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Complex Numbers APPENDIX Complex Numbers The complex Bumbers are a set of objects which can be added and multiplied, the sum and product of two complex numbers being also a complex number, and satisfy the following conditions. (1) Every real number is a complex number, and if a, ß are real numbers, then their sum and product as complex numbers are the same as their sum and product as real numbers. (2) There is a complex number denoted by i such that i2 = - 1. (3) Every complex number can be written uniquely in the form a + bi where a, b are real numbers. (4) The ordinary laws of arithmetic concerning addition and multipli­ cation are satisfied. We list these laws: If a, ß, y are complex numbers, then (aß)y = a(ßy) and (a + ß) + Y = a + (ß + y). We have a(ß + y) = aß + ay, and (ß + y)a = ßa + ya. We have aß = ßa, and a + ß = ß + a. If 1 is the real number one, then 1a = a. If 0 is the real number zero, then Oa = o. We have a + (-l)a = o. We shall now draw consequences of these properties. With each complex number a + bi, we associate the vector (a, b) in the plane. Let a = a 1 + a2 i and ß = b1 + b2 i be two complex numbers. Then 278 COMPLEX NUMBERS [APP. I] Hence addition of complex numbers is carried out "componentwise" and corresponds to addition of vectors in the plane. For example, (2 + 3i) + ( - 1 + 5i) = 1 + 8i. In multiplying complex numbers, we use the rule i2 = - 1 to simplify a product and to put it in the form a + bio For instance, let a = 2 + 3i and ß = 1 - i. Then aß = (2 + 3i)(1 - i) = 2(1 - i) + 3i(1 - i) = 2 - 2i + 3i - 3i 2 = 2 + i - 3( -1) =2+3+i = 5 + i. Let a = a + bi be a complex number. We define ii to be a - bio Thus if a = 2 + 3i, then ii = 2 - 3i. The complex number iX is called the conjugate of a. We see at once that With the vector interpretation of complex numbers, we see that aii is the square of the distance of the point (a, b) from the origin. We now have one more important property of complex numbers, which will allow us to divide by complex numbers other than 0. If a = a + bi is a complex number =I- 0, and if we let then aA = Aa = 1. The proof of this property is an immediate consequence of the law of multiplication of complex numbers, because The number A above is called the inverse of a, and is denoted by a- I or l/a. If a, ß are complex numbers, we often write ß/a instead of a -1 ß (or ßa- 1), just as we did with real numbers. We see that we can divide by complex numbers f= 0. [APP. I] COMPLEX NUMBERS 279 We define the absolute value of a complex number a. = al + ia 2 to be la.I = Jai + a~. This absolute value is none other than the norm of the vector (al' a2 ). In terms of absolute values, we can write provided a. "# o. The triangle inequality for the norm of vectors can now be stated for complex numbers. If a., P are complex numbers, then Ia. + PI ~ la.I + IPI· Another property of the absolute value is given in Exercise 5. Using some elementary facts of analysis, we shall now prove: Theorem. The complex numbers are algebraically closed, in other words, every polynomial fE C[t] of degree ~ 1 has a root in C. Proof We may write f(t) = a"t" + a"_lt"-l + ... + ao with a" "# O. For every real R > 0, the function Ifl such that t ~ If(t)1 is continuous on the closed disc of radius R, and hence has a minimum value on this disco On the other hand, from the expression f (t ) = a"t,,( 1 + --a"-l + ... + -ao ) a"t a"t" we see that when Itl becomes large, then If(t)1 also becomes large, i.e. given C > 0 there exists R > 0 such that if Itl > R then I f(t) I > c. Con­ sequently, there exists a positive number Ro such that, if Zo is a mini­ mum point of Ifl on the closed disc of radius Ro, then If(t)1 ~ I f(zo) I for all complex numbers t. In other words, Zo is an absolute minimum for Ifl. We shall prove that f(zo) = O. 280 COMPLEX NUMBERS [APP. I] We express f in the form with constants Ci' (We did it in the text, but one also sees it by writing t = Zo + (t - zo) and substituting directly in f(t).) If f(zo) i= 0, then Co = f(zo) i= O. Let z = t - zo, and let m be the smallest integer > 0 such that Cm i= O. This integer m exists because f is assumed to have degree ~ 1. Then we can write for some polynomial g, and some polynomial f1 (obtained from f by changing the variable). Let Z1 be a complex number such that and consider values of z of type where A is real, 0 ~ A ~ 1. We have f(t) =f1(AZ1) = Co - AmCo + Am+1z,;,+1g(AZ1) = co[l - 2m + Am+1z';'+1cö1g(AZ1)]. There exists a number C > 0 such that for all A with 0 ~ A ~ 1 we have IZ,;,+1cö 1g(2z1)1 ~ C, and hence If we can now prove that for sufficiently small A with 0 < A < 1 we have then for such A we get I f1(.A,Z1) I < !col, thereby contradicting the hypoth­ esis that If(zo)1 ~ If(t)1 for all complex numbers t. The left inequality is of course obvious since 0< A < 1. The right inequality amounts to CAm+ 1 < 2m, or equivalently CA< 1, which is certainly satisfied for suffi­ ciently small A. This concludes the proof. [APP. I] COMPLEX NUMBERS 281 APP. EXERCISES 1. Express the following eomplex numbers in the form x + iy, where x, y are real numbers. (a) (-1 + 3i)-1 (b) (l + i)(1 - i) (e) (1 + i)i(2 - i) (d) (i - 1)(2 - i) (e) (7 + ni)(n + i) (f) (2i + 1)ni (g) (J2 + i)(n + 3i) (h) (i + 1)(i - 2)(i + 3) 2. Express the following eomplex numbers in the form x + iy, where x, y are real numbers. 1 2 + i 1 (a) (l + i)-I (e) -. (d)-. (b) 3 + i 2 -I 2-1 1 + i i 2i 1 (e) -. (g) -3-. 1 (f) 1 + i -I (h) -1 + i 3. Let IX be a eomplex number # O. What is the absolute value of lXiii? What is ~? 4. Let IX, ß be two eomplex numbers. Show that IXß = iiß and that IX + ß = ii + ß. 5. Show that IIXßI = IIXIIßI. 6. Define addition of n-tuples of eomplex numbers eomponentwise, and multipli­ eation of n-tuples of eomplex numbers by eomplex numbers eomponentwise also. If A = (lXI"" ,IX.) and B = (ßI"" ,ß.) are n-tuples of eomplex numbers, define their produet <A, B) to be (note the eomplex eonjugation!). Prove the following rules: HP 1. <A, B) = <B, A). HP 2. <A, B + C) = <A, B) + <A, C). HP 3. If IX is a comp/ex number, then <IXA, B) = IX<A, B) and <A, IXB) = ii<A, B). HP 4. If A = 0 then <A, A) = 0, and otherwise <A, A) > O. 7. We assume that you know about the funetions sine and eosine, and their addition formulas. Let (J be areal number. (a) Define ei8 = eos (J + i sin (J. Show that if (JI and (J2 are real numbers, then 282 COMPLEX NUMBERS [APP. I] Show that any complex number of absolute value 1 can be written in the form eil for some real number t. (b) Show that any complex number can be written in the form reiB for some real numbers r, () with r ~ o. (c) If Zl = r 1e iBl and Z2 = r 2 e iB2 with real r1, r 2 ~ 0 and real (}1, (}2' show that (d) If z is a complex number, and n an integer > 0, show that there exists a complex number w such that w' = z. If z # 0 show that there exists n dis­ tinct such complex numbers w. [Hint: If z = reiB, consider first rl/neiB/n.] 8. Assuming the complex numbers algebraically closed, prove that every ir­ reducible polynomial over the real numbers has degree 1 or 2. [Hint: Split the polynomial over the complex numbers and pair off complex conjugate roots.] APPENDIX " Iwasawa Decomposition and Others Let SLn denote the set of matrices with determinant 1. The purpose of this appendix is to formu1ate in some general terms results about SLn . We shall use the language of group theory, which has not been used previously, so we have to start with the definition of a group. Let G be a set. We are given a mapping G x G --+ G, which at first we write as a product, i.e. to each pair of elements (x, y) of G we associate an element of G denoted by xy, satisfying the following axioms. GR 1. The product is associative, namely for all x, y, Z E G we have (xY)Z = x(yz). GR 2. There is an element e E G such that ex = xe = x for all x E G.
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