An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

~ 67-71.

Multiplication. Division.

115

the sum or of the difference of two complex numbers may have any
value not greater than the sum, and not less than the difference of
the moduli of the summands. For, putting
(r cos (p + r' cos 9') + i (r sin p + r' sin T') = R (cos - + i sin ),
if we identify the real parts and also the imaginary parts
R cos =- r cosT + r' cos p', R sin = r sin ~p + r'sin q',
we find for the required modulus B that:
R2 = r2 + r'2 + 2rr' cos(qp - q'); but, - < cos(q - q') - + 1.
70. Multiplication.
If m be a positive integer, (a + ib) * m must be understood to mean
that a + ib is to be put m times as summand; from this follows
(a + ib) * m = am + ibm;
in agreement with this we define in general:
(a + ib) (a' + ib') a' ( + i b) + b' (a + i b)= c a' + it C '+ iab' + i'2 b
=-a aa' - bb' + i (ba' + ab').
This keeps up the proposition of the interchangeability of factors.
In the second form we obtain:
r (cos p +- i sin rp). r'(cos 9' + i sin T ')
= rr' cos gq cos p' - sin qp sin ' +- i (sin q cos g' + cos s sin q')}
=rr'{cos ( cp + 9 ') + isin (ip + cp')} = rr'p+s',.
The modulus of the product is equal to the product of the moduli
of its factors; the amplitude of the product is equal to the sum of
their amplitudes.
The product vanishes only when one of its factors vanishes.
We have in general:
r, r' p. *.r, = (r r' r. ),(p +,+....
In the third form the equation is written:
reiTP. r'ei' r"ei " * * rve(v =- (rr'.. rv) ei('P+'++..p );
showing that here also: Powers of the same base e with imaginary
exponents are multiplied by adding the exponents.
71. Division is defined as inverse of multiplication: The meaning
of a +- ib: a' - ib' is that a number should be determined, which,
multiplied by a'- ib' shall give a product equal to a + ib. As
a+ib - =1, the calculation of the quotient can be reduced to the
a - i b
multiplication of two complex numbers.
a+-ib a+- ib a'-ib' (a ib) (a'- b') aa'+ bb' ba'- ab'
a'-Ja'ib' — a'+ib' ' —ib' -+- a'-b a+ a.'2 a8
8*