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THE ALGEBRA OF COMPLEX QUANTITIES.
x1. Geometr1c Add1t1on And Mult1pl1cat1on.
57. Classification of Magnitudes: Definitions.
It was pointed out in Art. 2 that any one straight line suffices for the complete characterization of all so-called real quantities; in fact the real magnitudes of algebra were defined as lengths set off upon such a line. But because, in this representation, no distinction in direction was necessary, all line-segments were taken to be real magnitudes, and comparisons of direction were made, by means of the principles of geometrical similarity, for the sole purpose of determining lengths. Such comparisons will still be necessary whenever the product or quotient of two real magnitudes is called for, but into the real magnitudes themselves no element of direction enters; their sole characteristics are length and sense, that is, length and extension forwards or backwards.
If the attempt be made to apply the various algebraic processes to all real magnitudes, negative as well as positive, another kind of magnitude, not yet considered, is necessarily introduced. For example, if x be positive, no real magnitude can be made to take the place of either (— xyl2 or logm (— x); for the square of a real quantity is always positive (Art. 14), and the definition of an exponential given in Art. 23 precludes its ever assuming a negative value. In order that forms like these may be admitted into the category of algebraic quantity, a new kind of quantity must therefore be defined, or more properly, a new definition of algebraic quantity in general must be given.
Having assigned some fixed direction as that in which all real quantities are to be taken, we adopt a straight line having this direction as a line of reference, call it the real axis, and determine the directions of all other straight lines in the plane by the angles they make with this fixed one. Line-segments having directions other than that of the real axis are the new magnitudes that now demand consideration. They are called vectors. They have two determining elements: length and the angle they make with the real axis.
(i) . Its length, taken positively, is called the tensor of the magnitude, and the arc-ratio of the angle it makes with the real axis is called its amplitude (or argument}.
Classified and defined with respect to amplitude, the magnitudes themselves are:
(ii) . Real, if the amplitude be o or a multiple of n;
(iii) . Imaginary, if the amplitude be x/2 or an odd multiple of ir / 2.
(iv) . Complex, for all other values of the amplitude. In general, therefore, vectors in the plane represent
complex quantities, but in particular, when parallel to the real axis they represent real quantities; when perpendicular to it, imaginary.
Any quantity is by definition uniquely determined by its •tensor and amplitude, and hence:
(v) . Two quantities are equal if their tensors and their amplitudes are respectively equal, the geometrical rendering of which is: two magnitudes, or vectors, are equal if (and only if) they are at once parallel, of the same sense, and of equal lengths.
•The algebra of complex quantities, like that of real quantities, is developed from the definitions of the fundamental algebraic operations: addition and subtraction, multiplication and division, exponentiation and the taking of logarithms. These operations applied to magnitudes represented by straight lines in the plane are called algebraic by reason of their identity with those of the analysis of real quantities, but specifically geometric, because each individual operation has its own unique geometrical configuration. On the other hand, the algebraic processes applied to real quantities may be described as geometric addition, multiplication, involution, etc., in a straight line.
58. Geometric Addition. Regarding lines for our present purpose as generated by a moving point, the operation of addition is defined to mean that a point P, free to move in any direction, is successively transferred forwards or backwards, that is, in the positive or negative sense as marked by the signs + and —, through certain distances designated b^y appropriate symbols a, /?, y . . Thus the sum -f- a *— /3 + y, in which a, /?, y represent vectors in the plane (or in space), joined to form a zig-zag, as shown inl'the accompanying figure, may be read off as R follows, the arrow-heads indicating
, then backwards through distance /8 to B'', then forwards through distance y to C, and the result is
. the same as if the motion had taken place in a direct line from O to C'\ this fact is expressed in the equation
direction of motion forwards: Move forwards through distance a to A,
llf not already contiguous, the magnitudes that form the ^terms of a sum, by changing the positions of such as require -it without changing their direction, may be so placed that all the intermediate extremities are conterminous. Geometric addition may therefore be defined as follows: /'The sum of two or more magnitudes, placed for the f /purpose of addition so as to form a continuous zig-zag, is the single magnitude that extends from the initial to the terminal extremity of the zig-zag. /
59. The Associative and Commutative Laws for
geometric addition in the plane are deduced as immediate consequences of its definition. For in the first place, the ultimate effect i.; the same whether a transference is made from O direct to B then to C as expressed by (o -)- /J) -f- y, in the subjoined figure, or from O to A then direct from A to C as expressed by a + (/? + y), or) from O to A to B to C as expressed by a + + y; hence
(« + /?).+ y = «+(/? + y)
and in the second place, by (v) of ABCD, a parallelogram, AD — BC=y, - /?, and by the definition of addition AB + BC = AC= AD + DC, whence
One or more of the terms may be negative. Expressing this fact by writing ± a, ± /?, ± y in place of a, p, y, the two resultant equations of the last paragraph become (±a±/3)±y=±a+(±/3±y)=±a±/3±yl ±y±P=±p±y,