3. ALGEBRAIC PROPERTIES OF THE FIELD C
A, BEC,
2. Multiplication of Complex Numbers in Polar Coordinates. Since
c"2 =
i,e" =
—1, g$3T/2 = —1,=
1;these are particular cases of the identity
jm = (eul12)m = m E Z.
The representation of conjugate complex numbersand of inverses is sim- ple in polar coordinates. Since coa(—ço) = cos and sin(—ço) = — sinço, it follows that:
liz =
= Izt(cosw+isinsp), then (2)= =
1z1'e" =
The second equation follows from the first since z1 = 1z12i.
The real polar coordinate mapping
{r E R:r>0} x R —.
Cx, (r,çø) i—+(z,y) := (rcosço,rsinço)is differentiable arbitrarily often in the real domain. We have det
(Zr
= det (c?8 çø —r sin çO = rYcoJ \SIflçO rCO64QJ
and therefore there exists everywhere a real differentiable inverse mapping (which is given by
(x,y)i—'
x)
assuming the appropriate branch of the arccosine function is chosen).
The products and quotients of two complex numbers are therefore ob- tained by respectively multiplying and dividing their absolute values, and respectively adding and subtracting their amplitudes (see Fig. a). The equa- tion (1) is fundamental and far more than simply a convenient calculating rule which makes the use of polar coordinates obviously advantageous in multiplying complex numbers. It is a profound and unexpected justifica- tion for the geometric interpretation of complex numbers in the plane. The mathematical power of this equation was already known to EuLER.15
w— + isin*)
z + iSnQ)
x
Fig. b
Thescalar product (w, z) = Re(wi) takes the well known form (w, z) =
Iwl Izi cos where x := —ço is the "angle between the vectors w and z,"
as in seen by using the equation (1) in the form
= Iwl IzI(cos(%b — cp) + — p))
(see Fig. b). It now becomes clear why the equation lw+z12 = 1w12+1z12+
2Re(wi) was referred to as the cosine theorem in 4.2; since a + x =
(see Fig. b) we have coax = —cosa and hence 1w + z12 = 1w12 + 1z12
2lwllzlcosa.
3. de MOIVRE's Formula.
+ = + isinnçp for nZ. This is clear from
= more generally, we have the followingTheorem. For every compler number z = = r(cos ço + isin E
the eqsalion z" = =
r'(cosnço+
holds/or all n Z.The French huguenot mathematician Abraham DE M0IvRE (1667—1754) emigrated to London after the revocation of the Edict of Nantes in 1685. He became a member of the Royal Society in 1697 and later of the Academies 150n page 154ofCauchy's Cours d'analyseof 1821, we read however the sen- tence, so astounding to modern ears "L'équation cos(a + b) + + b) =
(coss+VCTsino)(cosb+.vCisjnb) elle-même, prise àlalettre, se trouve inex- acts et n'a pas de seas."
wz — IwlI:Kcos(* +
+isin(* + w+z
Fig. a
in Paris and Berlin. His famous book on probability theory, the Dochine of chances was published in 1718; he discovered the well known "Stirling's formula" n! before Stirling; and in 1712 he was appointed by the Royal Society to adjudicate on the merits of the rival claims of NEWTON and LEIBNIZ in the discovery of the infinitesimal calculus. NEWTON in his old age, is said to have replied, when asked about anything mathematical
"Go to Mr. DE MOJVRE; he knows these things better than I do." DE MOIVRE gave the first indication in 1707 of his "magic" formula by means of some numerical examples. By 1730 he seems to have been aware of the general formula
cosço= isinn%o,
n>O.
In 1738 he describes (in a rather long-winded fashion) a procedure for finding roots of the form + ib, which is equivalent as far as content goes, to the formula now known by his name. The formula in the form in which it is now usually expressed is first found in EULER in Chapter VIII of his Introduciio in analysin infinitorum published in 1748. It was also EULER who, in 1749, gave the first valid proof of the formula for all n Z and who stripped DE MOIVRE'S formula of all its mystery by the equation
=
DE MOIVRE'S formula provides a very simple method of expressing cos nw and sin nio as polynomials in and 5mw, for all n 1. Thus for exam- ple, we obtain for n = 3, by separating the real and imaginary parts:
cos3w= cos3w— 3coswsin2w, sin3w= 3cos2wsinw—sin3w.
The trigonometrical representation of the solutions of the quadratic equa- tion z2 + az + b = 0 foreshadowed in 3.5 arises in the following way: we write —46) = r(cos + i sin w) and the roots then take the form
z1
4. Roots of Unity. As one of the most important applications of polar coordinates, we shall demonstrate the following.
Lemma. Let n I be a natural number. Then there are precisely n differ- ent complex numbers z, such that z" = 1, namely
CXI) 2xi In particular (v = where
(
:=Proof. The equations = and = 1 clearly hold (DE M0IvRE). Since c1,ç
=exp—(v—p),
2,riit follows that = if and only if — p) Z because the kernel of p is 2,r7L. Since —n < v — p
< n it follows that ii = p, or in other
words . . are all distinct from each other. For z =
I if and only if IzI =
1 and es" = 1, that is,
if so = with kZ. As 0 <
<it follows that k E (0,1,... ,n —
1), thatz =
Accordingly there are no other complex numbers z, apart fromCa, . . , satisfying the equation = 1. 0
The n numbers 1,(,(2,..
are called the nIh roots of unity. Ge- ometrically, they represent the vertices of a regular n-sided polygon (the figure shows the fifth roots of unity). An nth root of unity is said to be primitive if all the other nth roots can be represented by one of its powers;the root ( is always a primitive nth root, that is, for n = 5
/
The lemma above can be immediately generalized. Writing for
where denotes the positive real nth root of (ci, we have the following:
Existence and
UniquenessTheorem for nth Roots. Every complex
number c = has precisely n different complex nih roots, forevery n E N, n
1, namely the roots ... where( :=
exp
This provides a new proof of the theorem 3.6.
x /
Realization of the many-valuedness of roots gradually developed during the 17th century. For example, the theorem that nth roots have n distinct values was, by 1690, already very familiar to Michael ROLLE (1652—1719), a mathematician who worked in Paris and was a member of the Académie Française. Incidentally R0LLE found the well known theorem in the differ- ential calculus which bears his name in the course of researches into the roots of polynomials, when he observed that between any two neighboring real roots of a real polynomial, there must always lie a root of the first derivative.
The British mathematician Roger COTES (1682—1716) who was a student and then Professor at Cambridge, and a friend of NEWTON, investigated in 1714 the factorizat ion of the polynomials —1 and + aZ' +1 into real quadratic factors, in connection with his researches into the integration of rational functions by the method of decomposition into partial fractions.
He was aware for example of the formula
Z2"-i- 1 =
(Z2_2Zcos2 1+i)
COTES'S results were first published posthumously in 1722 under the title Harmonia mensurarum. It was the desire to round off and improve upon these results which motivat.ed DE M0IvRE among others.