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We note that those inequalities may also be written as -lzl < x < lzl and - lzl < Y and lmz < l zl < (1. 6-2) lzl respectively. 6-3) It is clear that Re z lzl holds iff z is real and nonnegative. On the other hand, Imz iy with y > 0. , we have = lz1 Ilz2I are nonnegative. 6-4) where n is any positive integer. 6-5) iff Re(z1 z2) lz1z21, which is true iff z1z2 > 0. This condition is trivially 0 or z2 0. Assuming that z2 f= 0, the condition satisfied if either z1 = = = 21 Complex Numbers 2 lz2l (zi/ z2) > 0, or simply as z1/ z2 > 0.

X+y is uniquely defined and x+y 2. x+ y = y + x. 3. (x+y)+z=x+(y+z). 4. There exists an element 8 E X such that 5. For every x+(-x) 6. For every x+ 8=x for all x E X there exists an element, denoted = 8. x E X and every a E F, the product ax -x, x E X. such that is uniquely defined E X. and ax a(f3x) =(af3)x. (a+f3)x=ax+f3x. 9. a(x+ y)=ax +ay. 10. µx=x, where µ is the 7. 8. unit of F. It is clear that the complex number system C satisfies all of conditions to 10, with F 1 the commutative field of the reals.

3, and we omit them. 2, and 3 follow easily from 20 Chapter 1 Property 4 can be written as Vx2 + Y2 < lxl + IYI which is equivalent to x2 + y2 < x2 + 2lxl IYI + y2 , since both sides of the inequality are nonnegative real numbers. The last inequality reduces to 0 < lxll YI, which is obviously true. For the first inequality in property 5 , we have lxl � < Jx2 + Y2 lzl = = and similarly for the second. We note that those inequalities may also be written as -lzl < x < lzl and - lzl < Y and lmz < l zl < (1.