Uniplanar Algebra

that is, by virtue of the definition of Art. 6, cis and cis (4) are reciprocal to each other and we may write /cis cis (-).

In like manner, since it is now evident that

(a cis ) X (/a/cis ) = a / acis o = 1

... / (a·cis &) = / a ⋅ / cis 4.

If now in the formula expressing the law of multiplication we write / band — for b and

the law of geometric division,

62.

prove:

(1).

(2).

respectively, we have, as

(a cis) / (bcis y) = a/b cis (☀ — ¥).

Agenda. Properties of cis . If n be an integer

(3). cis ([2 n + 1] π) = − cis &.

(4). cis (+ [2 n ±}] π) = ± i cis .

(5). cis (+ [2 n± } ] π) == ¿ cis &.

(6). Show that the ratio of two complex quantities having the same amplitude, or amplitudes that differ by ± 27, is a real quantity.

(7). Show that the ratio of two complex quantities having amplitudes that differ by ± 1 is a purely imaginary quantity.

63. The Imaginary Unit. By definition (iii) of Art. 57 cis /2i is an imaginary having a unit tensor; it is therefore called the imaginary unit. Its integral powers form a closed cycle of values; thus:

and the higher powers of i repeat these values in succession; that is, if n be an integer,

and these are the only values the integral powers of i can acquire.

64. The Associative and Commutative Laws. Let a, b, c be the tensors, 4, 4, x the amplitudes of a, ß, γ respectively; that is,

Then, by the law of geometric multiplication,

a X (BX y) = (a · cis ) × ([b · cis ] × [ccis x])
(a cis ) X (b × c• cis [+x])

and by the same process,

(a × ẞ) × y=(a × b) × c • cis ([$ +4] +x).

X B)

But, by the rules of algebraic multiplication and addition, ax (b×c)=(a × b) × c and ☀+ (y + x)=($+4)+x; .'. a X (BX y) = (a X B) Xy,

which is the associative law for multiplication.

And again, by the law of geometric multiplication,

a XB (a cis ) × (b⋅ cis 4)
β · $) (b·cis ¥)

=aXb、cis(+),

and similarly ẞxa=bacis ( X ).

But, by the rules of algebraic multiplication and addition,

ax b b xa and +4=4+;

.'.

which is the commutative law for multiplication.

The letters here involved may obviously represent either direct factors or reciprocals, and the sign X may be replaced at pleasure by the sign /, without affecting the proof here given (see Art. 61). Hence, as in Arts. 32, 33, with real magnitudes, so with complex quantities the associative and commutative laws for multiplication and division have their full expression in the formula,

X(Xax B) = × (a) × (B),

YaxB=BY a.

65. The Distributive Law. From the definitions of geometric addition and multiplication (Arts. 58, 60) the law of distribution for complex quantities is an easy conse

quence. The constructions of the subjoined figure, in which a, ß and mcis represent complex quantities will B bring this law in direct evidence. The operation m cis changes. OA into OA', AB= A'Finto AE-A'B', and OB into OB', that is, turns each side of the triangle OAB through

the angle AOA' and changes its length in the ratio of m to 1, producing the similar triangle OA'B', in which

OA'm'cis Xa,

A'B' =mcis @ X ß,

OB'm cis X (a + B).

But, by the rule of geometric addition,

OB' = OA' + A'B';

mcise × (a +ß): =mcis @ Xa+m · cis 0 × ß.

This demonstration is in no way disturbed by the introduction of negative and reciprocal signs. The last equation above written is, in fact, the first equation of page 37, and the subsequent equations of Arts. 20, 21 and their proofs remain intact when for the real quantities a, b, c, d, etc., complex quantities are substituted. Hence, writing mcis @=y, (±a±ẞ) × (±y) =+ (± a) × (± y) + (± ẞ) ×(±y). Here, as in Arts. 20, 21, the sign X is distributive over two or more terms that follow it, but not so the sign /.

66. Argand's Diagram. It is obvious from its definition as here given (Art. 57) that to every complex quantity there corresponds in the plane a unique geometrical figure which completely characterizes it. This figure is known as Argand's diagram,* and consists of the

-X

real axis OX with reference to which the arc-ratio is estimated, the imaginary axis OY perpendicular to OX, the directed line OP that represents the complex quantity and the perpendicular PA from P to OX. The axes OX and OY are supposed to be fixed in position and direction for all quantities. Any point P in the plane then determines one and only one complex quantity and one set of line-segments OA, AP, OP, different from every other set.

χ A Fig.35.

*First constructed for this purpose by Argand: Essai sur une manière de représenter les quantités imaginaires dans les constructions géométriques; Paris, 1806. Translated by Prof. A. S. Hardy, New York, 1881.

If OA=x and AP=y, the complex quantity appears in

the form xiy, and if OP=a and arc-ratio of AOP=0, the relations

x= a cos, ya sin 0,
a2x2+y, tan 0=y/x

are directly evident from the figure and we have

xiya (cos +isin 0),

tsr (x+y)= + √x2 + y2,

vsr (x+y)= cos + i sin 0,

=cos

amp (x+y)=arc-ratio whose tangent is y/x.

In analysis the complex quantity most frequently presents itself either explicitly in the form x+iy, or implicitly in some operation out of which this form issues.

67. Agenda. Multiplication, Division and Construction of Complex Quantities. Prove the following:

(1). (a+ib) × (x + iy) = ax — by +i (ay + bx).

(2). (a+ib)/(x+iy) = ax + by + i (bx — ay).