14 ·Chapter^1Letting a+ bi= A, 2c = C, the foregoing equation can be written as
Az+Az+C= o
where A i= 0 is a complex constant and C is a real constant. This is the
complex form of the equation of the straight line.
In a similar way, it is easily seen that the equation of a circle
x^2 + y^2 + ax + by + c = 0
has the following equivalent representation in the complex form:zz + Az + Az + C = 0
with A a complex constant and C a real constant.
The numbers z and z are called the complex conjugate coordinates, or
simply the conjugate coordinates corresponding to the point (x, y). Also,
they have been called the isotropic coordinates and the minimal coordinates
of the point. As will be seen later, for many purposes the system (z, z) is
more convenient and elegant than the system (x, y).
In matrix form the transformation from real to conjugate coordinates
is given by
Exercises 1.1
- Perform the following op@rEttioi;:iil.
(a) (2,3) + (4,-2)
(c) (2,0)-(1,-5)
(b) ci. 0) + (2, 4v'3)
(d) (5,1/1) - (-2, %)
- Perform the following gperations.
(a) (3,2)(1,4) -. (b) (-1 1 3)(2,-5)
(c) i-1,2) (d) (1/ 2 ~3)
(3,1) (41=6)- Perform the indicated Qperations.
(a) (2-3i)+(4+5i) (b) (6+5i)-(-2+3i)
(c)(8-i)(4+3i) (d) l-
2
~1 +Si
- Show that:
(a) i^3 = -i (b) i^4 = 1
(c) i^4 n = 1, i^4 n+l = i, i4'n+2 = -1, i^4 n+^3 ;:::: -i
d
1 + i + i^3 + i^5
( ) 1 + i^2 + i4 + i6 + i8 ::;:::^1 + i- Prove that addition and multiplication of complex numbers are
comm.utative and associii,tiv~.