24 CHAPTER l • COMPLEX NUMBERS• EXAMPLE 1.9 Arg (1 + i) = ~·
Remark 1.1 Clearly, if z = x + iy = r(cos8 + isin8}, where x =f 0, then
arg z c arctan !!. ,
x
where arctan ~ = { 8 : tan 8 = ~}. Note that, as with arg, arctan is a set (as
opposed to Arc tan, which is a number). We specifically identify arg z as a
proper subset of arctan ~ because tan e has period 1r' whereas cos e and sine
have period 2tr. In selecting the proper values for arg z, we must be careful in
specifying the choices of arctan ~ so that the point z associated with r and () lies
in the appropriate quadrant. •
•EXAMPLE 1.10 If z = -J3 - i = r(cosO + isinB), then r = lzl =
l- v'3-ii = 2 and 0 E arctan ~ = arctan _-}a = { ~ + ntr: n is an integer}.
It would be a mistake to use ~ as an acceptable value for 8, as the point z asso-
ciated with r = 2 and e = ~ is in the first quadrant, whereas -v'3 -i is in thethird quadrant. A correct choice for 8 is 8 = ~ -tr = -:". Thus,
- v.:i-ir;;. =^2 cos--+i -^5 tr ·2 sm--. -51r
6 6
(
= 2cos --571" ) ( -571" )
6
- 2n7r + i2sin 6-+ 2n7r ,
where n is any integer. In this case,Arg (-v'3-i) = -;7r, and
arg (-v'3 -i) = {-;7r + 2ntr: n is an integer}.
Note that arg (-v'3 -i) is indeed a proper subset of arctan _-Ja.
- EXAMPLE 1.11 If z = x + iy = 0 + 4i, it would be a mistake to attempt to
find Arg z by looking at arctan ~, as x = 0, so ~ is undefined. If z =f 0 is on the
y-axis , then
7r.
Argz =2
1f Imz > 0, and
Argz = -~ if Imz < O.
2