Thread: Need help with a question on complex number

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

Originally Posted by leonheart1234

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

This question was already solved some 1 week ago.

Tonio

Yes,I know its similar.But the notation is different.
In the other post it says Re(z-1) and Im(z-1)
Mine is Re(z^(-1)) and Im(z^(-1)).

Originally Posted by leonheart1234

Yes,I know its similar.But the notation is different.
In the other post it says Re(z-1) and Im(z-1)
Mine is Re(z^(-1)) and Im(z^(-1)).

Well, it's pretty similar, and if you did the past question and understood it then

it should be clear what to do in this case.

If , what is then ? After you answer this then try your problem.

Tonio

Originally Posted by leonheart1234

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

Try:

Why do they work?

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.

Originally Posted by Krahl

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 .

You should say a bit more: either |x|= 1 and y= 0 or |y|= 1 and x= 0. Thus giving the four solutions Archie Moore gave.