The System of Complex Numbers
The development and expansion of the number concept through the ages has
its root in the desire to accommodate various operations without restriction
within the system of numbers. Thus the unit number 1 in conjunction with the
unlimited operation of addition at once yields as a system of numbers the
positive integers without upper limit. This system must be extended, how-
ever, to include the negative integers if one wishes to accommodate the
—I 1 r
i 2 +3 +4 +5 4-6
Representation of +5 and —3 by vectors in a one-dimensional space.
operation of subtraction without restriction. One can then operationally
redefine "subtraction" as "addition of a negative number."
Geometrically, any integer n may be represented as a vector in a one-
dimensional space, the length of the vector being equal to the absolute value
\n\, and the vector pointing to the right if n is >0 and to the left if n is <0,
as drawn in Figure A2.1. Addition of such numbers means, then, addition of
vectors, which is carried out by joining the origin of any vector in the sum
to the endpoint of the preceding vector, the resultant being the vector joining
the origin of the first with the endpoint of the last vector, as drawn, for
example, in Figure A2.2.
The operation of multiplication when applied without restriction to the
system of integers does not necessitate any enlargement of that system.
Geometrically, one can formulate the result of any multiplication m a very
picturesque manner by introducing the concept of the phase of a number.
-3 -2 1 4-1 12 +3 4-4
Reprcscntadon of the sum 4-3 — 5 4- 1 by vector addition, with the resultant —1.
The phase of a number is the angle formed by the vector representing that
number with the unit vector 4- 1 . Thus any positive number has the phase
and any negative number has the phase ir. It is understood that angles are
measured in radians, i.e., in terms of the length along the unit, circle cut out
by the angle (sec Figure A2.3).
Strictly speaking, the phase of a positive number can be any even multiple
of - (i.e., it can be . . . — 4tt, — 2tt, 0, 2n, 4V, . , , ,) and the angle of phase
of a negative number can be any odd multiple of w (i.e., . . . — 3tt, —tt, tt,
3tt, ) but for the purpose of this appendix it. is sullicient to consider only
the phases and tt.
Now the operation of multiplication can be defined geometrically by
stating that multiplying two numbers results in a number whose vector has a
length equal to the product of the lengths of the factor vectors, and whose phase
is the sum of the phases of the factor vectors. Thus a product containing an
even number of negative factors will be a positive number, the phase of the
product being an even multiple of 77, and a product containing an odd number
of negative factors will be a negative number, the phase of the product being
an odd multiple of tt.
The operation of division cannot be applied to the system of integers without
restriction unless one enlarges the number system to encompass all rational
numbers. Any number r which can be written as a ratio of two integers n
and n 1,
r =— (11 and m integers) (A2.1.)
is called a rational number. Thus the so-called arithmetic operations of
addition, subtraction, multiplication, and division define the system of rational
+ 1 4-2
The phase of any negative number, e.g., —2, is -n (or any odd multiple of tt).
THE ELEMENTS OF PHYSICS
Since ancient times operations have been known that arc not arithmetic.
An example is the taking of the square root of a number,
which is a special case of a more general type of "algebraic" operations.
One defines, quite generally, an algebraic number as a solution .y of an
equation of the form
a n x" + c^-i-v"- 1 + ■■• + «!* + a a = 0,
where the %, a^ . . , , a r , are integers. The example (A2.2) is now a special
case obtained if one puts all coefficients except a n and a 2 equal to zero and
writes a — aja 2 so that Equation (A2.3) reads
a = 0.
Now the interesting question arises: Are all algebraic numbers rational
numbers? The answer to this has been known since ancient times: one must
go beyond the system of rational numbers to accommodate all algebraic
numbers. In fact, one requires for the unlimited use of algebraic operations
two extensions of the system of real rational numbers:
(1) An extension to real, but irrational, numbers is already demanded by
the operation (A2.2), if the radicand a is allowed to be any positive integer.
In particular, it was already known to Pythagoras that the square root of 2
x = V 2
cannot be written as a rational number.
Indeed, if two integers/?' and m existed such that (2)' a = njin, one would
have after elimination of all common factors of n' and m' an equation of the
form 2m' 1 — ri 1 , where m and n are integers which have no common factor.
This means that n~ is an even number and therefore n is also an even number,
because only squares of odd numbers are odd. Thus one can write n — 2c,
where c is some integer. Now one has the equation 2m 2 — Ac- or 2c' 2 =
m*. from which follows that m must be an even number too. This means n
and m have a common factor, 2, contrary to the premise implied by the ration-
ality of (2)' A . This contradiction can be avoided only if (2)' A cannot be repre-
sented by a rational number.
(2) An extension to so-called imaginary numbers is already demanded by
the observation that the square root of any negative number cannot be any
APPENDIX TWO 281
of the previously defined real numbers. Indeed, the number
i = \'-l (A2.6)
i' 2 =-l (A2.7)
is negative, cannot be either a positive or negative real algebraic number
because the square of any real number is positive.
*- Real axis
The representation of the imaginary unit i = V —1 as a vector of unit length along
the imaginary axis, giving it a phase w/2.
The geometric representation of the law of multiplication contains the
clue to representing this imaginary unit number i geometrically. If one
wants to maintain the multiplication rule of adding the phases offactors, one
must assign the number i a phase of w/2, in order that i? has the phase it in
accordance with Equation (A2.7). This amounts to representing the number
i by a vector of unit length pointing in a direction at right angles to the real
axis, as drawn in Figure A2.4. Any multiple of i can then be located along
the so-called imaginary axis, drawn at right angles to the real axis with an
intersection at the origin O.
By insisting on the unlimited use of the operation of addition, defined as
vector addition, one is then committed to enlarge the number system to the
system of complex numbers: Any vector in the plane formed by the real and
THE ELEMENTS OF PHYSICS
imaginary axis is called a. ''complex'' number, denoted c. The length of the
vector is called the "absolute value" of r, denoted |c|, and the angle
formed by c with the positive real axis is called the ''phase" of c, as indicated
in Figure A2.5.
\ c \.
-»- Real axis
The plane of complex numbers. Any complex number can be represented by a
vector of length |r|, having the phase <f>.
Adding complex numbers means adding them vcetorially: Cj + c, is
obtained by joining the origin of c% to the endpoint off, and connecting the
end point of c 2 with the origin, as drawn in Figure A 2, 6.
Multiplying complex numbers means multiplying the absolute values and
adding the phases: c = e a : £- 4 has absolute value \c\ = |c T | ■ \cr,\ and phase
t$> = 4>, + <£;>, as drawn in Figure A2.7,
The addition of complex numbers.
Of particular interest for the application of complex numbers to physics
is the square of the absoltite value of the stun of two complex numbers,
P = | Cl + c.
P is always a real, positive number. If c± and c, differ in phase by an odd
integer multiple of jt, P has its minimum value, P = (\t\\ — \c. 2 \y l and one
says 'Y 1 ! and <? 3 interfere destructively." But if c L and c\, differ in phase by an
even multiple of 77, P has its maximum value, P = Qfijj + |c' 2 Q- and one says
"c ± and c.„ interfere constructively."
The multiplication of complex numbers.
For the sake of completeness it should be mentioned here that the real
algebraic numbers do not comprise the totality of all possible numbers
making up the so-called continuum of real numbers. There are numbers that
cannot be represented as solutions of an algebraic equation such as (A2.3).
Such numbers are called transcendental numbers. An example of a trans-
cendental number is 77 = 3.14 ... . defined as half the circumference of a
circle of radius unity. In fact, George Cantor (1845-1918) discovered that
almost all real numbers are transcendental, in the following sense. According
to Cantor's definition the set of real numbers is identical with the set. of
points making up the one-dimensional continuum. Among these the
algebraic numbers form a subset, that is "enumerable," i.e., all its members
can be arranged in some sequence a-,, x\, . . . , x n , . . . and thus correlated to
the integers 1, 2, ...,«,... , allowing one to call .^ the "firsL." .\ 2 the
"second," . . . , ,v„ the "nth," . . , algebraic number. The transcendental
numbers, on the other hand, arewai enumerable. This discovery of Cantor's
THE ELEMENTS OF PHYSICS
is perhaps the most astonishing of the many great contributions he made to
the apprehension of the continuum of real numbers.
Additional Suqaested Reading
Cantor, G. Contributions to the Founding of the Theory of Trimsfiriite Numbers,
1895-1897; Dover Publications, Inc., New York, 1915.
Dantzig, T. Number, Doubled ay & Co., Inc., New York, 1956.
Davis, P. J. "Number," Scientific American, September 1964, 50-59.
Kamke, E. Theory of Sets, Dover Publications, Inc., New York, 1950.
Absolute future and past, 113
Absolute temperature, 121
Absorption probability, 241
Acceleration, central, 18
Action at a distance, 41, 49, 146
Action principle, 192
Addition, of entropies, 135
of vectors, 18
of velocities, 107, 115
Aerodvnannic effects, 97
Angular momentum, 77
Angular velocity, 69, 82
Antineutrino, 90, 115, 174,252
Archimedes' principle, 93
Arrow of lime, 1 34
Axes of inertia, 84
Karyon number, 254
Bernoulli's principle, 94, 97
Bit, of information, 20, 139
Blackbody radiation, 240
Bohr's model of the atom, 223
Boiling line, 13 f
Boltzmann's theorem, 238
Bosons, 80, 254
Camot's cycle, 121, 132
Cavendish experiment, 26
Celsius scale, 130
Center of mass, 41 , 47
Central acceleration, 18
Central force, 153
Centrifugal force, 67
Cerenkov radiation, 186
Cesium clock, 8
CG5 system, 6
Charge, concept, 143
conservation, 178, 264
Circular motion, 18
Circulation, 158, 162, I7S
Coexisting phases, 132
Coherence length, 182
Collision, 46, 1 14
Combination principle, 216
Combined inversion, 171
Compound nucleus, 118
Conductors, extended, 165
Configuration space, 193
Conservation laws, 263
Conservation, of angular momentum, 77
of charge, 1 78, 264
of energy, 87
of momentum, 45
Coordinate concept, 1
Coordinate system, 2
Coriolis, effect, 68
force, 69, 81
Coulomb's law, 144
Current, electric, 151
Curve ball, 97