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Fig 24 is usually known as argands diagram 45 de

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the principal value of the amplitude. Fig. 24 is usually known as Argand’s diagram. 45. De Moivre’s Theorem. The following statements follow im- mediately from the definitions of addition and multiplication. (1) The real (or imaginary) part of the sum of two complex numbers is equal to the sum of their real (or imaginary) parts. (2) The modulus of the product of two complex numbers is equal to the product of their moduli. (3) The amplitude of the product of two complex numbers is either equal to the sum of their amplitudes, or differs from it by 2 π . It should be observed that it is not always true that the principal value of am( zz 0 ) is the sum of the principal values of am z and am z 0 . For example, if z = z 0 = - 1 + i , then the principal values of the amplitudes of z and z 0 are each 3 4 π . But zz 0 = - 2 i , and the principal value of am( zz 0 ) is - 1 2 π and not 3 2 π . * It is evident that | z | is identical with the polar coordinate r of P , and that the other polar coordinate θ is one value of am z . This value is not necessarily the principal value, as defined below, for the polar coordinate of § 22 lies between 0 and 2 π , and the principal value between - π and π .

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[III : 45] COMPLEX NUMBERS 102 The two last theorems may be expressed in the equation r (cos θ + i sin θ ) × ρ (cos φ + i sin φ ) = { cos( θ + φ ) + i sin( θ + φ ) } , which may be proved at once by multiplying out and using the ordinary trigonometrical formulae for cos( θ + φ ) and sin( θ + φ ). More generally r 1 (cos θ 1 + i sin θ 1 ) × r 2 (cos θ 2 + i sin θ 2 ) × . . . × r n (cos θ n + i sin θ n ) = r 1 r 2 . . . r n { cos( θ 1 + θ 2 + · · · + θ n ) + i sin( θ 1 + θ 2 + · · · + θ n ) } . A particularly interesting case is that in which r 1 = r 2 = · · · = r n = 1 , θ 1 = θ 2 = · · · = θ n = θ. We then obtain the equation (cos θ + i sin θ ) n = cos + i sin nθ, where n is any positive integer: a result known as De Moivre’s Theorem . * Again, if z = r (cos θ + i sin θ ) then 1 /z = (cos θ - i sin θ ) /r. Thus the modulus of the reciprocal of z is the reciprocal of the modulus of z , and the amplitude of the reciprocal is the negative of the amplitude of z . We can now state the theorems for quotients which correspond to (2) and (3). (4) The modulus of the quotient of two complex numbers is equal to the quotient of their moduli. (5) The amplitude of the quotient of two complex numbers either is equal to the difference of their amplitudes, or differs from it by 2 π . * It will sometimes be convenient, for the sake of brevity, to denote cos θ + i sin θ by Cis θ : in this notation, suggested by Profs. Harkness and Morley, De Moivre’s theo- rem is expressed by the equation (Cis θ ) n = Cis .
[III : 45] COMPLEX NUMBERS 103 Again (cos θ + i sin θ ) - n = (cos θ - i sin θ ) n = { cos( - θ ) + i sin( - θ ) } n = cos( - ) + i sin( - ) . Hence De Moivre’s Theorem holds for all integral values of n , positive or negative . To the theorems (1)–(5) we may add the following theorem, which is also of very great importance.

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