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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 

-6 -5 


— r 

+ 5 

—I 1 r 

i 2 +3 +4 +5 4-6 

Figure A2.1 
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. 




-5 -4 

-3 -2 1 4-1 12 +3 4-4 


Figure A2.2 
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 

Figure A2.3 
The phase of any negative number, e.g., —2, is -n (or any odd multiple of tt). 



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 


of the previously defined real numbers. Indeed, the number 

i = \'-l (A2.6) 

whose square 

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. 

Imaginary axis 

*- Real axis 

Figurl A2.4 

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 





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. 

Imaginary axis 

\ c \. 

-»- Real axis 

Figure A2.5 
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, 

Imaginary axis 

Real axis 

Figl-rl A2.6 
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." 

Imaginary axis 

Real axis 

Figure A2.7 
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 



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. 

Subject Index 

Absolute future and past, 113 
Absolute temperature, 121 
Absorption probability, 241 
Acceleration, central, 18 

Coriolis, 69 

gravitational, 15 

instantaneous, 14 
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 
Antigravity, 143 
Antineutrino, 90, 115, 174,252 
Antiparticles, 261 
Archimedes' principle, 93 
Arrow of lime, 1 34 
Asteroids, 42 
Astronauts, 82 
Axes of inertia, 84 

Karyon number, 254 
Harvons, 257 

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 

Calorie, 95 

Camot's cycle, 121, 132 

Causality, 2 

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 
Clocks, 8 

Coexisting phases, 132 
Coherence length, 182 
Collision, 46, 1 14 
Combination principle, 216 
Combined inversion, 171 
Compound nucleus, 118 
Conductivity, 233 
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 
Correlations, 271 
Coulomb's law, 144 
Current, electric, 151 
Curve ball, 97 

Decay curve, 
Density, 92