It says to find 4 different complex number z1,z2,z3,z4 such that each complex number through these 4 properties will produce an integer.
property 1) Re(z) = integer
property 2)Im(z) = integer
property 3) Re(z^(-1)) = integer
property 4)Im(z^(-1)) = integer
where Re and Im denotes the real and imaginary of the complex number respectively.
Please help.
I came up with only 1, i need 3 more.
The one i came up with is
z1=1+1i
let
then
then ----------- (1)
and
--------------- (2)
You need to find integers x and y such that (1) and (2) are also integers. Your solution of z1=1+1i with x = 1 and y = 1 is incorrect since (1) gives .
In fact you can go further and find all the possible solutions to your question.
For x and y integers, must also be an integer. This equation is a circle centered about the origin and only has 4 solutions (along the x and y axes). x = k, y = 0 and x= -k, y=0 etc...
for or , or so one of (1) or (2) will be a fraction. So
or .
HallsofIvy: either |x|= 1 and y= 0 or |y|= 1 and x= 0. Thus giving the four solutions Archie Moore gave.