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.
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 B, 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 G in Fig. 3. In the latter case,
2 l o i 2
cb i j) I 1
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-b = a,
and, if b 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
In the application 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.
i THE GENERAL NUMBER. 5
In the same way, an illumination of 5 foot-candles, lowered
by 3 foot-candles, gives an illumination of 2 foot-candles, thus,
. o 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:
(-l)X(-lH(-fl), 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
+5 deg. latitude -7 deg. latitude = -2 cleg, 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 liabilities, etc. r
there may be application 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 applied, as the imaginary
number v~l, and the only difference is, that we have become
familiar with the negative number at an earlier age, where we
wer? 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. When
we first learned it, however, it was quite a step to become
accustomed to saying, 5 7 =2, and not simply, 57 is
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-
. . . (b terms) =c.
Just like addition, multiplication can always be carried
Three groups 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 division, which is written, thus :
5 = 4 horses ;
or, in general,
If we have a bunch of 12 horses, and divide it into two equal
groups, we get 6 horses in each group, thus:
if we divide unto 4 equal groups,
r = 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
-=- = 2, remainder 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.
-5- = 3, remainder 1,
we may now cut the left-over apple into 3 equal parts, in which
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^=21.
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 applicable, and sometimes
not, as the negative number. In the above instance of 12
horses, divided into 5 groups, it is not applicable.
^ - = 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 3 = 64 ;
or, in general, a b = 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, denned as multiple multiplica-
tion, requires the exponent b to be an integer number; b is the
number of factors.
Thus 4~ 3 has no immediate meaning; it would by definition
be 4 multiplied ( 3) times with itself.
Dividing continuously by 4, we get, 4 6 n-4=4 5 ; 4 5 ^4=4 4 ;
4 4 -^4 = 4 3 ; etc., and if this .successive division by 4 is carried
still further, we get the following series:
4 3 4X4X4
-j - = 4X4 =4 2
4 2 4X4
4 ~ 4
= 4-2 =
iL!_ 4 _J_ =4 -3_i
4 4 2 ' 4X4X4 4 3 '
or, in general, a- 6 = -,
10 ENGINEERING MATHEMATICS.
Thus, powers with negative exponents, as a~ 6 , are the
reciprocals of the same powers with positive exponents : -^.
7. From the definition of involution then follows,
because a 6 means the product of b equal factors a, and a n the
product of n equal factors a, and a b Xa n thus is a product hav-
ing b+n equal factors a. For instance,
4 3 X4 2 =(4X4X4)X(4X4)=4 5 .
The question now arises, whether by multiple involution
we can .reach any further mathematical operation. For instance,
may be written,
(43)2 = 4 3 X 43
= 4 6.
and in the same manner,
that is, a power a b is raised to the n th power, by multiplying
its exponent, Thus also,
that is, the order of involution is immaterial.
Therefore, multiple involution leads to no further algebraic
8. 4 3 = 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
or, more general,
\ / c=a.
THE GENERAL NUMBER. 11
%/c thus is defined as that number a, which, raised to the power
b, 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 c*> or. c^, have any meaning.
it is seen that c 6 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,
n / ln
Cb = ( C n) 6 = vV 1 ,
9. Involution with integer exponents, as 4 3 = 64, can always
be carried out. In many cases, evolution can also be carried
out. For instance,
while, in other cases, it cannot be carried out. For instance,
12 ENGINEERING MATHEMATICS.
Attempting to calculate A/2, we get,
and find, no matter how far w r e 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
^/2, but we can only approximate it, and carry the approxima-
tion to any desired degree; some such numbers, as TT, 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 ife. If we draw a square, with 1 foot as side, the length
of the diagonal is $2 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
1000s = 2138.7387387 ,
999z = 2136.6
2136.6 21366 1187 77
999 " 9990 ~ 555 555*
THE GENERAL NUMBER. 13
10. The following equation,
may be written, since,
hut also the equation,
may be written, since
Therefore, -<T+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,
*(f 4 cannot be 2, as 2 squared gives ^4, nor can it be +2.
11-4= ^4X(-1)= i-2A/-l, and the question thus re-
solves itself into: What is A/^l?
We have derived the absolute numbers from experience,
for instance, by measuring distances on a line Fig. 4, from a
starting point A.
i -f -p -I -i
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
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"^T, a quantity
of which we do not yet know the meaning. Multiplying once
more by V-l, we' get +2xv^xV~l"= -2; that is,
multiplying a number +2, twice by \/-l, gives a rotation of
180 deg., and multiplication by V^T thus means rotation by
half of 180 deg.; or, by 90 deg., and +2\^ r T thus is the dis-
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 2j, a new kind of number, having a direction 90 deg.
THE GENERAL NUMBER.
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 the' 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 -3 -2 -1
+ 1 +2
For instance, in problems dealing with plain geometry, as in
electrical engineering when 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
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 multiplication by the negative unit (-1) means
rotation by 180 deg., or reverse of direction, so multiplication
by the quadrature unit, f, means rotation by 90 deg., or change
from the horizontal to the vertical direction, and inversely.
12. By the positive and negative numbers, all the points of
a line could be represented numerically as distances from a
chosen point A.
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, OB= +3, and a
vertical distance, BP= +2f, and therefore is given by a
combination of the distances, OB= +3 and BP=+2j. For
convenience, the act of combining two such distances in quad-
rature with each other can be expressed by the plus sign,
and the result of combination thereby expressed by OB+BP
THE GENERAL NUMBER.
Such a combination of an ordinary number and a quadra-
ture number is called a general number or a complex quantity.
The quadrature number jb 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 + jb. They are especially useful and impor-
tant in electrical engineering, as most problems of alternating
currents lead to vector representations in the plane, and there-
fore can be represented by the general number a+jb] that is,
the combination of the ordinary number or horizontal distance
a, and the quadrature number or vertical distance jb.
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+jb both coordinates are combined, a being
the x coordinate, jb the y coordinate.
Thus in Fig. 8, coordinates of the points are,
PI: x=+3, y=+2 P 2 : x = +3 y= -2,
P 3 : x=-3, y=+2 P 4 : x= -3 y = -2,
and the points are located in the plane by the numbers:
Pi=3+2/ P 2 = 3-2/ P 3 =-3+2/ P 4 =-3-2/
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
The meaning of l we find in the same manner as that
A positive number a may be represented on the horizontal
axis as P*
Multiplying a by ^ 1 gives ajjl, whose meaning we do
not yet know. Multiplying again and again by ^ 1, we get, after
four multiplications, a(^-l) 4 = a; that is, in four steps we
have been carried from a to a, a rotation of 180 deg., and
iT-1 thus means a rotation of = 45 dcg., therefore, a-v' 1
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 y = ^=j m j or, is represented by the general number,
-$( 1, however, may also mean a rotation by 135 deg. to P% 9
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 <$ 1;
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.
In the same manner, V + l 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,
the x coordinate is cos qX , the y coordinate sin qX ,
n/ T 360 360
v -f l=cos qX +] sin qX ,
where q is any integer number.
FIG. 9. Vector Diagram a^ 1.
There are therefore n different values of a v+1, which lie
equidistant on a circle with radius 1, as shown for n = 9 in
14. In the operation of addition, a + b = c, the problem is,
a and b being given, to find c.
The terms of addition, a and 6, are interchangeable, or
equivalent, thus: a + b = b + a, and addition therefore has only
one reverse operation, subtraction; c and b being given, a is
found, thus; a = cb, and c and a being given, b is found, thus:
b=ca. Either leads to the same operation subtraction.
The same is the case in multiplication-; aXb = c. The
factors a and b are interchangeable or equivalent; aXb=bXa
and the reverse operation, division. a = 7- is the same as 6=.
In involution, however, a b = c, the two numbers a and b
are not interchangeable, and a b is not equal to b a . For instance
4 3 = 64and3 4 = 81.
Therefore, involution has two reverse operations:
(a) c and b given, a to be found,
FIG. 10. Points Determined by v-!.
(6) c and a given, b to be found,
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 6 Xa n = a 6+n . Therefore, the logarithm of a
THE GENERAL NUMBER. 21
product is the sum of the logarithms of the factors, thus log a c Xd
= loga c+ loga d.
2. A power is raised to a power by multiplying the exponents :
Therefore the logarithm of a 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 =n log a C,
loga 1=0, because a = 1. If the base a > 1, Iog c is positive,