APPENDIX TWO 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 ■1 — r +i + 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. 278 APPENDIX TWO 279 -5 -4 -3 -2 1 4-1 12 +3 4-4 4-5. 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.) m is called a rational number. Thus the so-called arithmetic operations of addition, subtraction, multiplication, and division define the system of rational numbers. + 1 4-2 Figure A2.3 The phase of any negative number, e.g., —2, is -n (or any odd multiple of tt). 280 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, Vfl, (A2.2) 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, (A2.3) 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. (A2.4) 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 (A2.5) 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) 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 282 THE ELEMENTS OF PHYSICS APPENDIX TWO 283 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. (A2.8) 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 1 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 284 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. 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, 92 249 285