8 CHAPTER 1 • COMPLEX N UMBERS- EXAMPLE 1.1 If z 1 = (3, 7) and Z2 = (5, -6), then
z1 + Z2 = (3, 7) + (5, -6) = (8, 1) and
ZJ - Z2 = (3, 7) - (5, -6) = (-2, 13).We can also use the notation z 1 = 3 + 7i and z2 = 5 - 6i:
z1 + z2 = (3 + 7i) + (5 - 6i) = 8 + i and
Zt - Z2 = (3 + 7i) - (5 - 6i) = - 2 + 13i.
Given the rationale we devised for addition and subtraction, it is temptingto define the product z1z 2 as Z1Z2 = (x1x2, Y1!f2). It turns out, however, that
this is not a good definition, and we ask you in the exercises for t his section to
explain why. How, then, should products be defined? Again, if we equate (x, y)with x + iy and assume, for the moment, that i = A makes sense (so that
i^2 = - 1), we have
Z1Z2 = (x1, Y1)(x2, Y2)
= (x1 + iy1)(x2 + iy2)
.. .2
= X1X2 + ix1112 + ix2y1 + t Y1Y2
= X1X2 - Y1Y2 + i(X1Y2 + X2Y1)
= (x1X2 - y1y2, X1Y2 + x2y1).
T hus, it appears that we a.re forced into the following definition.Definition 1.3: Multiplicat ionz1z2 = (x1, y1)(x2, y2)
= (x1X2 -y1y2, X1Y2 + X2Y1).
• EXAMPLE 1.2 If z1 = (3, 7) and z2 = 5 - 6i, then
Z1Z2 = (3, 7)(5, -6)
= (3. 5 - 7. (- 6), 3. (- 6) + 5. 7)