in which we step when subtracting, and 
it thus appears natural to distinguish it from the distance 
2 in Fig. 2, by calling the former— 2, while we call the distance 
AC in Fig. 2: +2, since it is in the direction from A, in which 
we step in adding. 

This leads to a subdivision of the system of absolute numbers, 

1, 2, 3, . . . 

into two classes, positive numbers, 

+ 1, +2, +3, ...: 
and negative numbers, 

-1,-2, -3, ...; 

and by the introduction of negative numbers, we can always 
carry gut the mathematical operation of subtraction: 

c—h = a, 

and, if h is greater than c, a merely becomes a negative number. 

3. We must therefore realize that the negative number and 
the negative unit, —1, is a mathematical fiction, and not in 
universal agreement with experience, as the absolute number 
found in the operation of counting, and the negative number 
does not always represent an existing condition in practical 
experience. 

In the apphcation of numbers to the phenomena of nature, 
we sometimes find conditions where we can give the negative 
number a physical meaning, expressing a relation as the 
reverse to the positive number; in other cases we cannot do 
this. For instance, 5 horses -7 horses = -2 horses has no 
physical meaning. There exist no negative horses, and at the 
best we could only express the relation by saying, 5 horses -7 
horses is impossible, 2 horses are missing. 


D 


THE GENERAL NUMBER. 


In the same way, an illumination of 5 foot-candles, lowered 
by 3 foot-candles, gives an illumination of 2 foot-candles, thus, 

. b foot-candles —3 foot-candles = 2 foot-candles. 

If it is tried to lower the illumination of 5 foot-candles by 7 
foot-candles, it will be found impossible; there cannot be a 
negative illumination of 2 foot-candles ; the limit is zero illumina- 
tion, or darkness. 

From a string of 5 feet length, we can cut off 3 feet, leaving 

2 feet, but we cannot cut off 7 feet, leaving —2 feet of string. 

In these instances, the negative number is meaningless, 
a mere imaginary mathematical fiction. 

If the temperature is 5 deg. cent, above freezing, and falls 

3 deg., it will be 2 deg. cent, above freezing. If it falls 7 deg. 
it will be 2 deg. cent, below freezing. The one case is just as 
real physically, as the other, and in this instance we may 
express the relation thus: 

+ 5 deg. cent. —3 deg. cent. = +2 deg. cent., 

-}-5 deg. cent. —7 deg. cent. = — 2 deg. cent.; 

that is, in temperature measurements by the conventional 
temperature scale, the negative numbers have just as much 
physical existence as the positive numbers. 

The same is the case with time, we may represent future 
time, from the present as starting point, by positive numbers, 
and past time then will be represented by negative numbers. 
But we may equally well represent past time by positive num- 
bers, and future times then appear as negative numbers. In 
this, and most other physical applications, the negative number 
thus appears equivalent with the positive number, and inter- 
changeable: we may choose any direction as positive, and 
the reverse direction then is negative. Mathematically, how- 
ever, a difference exists between the positive and the negative 
number; the positive unit, multiplied by itself, remains a pos- 
itive unit, but the negative unit, multiplied with itself, does 
not remain a negative unit, but becomes positive: 

( + 1)X( + 1) = ( + 1); 

(-1)X(-1) = ( + 1), andnot =(-1). 


6 ENGINEERING MATHEMATICS. 

Starting from 5 deg. northern latitude and going 7 deg. 
south, brings us to 2 deg. southern latitude, which may be 
expresses thus, 

+5 deg. latitude —7 deg. latitude = —2 deg. latitude. 

Therefore, in all cases, where there are two opposite direc- 
tions, right and left on a line, north and south latitude, east 
and west longitude, future and past, assets and liabihties, etc., 
there may be apphcation of the negative number; in other cases, 
where there is only one kind or direction, counting horses, 
measuring illumination, etc., there is no physical meaning 
which would be represented by a negative number. There 
are still other cases, where a meaning may sometimes be found 
and sometimes not; for instance, if we have 5 dollars in our 
pocket, we cannot take away 7 dollars; if we have 5 dollars 
in the bank, we may be able to draw out 7 dollars, or we may 
not, depending on our credit. In the first case, 5 dollars —7 
dollars is an impossibility, while the second case 5 dollars —7 
dollars = 2 dollars overdraft. 

In any case, however, we must realize that the negative 
number is not a physical, but a mathematical conception, 
which may find a physical representation, or may not, depend- 
ing on the physical conditions to which it is applied. The 
negative number thus is just as imaginary, and just as real, 
depending on the case to which it is appHed, as the imaginary 
number V — 1; and the only difference is, that we have become 
familiar with the negative number at an earlier age, where we 
wera less critical, and thus have taken it for granted, become 
familiar with it by use, and usually do not realize that it is 
a mathematical conception, and not a physical reality. AVhen 
we first learned it, however, it was quite a step to become 
accustomed to saying, 5— 7 =—2, and not simply, 5—7 is 
impossible. 

Multiplication and Division. 

4. If we have a bunch of 4 horses, and another bunch of 4 
horses, and still another bunch of 4 horses, and add together 
the three bunches of 4 horses each, we get, 

4 horses +4 horses + 4 horses = 12 horses; 


THE GENERAL NUMBER, 7 

or, as we express it, 

3X4 horses = 12 horses. 

The operation of multiple addition thus leads to the next 
operation, multiplication. Multiplication is multiple addi- 
tion, 

bXa = c, 

thus means 

a + a + a + . . . (b terms) = c. 

Just like addition, multiphcation can always be carried 
out. 

Three gi'oups of 4 horses each, give 12 horses. Inversely, if 
we have 12 horses, and divide them into 3 equal groups, each 
group contains 4 horses. This gives us the reverse operation 
of multiplication, or (imsi'on, which is written, thus: 

12 horses ^ , 
5 = 4 horses ; 


or, in general, 


6=«- 


If we have a bunch of 12 horses, and divide it into two equal 
groups, we get 6 horses in each group, thus : 

12 horses ^ , 
:z = 6 horses, 


if we divide unto 4 equal groups, 
12 horses 


3 horses. 


If now we attempt to divide the bunch of 12 horses into 5 equal 
groups, we find we cannot do it; if we have 2 horses in each 
group, 2 horses are left over; if we put 3 horses in each group, 
we do not have enough to make 5 groups; that is, 12 horses 
divided by 5 is impossible; or, as we usually say; 12 horses 
divided by 5 gives 2 horses and 2 horses left over, which is 
written, 

12 ^ . . ^ 

-]p- = 2, remamder 2. 


8 ENGINEERING MATHEMATICS. 

Thus it is seen that the reverse operation of multiplication, 
or division, cannot always be carried out. 

5. If we have 10 apples, and divide them into 3, we get 3 
apples in each group, and one apple left over. 

-^ = 3, remainder 1, 

we may now cut the left-over apple into 3 equal parts, in which 
case. 

In the same manner, if we have 12 apples, we can divide 
into 5, by cutting 2 apples each into 5 equal pieces, and get 
in each of the 5 groups, 2 apples and 2 pieces. 

| = 2+2x^=2i 

To be able to carry the operation of division through for 
all numerical values, makes it necessary to introduce a new 
unit, smaller than the original unit, and derived as a part of it. 

Thus, if we divide a string of 10 feet length into 3 equal 
parts, each part contains 3 feet, and 1 foot is left over. One 
foot is made up of 12 inches, and 12 inches divided into 3 gives 
4 inches; hence, 10 feet divided by 3 gives 3 feet 4 inches. 

Division leads us to a new form of numbers: the fraction. 

The fraction, however, is just as much a mathematical con- 
ception, which sometimes may be appUcable, and sometimes 
not, as the negative number. In the above instance of 12 
horses, divided into 5 groups, it is not appHcable. 

12 horses ^ , 
r = 2f horses 

is impossible; we cannot have fractions of horses, and what 
we would get in this attempt would be 5 groups, each com- 
prising 2 horses and some pieces of carcass. 

Thus, the mathematical conception of the fraction is ap- 
plicable to those physical quantities which can be divided into 
smaller units, but is not applicable to those, which are indi- 
visible, or individuals, as we usually call them. 


THE GENERAL NUMBER. 9 

Involution and Evolution. 
6. If we have a product of several equal factors, as, 

4X4X4 = 64, 
it is written as, 4^ = 64 ; 

or, in general, a!* = c. 

The operation of multiple multiplication of equal factors 
thus leads to the next algebraic operation — involution just as 
the operation of multiple addition of equal terms leads to the 
operation of multiplication. 

The operation of involution, defined as multiple multiplica- 
tion, requires the exponent h to be an integer number; h is the 
number of factors. 

Thus 4-^ has no immediate meaning; it would by definition 
be 4 multiplied ( —3) times with itself. 

Dividing continuously by 4, we get, 4^-^4 = 4^; 45-^4 = 44; 
44^4 = 43. g^(>^ 2,nd if this .successive division by 4 is carried 
still further, we get the following series: 


43 4X4X4 
4" 4 

= 4X4 

= 42 

42 4X4 
4~ 4 

= 4 

= 41 

41 4 
4 "4 

= 1 

= 40 

40 1 
4 ~4 

1 
~4 

= 4-1 

4 4 -^ 

1 
"4X4 

= 4-2 = i 
^ 42 

4-^1 .4 
4 42 • 

1 

4 43' 

4X4X4 

or, in general, 

10 ENGINEERING MATHEMATICS. 

ThuS; powers with negative exponents, as a-^, are the 
reciprocals of the same powers with positive exponents : -^. 
*j. From the definition of involution then follows, 

because a^ means the product of h equal factors a, and oT' the 
product of n equal factors a, and a^Xa^ thus is a product hav- 
ing h+n equal factors a. For instance, 

43X42 = (4X4X4)X(4X4)=45. 

The question now arises, whether by multiple involution 
we can reach any further mathematical operation. For instance, 

(43)2 = ?, 

may be written, 

(43)2^43x43 

= (4X4X4)X(4X4X4); 
= 46; 
and in the same manner, 

that is, a power a^ is raised to the n*^ power, by multiplying 
its exponent. Thus also, 

that is, the order of involution is immaterial. 

Therefore, multiple involution leads to no further algebraic 
operations. 

8. 43 = 64; 

that is, the product of 3 equal factors 4, gives 64. 

Inversely, the problem may be, to resolve 64 into a product 
of 3 equal factors. Each of the factors then will be 4. This 
reverse operation of involution is called evolution, and is written 
thus, 

^^^ = 4; 
or, more general, 

^/c==a. 


THE GENERAL NUMBER. 11 

^c thus is defined as that number a, which, raised to the power 
h, gives c; or, in other words, 

Involution thus far was defined only for integer positive 
and negative exponents, and the question arises, whether powers 

with fractional exponents, as ct> ov. c^, have any meaning. 
Writing, 

ii;V ^4 1 

1 
it is seen that c^ is that number, which raised to the power 6, 

gives c; that is, c^ is Vc, and the operation of evolution thus 
can be expressed as involution with fractional exponent, 

c~b = \/^ 
and 

n / 1\ » / h/—\ n 

or, 

- - b — 

Cb = (cn)& =\/c^^ 

and 

1 n b 

c^ ■ 
Irrational Numbers. 

Q. Involution with integer exponents, as 4^ = 64, can always 
be carried out. In many cases, evolution can also be carried 
out. For instance, 

'^^64 = 4, 

while, in other cases, it cannot be carried out. For instance, 

-n/2 = ?. 


Obviously then, 


12 ENGINEERING MATHEMATICS. 

Attempting to calculate a/2, we get, 

^2 = 1.4142135..., 

and find, no matter how far we carry the calculation, we never 
come to an end, but get an endless decimal fraction; that is, 
no number exists in our system of numbers, which can express 
^, but we can only approximate it, and carry the approxima- 
tion to any desired degree; some such numbers, as n, have been 
calculated up to several hundred decimals. 

Such numbers as a/2, which cannot be expressed in any 
finite form, but merely approximated, are called irrational 
numbers. The name is just as wrong as the name negative 
number, or imaginary number. There is nothing irrational 
about -^2. If we draw a square, with 1 foot as side, the length 
of the diagonal is ^j2 feet, and the length of the diagonal of 
a square obviously is just as rational as the length of the sides. 

Irrational numbers thus are those real and existing numbers, 
which cannot be expressed by an integer, or a fraction or finite 
decimal fraction, but give an endless decimal fraction, which 
does not repeat. 

Endless decimal fractions frequently are met when express- 
ing common fractions as decimals. These decimal representa- 
tions of common fractions, however, are periodic decimals, 
that is, the numerical values periodically repeat, and in this 
respect are different from the irrational number, and can, due 
to their periodic nature, be converted into a finite common 
fraction. For instance, 2.1387387. ... 
Let 

x= 2.1387387....; 
then, 

subtracting, 

Hence, 


1000a: = 2138.7387387.. 
999^ = 2136.6 


2136.6 21366 1187 77 
^~ 999 ~ 9990 ~ 555 "555* 


THE GENERAL NUMBER. 13 

Quadrature Numbers. 
lo. The following equation, 

^"+4 = (+2), 
may be written, since, 

(+2)X(+2)==(+4); 
but also the equation, 

^"4 = (-2), 
may be written, since 

(-2)>^-2) = (+4). 

Therefore, -V + 4 has two values, (+2) and (-2), and in 
evolution we thus first strike the interesting feature, that one 
and the same operation, with the same numerical values, gives 
several different results.' 

Since all the positive and negative numbers are used up 
as the square roots of positive numbers, the question arises, 
What is the square root of a negative number? For instance, 
'tf — 4 cannot be —2, as —2 squared gives -{*4, nor can it be +2. 

€^== iliX{-l)= ±2^~^, and the question thus re- 
solves itself into: What is aT^I? 

We have derived the absolute numbers from experience, 
for instance, by measuring distances on a line Fig. 4, from a 
starting point A. 

-5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 

1 1 1 ® 1 (p I 0 I 1 1 

CAB 

Fig. 4. Negative and Positive Numbers. 

Then we have seen that we get the same distance from A, 
twice, once toward the right, once toward the left, and this 
has led to the subdivision of the numbers into positive and 
negative numbers. Choosing the positive toward the right, 
in Fig. 4, the negative number would be toward the left (or 
inversely, choosing the positive toward the left, would give 
the negative toward the right). 

If then we take a number, as +2, which represents a dis- 
tance AB, and multiply by (—1)^ we get the distance AC= —2 


14 


ENGINEERING MATHEMATICS. 


in opposite direction from A, Inversely, if we take AC= -2, 
and multiply by (-1), we get AB=+2; that is, multiplica- 
tion by (-1) reverses the direction, turns it through 180 deg. 
If we multiply +2 by V^l~ we get +2V^, a quantity 
of which we do not yet know the meaning. Multiplying once 
more by V-l, we' get +2 X v^^X V~l= -2; that is, 
multiplying a number +2, twice by V-1, gives a rotation of 
180 deg., and multipHcation by V~^ thus means rotation by 
half of 180 deg.; or, by 90 deg., and +2\/^ thus is the dis- 


fT- 


OD 


-^ 


.90° 


■e- 


FiG. 5. 


tance in the direction rotated 90 deg. from +2, or in quadrature 
direction AD in Fig. 5, and such numbers as +2\/-l thus 
are quadrature numbers, that is, represent direction not toward 
the right, as the positive, nor toward the left, as the negative 
numbers, but upward or downward. 

For convenience of writing, V— 1 is usually denoted by 
the letter j. 

II. Just as the operation of subtraction introduced in the 
negative numbers a new kind of numbers, having a direction 
180 deg. different, that is, in opposition to the positive num- 
bers, so the operation of evolution introduces in the quadrature 
number, as 2f, a new kind of number, having a direction 90 deg. 


THE GENERAL NUMBER. 


15 


different; that is, at right angles to the positive and the negative 
numbers, as illustrated in Fig. 6. 

As seen, mathematically the quadrature number is just as 
real as thc^ negative, physically sometimes the negative number 
has a meaning — if two opposite directions exist — ; sometimes it 
has no meaning — ^where one direction only exists. Thus also 
the quadrature number sometimes has a physical meaning, in 
those cases where four directions exist, and has no meaning, 
in those physical problems where only two directions exist. 


+4/ 


•4 -3 


1 0 


h 

+ 1 


+3 


Fig. 6. 


For instance, in problems dealing with plain geometry, as in 
electrical engineering w^hen discussing alternating current 
vectors in the plane, the quadrature numbers represent the 
vertical, the ordinary numbers the horizontal direction, and then 
the one horizontal direction is positive, the other negative, and 
in the same manner the one vertical direction is positive, the 
other negative. Usually positive is chosen to the right and 
upward, negative to the left and downward, as indicated in 
Fig. 6. In other problems, as when dealing with time, which 
has only two directions, past and future, the quadrature num- 
bers are not applicable, but only the positive and negative 


16 


ENGINEERING MATHEMATICS. 


numbers. In still other problems, as when dealing with illumi- 
nation, or with individuals, the negative numbers are not 
applicable, but only the absolute or positive numbers. 

Just as multipUcation by the negative unit (—1) means 
rotation by 180 deg., or reverse of diree<tion, so multiplication 
by the quadrature unit, j, means rotation by 90 deg., or change 
from the horizontal to the vertical direction, and inversely. 

General Numbers. 

12. By the positive and negative numbers, all the points of 
a line could be represented numerically as distances from a 
chosen point A. 

y 


qP 


Fig. 7. Simple Vector Diagram. 


By the addition of the quadrature numbers, all points of 
the entire plane can now be represented as distances from 
chosen coordinate axes x and y, that is, any point P of the 
plane, Fig. 7, has a horizontal distance, 0B= +3, and a 
vertical distance, jBP= +2f, and therefore is given by a 
combination of the distances, 05= +3 and BP=- +2/. For 
convenience, the act of combining two such distances in quad- 
rature with each other can be expressed by the plus si^n, 
and the result of combination thereby expressed by OB^-BP 
= 3+2j. 


THE GENERAL NUMBER. 


17 


Such a combination of an ordinary number and a quadra- 
ture number is called a general number or a complex quantity. 

The quadrature number jh thus enormously extends the 
field of usefulness of algebra, by affording a numerical repre- 
sentation of two-dimensional systems, as the plane, by the 
general number a-\-jh. They are especially useful and impor- 
tant in electrical engineering, as mdst problems of alternating 
currents lead to vector representations in the plane, and there- 
fore can be represented by the general number a+jh; that is, 
the combination of the ordinary number or horizontal distance 
a, and the quadrature number or vertical distance jh. 


®Px 


p; 


Fig. 8. Vector Diagram. 

Analytically, points in the plane are represented by their 
two coordinates: the horizontal coordinate, or abscissa x, and 
the vertical coordinate, or ordinate y. Algebraically, in the 
general number a+jh both coordinates are combined, a being 
the X coordinate, jh the y coordinate. 

Thus in Fig. 8, coordinates of the points are. 

Pi: a:=+3, y= +2 P2: x= +S y= -2, 

Ps' x=-S, y=+2 P4: x=-S y=-2] 

and the points are located in the plane by the numbers: 
• Pi=3+2/ P2 = 3-2/ P3=-3+2/ P4=-3-2y 


18 ENGINEERING MATHEMATICS. 

13. Since already the square root of negative numbers has 
extended the system of numbers by giving the quadrature 
number, the question arises whether still further extensions 
of the system of numbers would result from higher roots of 
negative quantities. 

For instance, 

The meaning of a'^I we find in the same manner as that 
of ■€^. 

A positive number a may be represented on the horizontal 
axis as P^ 

Multiplying a by -^-1 gives a-^— 1, whose meaning we do 
not yet know. Multiplying again and again by 'C' — 1, we get, after 
four multiphcations, a(^— 1)^= —a; that is, in four steps we 

have been carried from a to —a, a rotation of 180 deg., and 

_ 10Q 

■>[^1 thus means a rotation of —— = 45 deg., therefore, a->i—l 

is the point Pi in Fig. 9, at distance a from the coordinate 
center, and under angle 45 deg., which has the coordinates, 

x= — = and 2/ = — =/; or, is represented by the general nuniber, 

\/2 

-yl —I, however, may also mean a rotation by 135 deg. to P2, 
since this, repeated four times, gives 4x135 = 540 deg., 
or the same as 180 deg., or it may mean a rotation by 225 deg. 
or by 315 deg. Thus four points exist, which represent a-yj —1; 
the points: 

Pi = — -^a, P2 = — 7-- a, 

v2 v2 

Therefore, ^—1 is still a general number, consisting of an 
ordinary and a quadrature number, and thus does not extend 
our system of numbers any further. 


THE GENERAL NUMBER. 


19 


In the same manner, ^^ + 1 can be found; it is that number, 
which, multiplied n times with itself, gives +1. Thus it repre- 

sents a rotation by — deg., or any multiple thereof; that is, 
III 

,. . . 360 , ■ . . 360 

the X coordmate is cos qX — , the y coordmate sm ^X — , 

n n 

and, 

n^ 360 . . 360 

V +l=cos qX + 7 sm qX — , 


where q is any integer number. 


Fig. 9. Vector Diagram a^ — 1. 


There are therefore n different valuesof av^ + 1, which lie 
equidistant on a circle with radius 1, as shown for n = 9 in 
Fig. 10. 

14. In the operation of addition, a + 6 = c, the problem is, 
a and 6 being given, to find c. 

The terms of addition, a and 6, are interchangeable, or 
equivalent, thus: a + h = h-^a, and addition therefore has only 
one reverse operation, subtraction; c and h being given, a is 
found, thus; a = c—b, and c and a being given, b is found, thus: 
b=c—a. Either leads to the same operation — subtraction. 

The same is the case in multipHcation; aXb = c. The 


20 


ENGINEERING MATHEMATICS. 


factors a and h are interchangeable or equivalent; aXh = hXa 
and the reverse operation, division, a = j- is the same as h=-. 

In involution, however, a^ = c, the two numbers a and h 
are not interchangeable, and a* is not equal to h^. For instance 
43 = 64 and 3^ = 81. 

Therefore, involution has two reverse operations: 

(a) c and b given, a to be found. 


or evolution, 


a = v^c ; 


Fig. 10. Points Determined by v^ — 1. 
(h) c and a given, h to be found, 

6=logac; 
or, logarithmation. 

Logarithmation. 

15. Logarithmation thus is one of the reverse operations 
of involution, and the logarithm is the exponent of involution. 

Thus a logarithmic expression may be changed to an ex- 
ponential, and inversely, and the laws of logarithmation are 
the laws, which the exponents obey in involution. 

1. Powers of equal base are multiplied by adding the 
exponents: a^Xa^^a^"*"". Therefore, the logarithm of a 


THE GENERAL NUMBER. 21 

product is the sum of the logarithms of the factors, thus loga cXd 

= l0ga C +\0ga d. 

2. A power is raised to a power by multiplying the exponents: 

Therefore the logarithm of z, power is the exponent times 
the logarithm of the base, or, the number under the logarithm 
is raised to the power n, by multiplying the logarithm by n: 

loga C"=n loga C, 

loga 1=0, because aP = l. If the base a > 1, loga c is positive, 
if c>l, and is negative, if c<l, but >0. The reverse is the 
case, if a<l. Thus, the logarithm traverses all positive and 
negative values for the positive values of c, and the logarithm 
q| 4J|iegative number thus can be neither positive nor negative. 

loga (-c)=loga c+loga (" 1), and the question of finding 
the logarithms of negative numbers thus resolves itself into 
finding the value of loga ( — 1)- 

There are two standard systems of logarithms one with 
the base £ = 2.71828. . .*, and the other with the base 10 is 
used, the former in algebraic, the latter in numerical calcula- 
tions. Logarithms of any base a can easily be reduced to any 
other base. 

For instance, to reduce 6 = loga c to the base 10: b = \ogaC 
means, in the form of involution : a^ = c. Taking the logarithm 
hereof gives, b logio a=logio c, hence, 

, logioc logio c 

o=T ; or loga c = -. . 

logio a logio a 

Thus, regarding the logarithms of negative numbers, we need 
to consider only logio ( — 1) or log, ( — 1). 

If jx = \og, (-1), then £^'^= -1, 

and since, as will be seen in Chapter II, 

£^'' = cos x + j sin Xf 


it follows that, 


cos x+y sin X = ~1, 


* Regarding e, see Chapter II, p. 71. 


ENGINEERING MATHEMATICS. 

x = 7t, or an odd multiple thereof, and 
log£(-l)=/7r(2n + l), 

where n is any integer number. 

Thus logarithmation also leads to the quadrature number 
j, but to no further extension of the system of numbers. 

Quaternions. 

i6. Addition and subtraction, multiplication and division, 
involution and evolution and logarithmation thus represent all 
the algebraic operations, and the system of numbers in which 
all these operations can be carried out under all condi^ons 
is that of the general number, a+j6, comprising the ordma?y 
number a and the quadrature number jh. The number a as 
well as h may be positive or negative, may be integer, fraction 
or irrational. 

Since by the introduction of the quadrature number jh, 
the application of the system of numbers was extended from the 
line, or more general, one-dimensional quantity, to the plane, 
or the two-dimensional quantity, the question arises, whether 
the system of numbers could be still further extended, into 
three dimensions, so as to represent space geometry. While 
in electrical engineering most problems lead only to plain 
figures, vector diagrams in the plane, occasionally space figures 
would be advantageous if they could be expressed algebra- 
ically. Especially in mechanics this would be of importance 
when dealing with forces, as vectors in space. 

In the quaternion calculus methods have been devised to 
deal with space problems. The quaternion calculus, however, 
has not yet found an engineering appHcation comparable with 
that of the general number, or, as it is frequently called, the 
complex quantity. The reason is that the quaternion is not 
an algebraic quantity, and the laws of algebra do not uniformly 
apply to it. 

17. With the rectangular coordinate system in the plane. 
Fig. 11, the X axis may represent the ordinary numbers, the y 
axis the quadrature numbers, and multipUcation by j=V-l 
represents rotation by 90 deg. For instance, if Pi is a point 


THE GENERAL NUMBER. 


23 


a+y6 = 3 4-2?s, the point P2, 90 deg. away from Pi, would 
be: 

P2 = jPi-Ka+jb)=j(3-V2j)= -2+3/, 

To extend into space, we have to add the third or z axis, 
as shown in perspective in Fig. 12. Rotation in the plane xy, 
by 90 deg., in the direction +x to +y, then means multiphca- 
tion by j. In the same manner, rotation in the yz plane, by 
90 deg., from -\-y to +z, would be represented by multiplica- 


Fig. 11. Vectors in a Plane. 

tion with h, and rotation by 90 deg. in the zx plane, from +z 
to +x would be presented by k, as indicated in Fig. 12. 

All three of these rotors, j, h, k, would be V — 1, since each, 
apphed twice, reverses the direction, that is, represents multi- 
plication by (—1). 

As seen in Fig. 12, starting from +x, and going to +y, 
then to +z, and then to +x, means successive multiplication 
by y, h and k, and since we come back to the starting point, the 
total operation produces no change, that is, represents mul- 
/tiplication by ( + 1). Hence, it must be, 

jhk= +1. 


24 


ENGINEERING MATHEMATICS. 


Algebraically this is not possible, since each of the three quan- 
tities is V-1, and V -ixV -IxV -1= -V^, and not 
( + 1). 


+z 


>^x 


Fig. 12. Vectors in Space, jhk=-\-l^ 

If we now proceed again from x, in positive rotation, but 
first turn in the xz plane, we reach by multiplication with k 
the negative z axis, —z, as seen in Fig. 13. Further multiplica- 

+y 


—a; 


+z' 


■>+« 


Fig. 13. Vectors in Space, khj= —1. 

tion by h brings us to +y, and multiplication by j to —x, and 
in this case the result of the three successive rotations by 


THE GENERAL NUMBER. 25 

90 deg., in the same direction as in Fig. 12, but in a different 
order, is a reverse; that is, represents (-1). Therefore, 

khj= -1, 
and hence, 

jhk= —khj. 

Thus, in vector analysis of space, we see that the fundamental 
law of algebra, 

aXh = bXa, 

does not apply, and the order of the factors of a product is 
not immaterial, but by changing the order of the factors of the 
product jhk, its sign was reversed. Thus common factors 
cannot be canceled as in algebra; for instance, if in the correct 
expression, jhk = khj, we should cancel by j, h and k, as could be 
done in algebra, we would get + 1 = —1, which is obviously wrong. 
For this reason all the mechanisms devised for vector analysis 
in space have proven more difficult in their application, and 
have not yet been used to any great extent in engineering 
practice. 

B. ALGEBRA OF THE GENERAL NUMBER, OR COMPLEX 

QUANTITY. 

Rectangular and Polar Coordinates. 

i8. The general number, or complex quantity, a+jb, is 
the most general expression to which the laws of algebra apply. 
It therefore can be handled in the same manner and under 
the same rules as the ordinary number of elementary arithmetic. 
The only feature which must be kept in mind is that f = — 1, and 
where in multiplication or other operations f occurs, it is re- 
placed by its value, — 1. Thus, for instance, 

(a + jh) (c + jd) =ac+ jad + jhc + fhd 
= ac+ jad + jbc — bd 
= (ac —bd) -\-j{ad-{-bc). 

Herefrom it follows that all the higher powers of j can be 
eliminated, thus: 

f =/, f= -1, f = -], f- +1; 
f = +y, f= -1, f= -/,, f = +i; 
f= +]', . . . etc. 


ENGINEERING MATHEMATICS. 

In distinction from the general number or complex quantity, 
th^oriinary numbers, • + « and -a, are occasionally called 

varsf, or real numbers. The general number thus consists 
theicombination of a scalar or real number and a quadrature 
number, or imaginary number. 

Since a quadrature number cannot be equal to an ordinary 
nun|bj^ it follows that, if two general numbers are equal, 
their /^eal components or ordinary numbers, as well as their 
qua(|ra|urenumbers or imaginary components must be equal, 

a-\-jh = c-\-jd, 
then, 

a = c and h = d. 

Every equation with general numbers thus can be resolved 
into two equations, one containing only the ordinary numbers, 
the other only the quadrature numbers. For instance, if 

x+jy = 5-Sj, 
then, 

x = 5 and y= —3. 

19. The best way of getting a conception of the general 
number, and the algebraic operations with it, is to consider 
the general number as representing a point in the plane. Thus 
the general number a+jh = 6+2.5j may be considered as 
representing a point P, in Fig. 14, which has the horizontal 
distance from the y axis, 0A = BP = a^6, and the vertical 
distance from the x axis, OB = AP = b=2.5. 

The total distance of the point P from the coordinate center 
0 then is 


= \/52+62 = 6.5, Z^yft^/^^ 

and the angle, which this distance OP makes with the x axis, 
is given by 


^ ^ AP 2.5 JJ .,....,-.^/" 
tan <? = :== =-77- fl/)^' 


OA 6 /^^ 


h 

- = 0.417. 
a 


i'X 


THE GENERAL NUMBER, 


27 


Instead of representing the general number by the two 
components, a and h, in the form a+jb, it can also be repre- 
sented by the two quantities: 

The distance of the point F from the center 0, 


and the angle between this distance and the x axis, 


tan 6=-. 
a 


H — I — h 


Fid. 14. Rectangular and Polar Coordinates. 

Then referring to Fig. 14, 

a = c con 0 and 6 = csin^, 

and the general number a+jh thus can also be written in the 
form, 

c(cos ^+y sin 6). 

The form a+jb expresses the general number by its 
rectangular components a and b, and corresponds to the rect- 
angular coordinates of analytic geometry; a is the x coordinate, 
b the y coordinate. 

The form c(cos^+y sin ^) expresses the general number by 
what may be called its polar components, the radius c and the 


28 ENGINEERING MATHEMATICS. 

angle 6, and corresponds to the polar coordinates of analytic 
geometry, c is frequently called the radius vector or scalar, 
0 the phase angle of the general number. 

While usually the rectangular form a-\-jb is more con- 
venient, sometimes the polar form c(cos d +j sin 6) is preferable, 
and transformation from one form to the otherA^'therefore fre- 
quently applied. 

Addition and Subtraction. 

20. If ai+y6i=6+2.5 J is represented by the point Pi; 
this point is reached by going the horizontal distance ai = 6 
and the vertical distance 6i=2.5. If a2+jb2 = S+4:j is repre- 
sented by th^ point P2, this point is reached by going the 
horizontal distance a2 = S and the vertical distance 62 = 4. 

The sum of the two general numbers (ai +/61) + (a2 +J&2) = 
(6 +2.5/) + (3 +4/), then is given by point Pq, which is reached 
by going a horizontal distance equal to the sum of the hor- 
izontal distances of Pi and P2: ao = ai+a2 = 6+3 = 9, and a 
vertical distance equal to the sum of the vertical distances of 
Pi and P2: 6o = &i+&2 = 2.5+4 = 6.5, hence, is given by the 
general number 

ao +jho = {ai +a2) +j(bi +62) 
= 9+6.5/. 

Geometrically, point Po is derived from points Pi and P2, 
by the diagonal OPq of the parallelogram OP1P0P2, constructed 
with OPi and OP2 as sides, as seen in Fig. 15. 

Herefrom it follows that addition of general numbers 
represents geometrical combination by the parallelogram law. 

Inversely, if Po represents the number 

ao+/&o=-9 + ().5/, 

and Pi represents the number 

ai+/6i = 6+2.5/, 

the difference of these numbers will be represented by a point 
P2, which is reached by going the difference of the horizontal 


THE GENERAL NUMBER. 


29 


distances and of the vertical distances of the points Po and 
Pi. P2 thus is represented by 


a2 = "ao— ai=9— 6=3, 


and 


b2 = bo-bi=6.o -2.5 = 4. 

Therefore, the difference of the two general numbers (ao+jho) 
and (ai +jbi) is given by the general number: 

a2 +jb2 = (ao -di) +j(bo -61) 
; =3+4i, 

as seen in Fig. 15. 


H 1 1 1 h 


Fig. 15. Addition and Subtraction of Vectors. 

This difference a2+j&2 is represented by one side OP 2 of 
the parallelogram OP1P0P2, which has OPi as the other side, 
and OPq as the diagonal. 

Subtraction of general numbers thus geometrically represents 
the resolution of a vector OPq into two components OPi and 
OP 2, by the parallelogram law. 

Herein lies the main advantage of the use of the general 
number in engineering calculation : If the vectors are represented 
by general numbers (complex quantities), conibination and 
resolution of vectors by the parallelogram law is carried out by 


30 ENGINEERING MATHEMATICS. 

simple addition or subtraction of their general numerical values, 
that is, by the simplest operation of algebra. 

21. General numbers are usually denoted by capitals, and 
their rectangular components, the ordinary number and the 
quadrature number, by small letters, thus : 

A = ai+ja2; 

the distance of the "point which represents the general number A 
from the coordinate center is called the absolute value, radius 
or scalar of the general number or complex quantity. It is 
the vector a in the polar representation of the general number: 

^ = a(cos 0 +j sin d), 


and is given by a = Var^ + a^r- 

The absolute value, or scalar, of the general number is usually 
also denoted by small letters, but sometimes by capitals, and 
in the latter case it is distinguished froni the general number by 
using a different type for the latter, or underlining or dotting 
it, thus: 

A=-ai+ja2', or A=ai+ja2; 

or A = ai+ja2; or A = ai+ja2 


a = Vai^ + a2^ ; or A = Var^ + a'/, 

and ai + ja2 = a (cos 6 + j sin d) ; 

or ai -\-ja2 = A(cos 0+j sin 6). 

22. The absolute value, or scalar, of a general number is 
always an absolute number, or positive, that is, the sign of the 
rectangular component is represented in the angle 0. Thus 
referring to Fig. 16, 

A =ai +/a2 = 4 4-3y; 


gives, a = v ai^ + a2^ = 5 ; 

tan ^ = } =0.75; 
6' = 37deg.; 
and A ==5 (cos 37 deg. H-j sin 37 deg). 


THE GENERAL NUMBER. 


31 


The expression 
gives 


A=ai-\-ja2 = 4:—Sj 


tan ^=-- = -0.75; 


d = -37 deg. ; or = 180 -37 = 143 deg. 


-AA-^ 


Fig. 16. Reptresentation of General Numbers. 

Which of the two values of 6 is the correct one is seen from 
the condition ai=acos^. As ai is positive, +4, it follows 
that cos 6 must be positive; cos (—37 deg.) is positive, cos 143 
deg. is negative; hence the former value is correct : 

^ -5(cos( -37 deg.) +/ sin( -37 deg.)} 
= 5(cos 37 deg. — ; sin 37 deg.). 

Two such general numbers as (4+3/) and (4-3/), or, 
in general, 

(a+jh) and (a—jh), 

are called conjugate numbers. Their product is an ordinary 
and not a general number, thus : (a+jb) (a —jb) = a^ + b^. 


32 ENGINEERING MATHEMATICS. 

The expression 


A = ai+ja2= -4+3y 


gives 


tan<9=--= -0.75; 
4 

d = -37 deg. or = 180 -37 = 143 cleg. ; 

but since ai = a cos 6 is negative, ~4, cos 6 must be negative, 
hence, 6 = 143 deg. is the correct vakie, and 

^=5(cos 143 deg. +jsin 143 deg.) 
= 5 (—cos 37 deg. +/ sin 37 deg.) 

The expression 

A = ai+ja2= -4:-3j 

gives 


^ = 37 deg.; or =180+37 = 217 deg.; 

but since ai=a cos 6 is negative, —4, cos 6 must be negative, 
hence ^ = 217 deg. is the correct value, and, 

4 = 5 (cos 217 deg. +j sin 217 deg.) 
= 5( — cos 37 deg. —j sin 37 deg.) 

The four general numbers, +4+3j, +4-3/, -4+3/, and 
—4 —3j, have the same absolute value, 5, and in their repre- 
sentations as points in a plane have symmetrical locations in 
the four quadrants, as shown in Fig. 16. 

As the general number A = ai-{-ja2 finds its main use in 
representing vectors in the plane, it very frequently is called 
a vector quantity, and the algebra of the general number is 
spoken of as vector analysis. 

Since the general numbers A=ai-\-ja2 can be made to 
represent the points of a plane, they also may be called plane 
numbers, while the positive and negative numbers, +a and —a, 


THE GENERAL NUMBER. 33 

may be called the linear numbers, as they represent the points 
of a line. 

Example: Steam Path in a Turbine. 

23. As an example of a simple operation with general num- 
bers one may calculate the steam path in a two-wheel stage 
of an impulse steam turbine. 


+2^ 


1 


«)))»»)) 

« »)>M») 


> +a; 


Fig. 17. Path of Steam in a Two-wheel Stage of an Impulse Turbine. 

Let Fig. 17 represent diagrammatically a tangential section 
through the bucket rings of the turbine wheels. Wi and W2 
are the two revolving wheels, moving in the direction indicated 
by the arrows, with the velocity s = 400 feet per sec. / are 
the stationary intermediate buckets, which turn the exhaust 
steam from the first bucket wheel TFi, back into the direction 
required to impinge on the second bucket wheel TF2. The 
steam jet issues from the expansion nozzle at the speed So = 2200 


34 


ENGINEERING MATHEMATICS. 


feet per sec, and under the angle ^o = 20 deg., against the first 
bucket wheel Wi. 

The exhaust angles of the three successive rows of buckets, 
Wi, I, and W2, are respectively 24 deg., 30 deg. and 45 deg. 
These angles are calculated from the section of the bucket 
exit required to pass the steam at its momentary velocity, 
and from the height of the passage required to give no steam 
eddies, in a manner which is of no interest here. 

As friction coefficient in the bucket passages may be assumed 
Ay = 0.12; that is, the exit velocity is 1-/^=0.88 of the entrance 
velocity of the steam in the buckets. 


•M^ 


> + x 


FiiJ. IS. Vector Diagram of Velocities of Steam in Turbine. 

Choosing then as x-axis the direction of the tangential 
velocity of the turbine wheels, as 2/-axis the axial direction, 
the velocity of the steam supply from the expansion nozzle is 
represented in Fig. 18 by a vector OSo of length So = 2200 feet 
per sec, making an angle ^o = 20 deg. with the x-axis; hence, 
can be expressed by the general number or vector quantity: 

^0 = So (cos do +j sin ^0) 

= 2200 (cos 20 deg. +; sin 20 deg.) 
= 2070 + 750yft. per sec 

The velocity of the turbine wheel "PTi is s = 400 feet per second, 
and represented in Fig. 18 by the vector OS, in horizontal 

direction. 


THE GENERAL NUMBER. 35 

The relative velocity with which the steam enters the bucket 
passage of the first turbine wheel Wi thus is : 

S,=So-s 

= (2070 +750y) -400 
= 1670 + 7^/ ft. per sec. 

This vector is shown as OSi in Fig. 18. 
The angle di, under which the steam enters th^ bucket 
passage thus is given by 

750 
isi,n 6i=j^ = 0A50, as ^i = 24.3deg. 

This angle thus has to be given to the front edge of the 
buckets of the turbine wheel Wi. 

The absolute value of the relative velocity of steam jet 
and turbine wheel Wi, at the entrance into the bucket passage, 
is 


si = \/l6702 + 7502 = 1830 ft. per sec. 

In traversing the bucket passages the steam velocity de- 
creases by friction etc., from the entrance value Si to the 
exit value 

S2 = si(l-/b^) = 1830X0.88 = 1610 ft. per sec, 

and since the exit angle of the bucket passage has been chosen 
as ^2 ==24 deg., the relative velocity with which the steam 
leaves the first bucket wheel Wi is represented by a vector 
OS^ in Fig. 18, of length §2 = 1610, under angle 24 deg. The 
steam leaves the first wheel in backward direction, as seen in 
Fig. 17, and 24 deg. thus is the angle between the steam jet 
and the negative x-axis; hence, ^2 = 180-24 = 156 deg. is the 
vector angle. The relative steam velocity at the exit from 
wheel Wi can thus be represented by the vector quantity 

♦^2 = S2(cos d2 +j sin dz) 

= 1610 (cos 156 deg. +/ sin 156 deg.) 
= -1470 + 655/. 

Since the velocity of the turbine wheel Wi is s = 400, the 
velocity of the steam in space, after leaving the first turbine 


36 ENGINEERING MATHEMATICS. 

wheel, that is, the velocity with which the steam enters the 
intermediate /, is 

= fi470+655y)+400 
= -1070 + 655y, 

and is represented by vector OSs in Fig. 18. 
The direction of this steam jet is given by 

tan ^3= -^^^=-0.613, 

as 

<?3= -31.6 deg.; or, 180-31.6 = 148.4 deg. 

The latter value is correct, as cos ^3 is negative, and sin 6^ is 
positive. 

The steam jet thus enters the intermediate under the angle 
of 148.4 deg.; that is, the angle 180 —148.4 = 31.6 deg. in opposite 
direction. The buckets of the intermediate / thus must be 
curved in reverse direction to those of the wheel Wi, and must 
be given the angle 31.6 deg. at their front edge. 

The absolute value of the entrance velocity into the inter- 
mediate / is 


S3 = Vl0702 + 6552 = 1255 ft. per sec. 

In passing through the bucket passages, this velocity de- 
creases by friction, to the value : 

S4 = S3(1 -A;/) = 1255X0.88 = 1105 ft. per sec, 

and since the exit edge of the intermediate is given the angle: 
^4 = 30 cleg., the exit velocity of the steam from the intermediate 
is represented by the vector OS 4: in Fig. 18, of length 84 = 1105, 
and angle ^4 = 30 deg., hence, 

/S4 = 1105 (cos 30 deg. +y sin 30 deg.) 
= 955 +550y ft. per sec. 

This is the velocity with which the steam jet impinges 
on the second turbine wheel W2, and as this wheel revolves 


THE GENERAL NUMBER. 37 

with velocity s = 400, the relative velocity, that is, the velocity 
with which the steam enters the bucket passages of wheel W2, is, 

' =(955+550/) -400 
= 555+550/ ft. per sec; 

and is represented by vector OS5 in Fig. 18. 
The direction of this steam jet is given by 

550 
tan 05=^ = 0.990, as 65 = 44.8 deg. 

Therefore, the entrance edge of the buckets of the second 
wheel W2 must be shaped under angle ^5 = 44.8 deg. 
The absolute value of the entrance velocity is 


S5 = ^5552 + 5502 = 780 ft. per sec. 

In traversing the bucket passages, the velocity drops from 
the entrance value ^5, to the exit valve, 

S6 = S5(1 -A;^) =780X0.88 = 690 ft. per sec. 

Since the exit angles of the buckets of wheel W2 has been 
chosen as 45 deg., and the exit is in backward direction, de = 
180-45=135 deg., the steam jet velocity at the exit of the 
bucket passages of the last wheel is given by the general number 

S6 = sq{cos Oq +/ sin Oq) 

= 690 (cos 135 deg. +/ sin 135 deg.) 

= -487+487/ ft. per sec, 

and represented by vector OSq in Fig. 18. 

Since s = 400 is the wheel velocity, the velocity of the 
steam after leaving the last wheel W2, that is, the^'losf 
or " rejected " velocity, is 

= (487 +487/) +400 

= -87+487/ ft. per sec, 

and is represented by vector OS7 in Fig. 18. 


38 


ENGINEERING MATHEMATICS. 


The direction of the exhaust steam is given by, 

487 
tan ^7= —37-= -5.0; as ^7 = 180-80 = 100 deg., 

and the absolute velocity is. 


S7 = V872+ 4872 = 495 f^. per sec. 

Multiplication of General Numbers. 

24. If A = ai+ja2 and B = hi-\-jb2, are two general, or 
plane numbers, their product is given by multiplication, thus : 

AB = {ai+ja2)(hi+jh2) 

= aihi +jaib2 +ja2hi +pa2b2, 
and since p= —1, 

AB = {aihi -a2?>2) +j{aih2 +^2*^1), 

and the product can also be represented in the plane, by a point, 

C = Ci+jC2, 

where, 

ci =0'ihi — a2&27 
and 

C2 = ai&2+a2^i. 

For instance, A=2+j multiplied by B=^l+1.5 j gives 

ci=2Xl-lXl.5 = 0.5, 
C2 = 2Xl.5 + lXl=4; 
hence, 

0 = 0.5+47, 

as shown in Fig. 19. 

25. The geometrical relation between the factors A and B 
and the product C is better shown by using the polar expression; 
hence, substituting, 


ai=a cos a 
a2 = a sin a 


, 61=6 cos/? 
■ and , r • n 
02 = 0 sm 


which gives 


a = Vai^+a2^ 


h = VbJ+b2^ 


(l2 

tan a = — 
ai 


and 


&2 

tan B = r~ 


THE GENERAL NUMBER. 


39 


the quantities may be written thus : 

A=a(cos a:+/sin a); 
5 = &(cos/9+ysin/?), 
and then, 

C = AB = ah (cos a+j sin a) (cos /?+ / sin /5) 
= ab \ (cos a cos /? —sin a sin /?) +j(cos a sin ^ +sin a cos /?)} 
= a6 {cos (a +/?) +j sin (a +/5)} : 


Fig. 19. Multiplication of Vectors. 

that is, two general numbers are multiplied by multiplying their 
absolute values or vectors, a and b, and adding their phase angles 
a and /?. 

Thus, to multiply the vector quantity, A = ai+ja2 = a (cos 
« + y sin^ by ^ = 61 +J62 = & (cos /? +f sin ^) the vector OA in Fig. 
19, which represents the general number A, is increased by the 
factor b = Vbi^ + b2^, and rotated by the angle ft which is given 

bv tan /5 = 7--. 
bi 

Thus, a complex multiplier B turns the direction of the 

multiplicand A, by the phase angle of the multiplier B, and 

increases the absolute value or vector of A, by the absolute 

value of B as factor. 


40 ENGINEERING MATHEMATICS. 

The multiplier B is occasionally called an operator, as it 
carries out the operation of rotating the direction and changing 
the length of the multiplicand. 

26. In multiplication, division and , other algebraic opera- 
tions with the representations of physical quantities (as alter- 
nating currents, voltages, impedances, etc.) by mathematical 
symbols, whether ordinary numbers or general numbers, it 
is necessary to consider whether the result of the algebraic 
operation, for instance, the product of two factors, has a 
physical meaning, and if it has a physical meaning, whether 
this meaning is such that the product can be represented in 
the same diagram as the factors. 

For instance, 3X4 = 12; but 3 horses X 4 horses does not 
give 12 horses, nor 12 horses^, but is physically meaningless. 
However, 3 ft. X4 ft. = 12 sq.ft. Th'us, if the numbers represent 

d) — I — 1 0 (D — I — I — I — I — I — I — I — © — I — • — I 
O A B C 

Fig. 20. 

horses, multiplication has no physical meaning. If they repre- 
sent feet, the product of multiphcation has a physical meaning, 
but a meaning which differs from that of the factors. Thus, 
if on the lin.e in Fig. 20, OA = S feet, OB = 4: feet, the product, 
12 square feet, while it has a physical meaning, cannot be 
represented any more by a point on the same line; it is not 
the point OC = 12, because, if we expressed the distances OA 
and OB in inches, 36 and 48 inches respectively, the product 
would be 36X48 = 1728 sq.in., while the distance OC would be 
144 inches. 

27. In all mathematical operations with physical quantities 
it therefore is necessary to consider at every step of the mathe- 
matical operation, whether it still has a physical meaning, 
and, if graphical representation is resorted to, whether the 
nature of the physical meaning is such as to allow graphical 
representation in the same diagram, or not. 

An instance of this general limitation of the application of 
mathematics to physical quantities occurs in the representation 
of alternating current phenomena by general numbers, or 
complex quantities. 


THE GENERAL NUMBER. 


41 


An alternating current can be represented by a vector 01 
in a polar diagram, Fig. 21, in which one complete revolution 
or 360 deg. represents the time of one complete period of the 
alternating current. This vector 01 can be represented by a 
general number, 

where ii is the horizontal, 12 the vertical component of the 
current vector 01. 


i. 

^l 

1 

■^E 

ij 

''y^ 

0 

2N. 

X 

1 

^f 

Fig. 21.. Current, E.M.F. and Impedance Vector Diagram. 

In the same manner an alternating E.M.F. of the same fre- 
quency can be represented by a vector OE in the same Fig. 21, 
and denoted by a general number, 

E = ei+je2, 

An impedance can be represented by a general number, 

Z = r—jx, 

where r is the resistance and x the reactance. 

If now we have two impedances, OZi and OZ2, Zi =ri —jxi 
and Z2 = r2—jx2, their product Zi Z^ can be formed mathema - 
ically, but it has no physical meaning. 


42 ENGINEERING MATHEMATICS. 

If we have a current and a voltage, I = ii + ji2 and E = ei -\-je2, 
the product of current and voltage is the power P of the alter- 
nating circuit. 

The product of the two general numbers I and E can be 
formed mathematically, IE, and would represent a point C 
in the vector plane Fig. 21. This point C, however, and the 
mathematical expression IE, which represents it, does not give 
the power P of the alternating circuit, since the power P is not 
of the same frequency as / and E, and therefore cannot be 
represented in the same polar diagram Fig. 21, which represents 
7 and E, 

If we have a current / and an impedance Z, in Fig. 21; 
I=ii+ji2 and Z = r—jx, their product is a voltage, and as the 
voltage is of the same frequency as the current, it can be repre- 
sented in the same polar diagram. Fig. 21, and thus is given by 
the mathematical product of / and Z, 

^E = IZ={H+ji2){r-jx), 

= (iir ■^i2X ) + jfer -iix). 

28. Commonly, in the denotation of graphical diagrams by 
general numbers, as the polar diagram of alternating currents, 
those quantities, which are vectors in the polar diagram, as the 
current, voltage, etc., are represented by dotted capitals: E, I, 
while those general numbers, as the impedance, admittance, etc. , 
which appear as operators, that is, as multiphers of one vector, 
for instance the current, to get another vector, the voltage, are 
represented algebraically by capitals without dot: Z = r-jx = 
impedance, etc. 

This Hmitation of calculation with the mathematical repre- 
sentation of physical quantities must constantly be kept in 
mind in all theoretical investigations. 

Division of General Numbers. 

29. The division of two general numbers, A=ai-]-ja2 and 
B = bi+jh2, gives, 

A ^ai+ja2 
'~B~bi+jh2 
This fraction contains the quadrature number in the numer- 
ator as well as in the denominator. The quadrature number 


THE GENERAL NUMBER. 43 

can be eliminated from the denominator by multiplying numer- 
ator and denominator by the conjugate quantity of the denom- 
inator, hi — jh2, which gives: 

(ai +yq2)(^i -jh2) _ (fli^i -\-a2h2) +/(a2&i -ai7>2) 
•~{h+jh2){h-jb2)~ 61^ + 622 


aibi -\-a2h2 . 0,2^1 —a\h2 


for instance, 

i_6+2.5/ 

'~B S+4:j 

_ (6 + 2.5.?) (3-^./) 
(3 + 4y)(3-4y) 
28-16.5/ 
25 
= 1.12-0.GG/. 

If desired, the quadrature number may be eliminated from 
the numerator and left in the denominator by multiplying with 
the conjugate number of the numerator, thus: 

^^A_a]+ja2 


B 61+ J 62 

{a\-^j(i2)(ai-ja2) 

{bi+jb2){ai-ja2) 

ai^+a2^ 


(aibi +a2b2) +j{aib2 —a2bi) 


for instance, 


^^^_6 + 2.5/ 
• B S+^i 

(6 + 2.5/) (6 -2.5/) 
(3 + 4/)(6-2.5/) 
29.75 
■"28 + 16.5/ 

30. Just as in multiplication, the polar representation of 
the general number in division is more perspicuous than any 
other. 


44 ENGINEERING MATHEMATICS. 

Let A = a(cos a+j sin a) be divided by J5 = 6(cos ,5+y sin /5), 
thus : 

A _a(cos a+j sin a) 
'^B^HcosJTJmij) 

a (cos a +f sin a) (cos /? — f sin /9) 
6 (cos /?+y sin /?)(cos /? — f sin /?) 

a{ (cos a cos /? + sin a sin /?) +/(sin a cos /? —cos a sin 5) } 
^ 6(cos2/?+sin2/?) 

= T-jcos («—/?)+/ sin (a— /?)}. 

That is, general numbers A and B are divided by dividing 
their vectors or absolute values, a and h, and subtracting their 
phases or angles a and /?. 

Involution and Evolution of General Numbers. 

31. Since involution is multiple multiplication, and evolu- 
tion is involution with fractional exponents, both can be resolved 
into simple expressions by using the polar form of the general 
number. 

A = ai +ja2 = a(cos a+j sin a), 
then 

C = A^ = a^ (cos na+j sin no). 

For instance, if 

A = 3 +4/ = 5(cos 53 deg. +/ sin 53 deg.); 
then, 

(7 = 44 = 54(cos 4 X53 deg. +j sin 4 X53 deg.) ■ 
= 625(cos 212 deg. +j sin 212 deg.) 
= 625( -cos 32 deg. -/ sin 32 deg.) 
= 625( -0.848 -0.530 j) 
= -529-331/. 

If, A=ai +ja2 = a (cos a+j sin a), then 

1 1/ a a\ 

(7 = A/A = A"=a"(cos-+7sin-) 

= V a I cos — 4- 7 sm - I . 
\ n '' n/ 


THE GENERAL NUMBER. 45 

32. If, in the polar expression of A, we increase the phase 
angle a by 2n, or by any multiple of 2;r : 2q7t, where q is any 
integer number, we get the same value of 4? thus: 

A = a\(ios{a-\-2q7:)-\-j ^m.{a+2qn)\, 

since the cosine and sine repeat after every 360 deg, or 2-. 
The nth root, however, is different: 

C = vA = va( cos ^+?sm ). 

We hereby get n different values of C, for ^ = 0, 1, 2. . .n-1; 
q = n gives again the same as ^ = 0. Since it gives 

a+2n7r a 

= — h27r: 

n n ' 

that is, an increase of the phase angle by 360 deg., which leaves 
cosine and sine unchanged. 

Thus, the nth. root of any general number has n different 
values, and these values have the same vector or absolute 

term \/a, but differ from each other by the phase angle — and 

its multiples. 

For instance, let 4= -529-33iy = 625 (cos 212 deg.+ 
j sin 212 deg.) then, 

/ 212+360^ . . 212 + 360g\ 
C= ■<lA= i625(cos ^ — --+jsm 1 ^) 

= 5(cos 53 + j sin 53) = 3 + ij 

= 5(cos 143 +j sin 143) =5( -cos 37 +f sin 37) = -4+3/ 
= 5(cos 233 + J sin 233) = 5( -cos 53 + / sin 53) = -3 + 4/ 
= 5(cos 323 +y sin 323) =5(cos 37 -j sin 37) =4-3/ 
= 5(cos 413 +/ sin 413) = 5(cos 53 +/ sin 53) =3 + 4/ 

The n roots of a general number A = a(cos a+j sin a) differ 

from each other by the phase angles — , or 1/nth of 360 deg., 

and since they have the same absolute value ^/a, it follows, that 
they are represented by n equidistant points of a circle with 
radius \^, as shown in Fig. 22, for n = 4, and in Fig. 23 for 


46 


ENGINEERING MATHEMATICS. 


n = 9. Such a system of n equal vectors, differing in phase from 
each other by 1/nth of 360 deg,, is called a po/i/p/iase system, or 
an n-phase system. The n roots of the general number thus 
give an n-phase system. 

33. For instance, \^1 = ? 

If A = a (cos a+j sin a) = l, this means: a=l, a=0; and 
hence, 

„/- 2q7T 2q7z^ 
vl =cos — + 7 sm , 

n '' n 


P,=-4+3;-. 


P3=-3-4i 


Fig. 22. Roots of a General Number, n=4. 
and the n roots of the unit are 
5 = 0 <M=1; 


q = 2 


360 . . 360 

cos \-i sm — : 


^360 . . ^ 360 

cos 2 X — + J sm 2 X — ; 


g = n — 1 cos (n-1) -^+;/ sm (n-1) — . 


However, 


n 


cos 


360 . . 360 / 360 . . 360\9 

?--+;sm5— = ^cos— +;sm--j; 


THE GENERAL NUMBER. 47 

hence, the n roots of 1 are, 

^/T ( 360^. . 360V 
VI = I cos (-? sin — I , 

where q may be any integer number. 

One of these roots is real, for ^ = 0, and is= +1. 

If n is odd, all the other roots are general, or complex 
numbers. 

71/ 

If n is an even number, a second root, for q = K, is also real: 
cos 180+ /sin 180= -1. 


Fig. 23. Roots of a General Number, n = 9. 

If n is divisible by 4, two roots are quadrature numbers, and 

are +/, for q = j, and -/, for <? = -^. 

34. Using the rectangular coordinate expression of the 
general number, A=cii +jci2, the calculation of the roots becomes 
more complicated. For instance, given -<& = ? 

Let C=4a=ci+jc2; 

then, squaring, 

4 = (ci+/c2)2; 
hence, 

ai +ja2 = (ci^ -C2^) -\-2jc\C2. 

Since, if two general numbers are equal, their horizontal 
and their vertical components must be equal, it is: 


= /.,2_ 


ai=ci 


C2^ 


and a2 = 2ciC2. 


48 ENGINEERING MATHEMATICS. 

Squaring both equations and adding them, gives, 

Hence : 


and since Ci^-C2^ = ai', ^/ \ 


then, ci2 = i(Vai2 + a22 + ai), 


2_l/^x/^, 21^.2 !^,^ ' - MV -/ 


and C2^ = i{Vai^+a2^-ai). 

Thus 


and 
and 


Ci=ViiVai2 + a22+ai 


C2 = i/j|V^?T^-a2}, 


which is a rather complicated expression. 

35. When representing physical quantities by general 
numbers, that is, complex quantities, at the end of the calcula- 
tion the final result usually appears also as a general number, 
or as a complex of general numbers, and then has to be reduced 
to the absolute value and the phase angle of the physical quan- 
tity. This is most conveniently done by reducing the general 
numbers to their polar expression. For instance, if the result 
of the calculation appears in the form. 


{ai + ja2){hi +jh2)^Vci -\-jc2 

• {d,+jd2y\e,-^je2) ' 

by substituting 

a = Vai^4-a2^; tana = — . 


h = Vb{^ + h2^; tan^ = r^ 
^1 


and so on. 

P a(cQs g+jsin a:)6^(cos^ + /sin./9)^Vc(cos ;-+/sin j^)^ 
d^^cos d+j sin ^)2e(cos e+j sin e) 

ab^Vc 

= -^^ lcos(a+3/9 + r/2 -2^ - s)+j sin (a'+3/?+r/2 -2^ -e)]. 


THE GENERAL NUMBER. 49 

Therefore, the absohite value of a fractional expression is 
the product of the absolute values of the factors of the numer- 
ator, divided by the product of the absolute values of the 
factors of the denominator. 

The phase angle of a fractional expression is the sum of 
the phase angles of the factors of the numerator, minus the sum 
of the phase angles of the factors of the denominator. 

For instance. 


^ (3-4/P(2 + 2y)^-2.5 + 6/ 
5(4 + 3/)2\/2 
25(cos307+/sin307)22\/2'(cos45+/sin45)-y6y5(cosll4+/sinll4)i 
125 (cos37+/sin37)2\/2 

= 0.4^6^|cosf2x307+45 + -^-2x37J 

+/sin (2x307+4o+-^-2x37J 

= 0.4 ^6^ 1 cos 263 + j sin 263 \ 

= 0.746{ -0.122-0.992/1 = -0.091 -0.74/. 

36. As will be seen in Chapter II: 

u^ u^ w^ 

^ x^ x'^ x^ x^ 

x^ x^ x^ 

sinx = x-^+-r--T-+ -. . . 
|3 |5 1 7 

Herefrom follows, by substituting, x = Q, u = jd, 
cos ^+y sin ^= £^^, 
and the polar expression of the complex quantity, 

A = a(cos a+ysin a), 
t}ius can also be written in the form. 


50 ENGINEERING MATHEMATICS. 

where e is the base of the natural logarithms, 

£ = 1+1+12 + 13 + 14+. . .=2.71828. .. 

Since any number a can be expressed as a power of any 
other number^ one can substitute, 

where ao = \o£[,e a = -r^ .and the general number thus can 

logio ^ ^ 

also be written in the form, 

. ^ ? 

that is the general number, or complex quantity, can be expressed 
in the forms, 

A=ai+ja2 
= a(cos a+j sin a) 

The last two, or exponential forms, are rarely used, as they 
are less convenient for algebraic operations. They are of 
importance, however, since solutions of differential equations 
frequently appear in this form, and then are reduced to the 
polar or the rectangular form. 

37. For instance, the differential equation of the distribu- 
tion of alternating current in a flat conductor, or of alternating 
magnetic flux in a flat sheet of iron, has the form : 

and is integrated by y = A£~^'^, where. 


V=\/-2jc^=±{l-j)c; 
hence, 

2/=^£+(i-^*)"^+A2£-^^~^*K 

This expression, reduced to the polar form, is 

y = Aie'^''^(cos ex -j sin ex) +A2£~''''(cos ex+j sin ex). 


THE GENERAL NUMBER. 51 

Logarithmation. 

38. In taking the logarithm of a general number, the ex- 
ponential expression is most convenient, thus : 

logs (ai ^-ja^) =log£ a(cos a +j sin a) 
=log£a£^" 

= l0gea+l0ge£'*'' 

= \oge a+ja; 
or, if 6 = base of the logarithm, for instance, 6 = 10, it is: 

log6(ai+/a2)=log6a£''« = log6 a + ja log^ e; 
or, if b unequal 10, reduced to logio; 

X , . X logio a . logio £