Complex number multiple choice

[alijan kk]

Homework Statement

If Z= (1)/(z conjugate) then Z : ?

Homework Equations

The Attempt at a Solution

let z= a+bi
the z conjugate= a-bi

(a+bi)=(1)/(a-bi)

(a+bi)(a-bi)=1

a²+b²=1

does it tell from this expresssion that the complex number is a pure real ?

 

[DrClaude]

does it tell from this expresssion that the complex number is a pure real ?

What is the condition on $a$ and $b$ for $Z$ to be real?

 

[mjc123]

No. What does a² + b² = 1 mean geometrically? Think of a graph in the complex plane.

 

[alijan kk]

What is the condition on $a$ and $b$ for $Z$ to be real?

b should be zero if the complex number is pure rule
can we put b= 0

 

[DrClaude]

b should be zero if the complex number is pure rule
can we put b= 0

We can set b = 0, but we don't have to, so this is not the answer to the problem.

 

[alijan kk]

We can set b = 0, but we don't have to, so this is not the answer to the problem.

so this a wrong question in my book ?

 

[DrClaude]

so this a wrong question in my book ?

No, the question asks about what you can say about a number for which $z = \bar{z}^{-1}$. See @mjc123's post above for a hint.

 

[alijan kk]

No. What does a² + b² = 1 mean geometrically? Think of a graph in the complex plane.

a unit circle ,

 

[alijan kk]

the inverse is 1/a²+b² of a²+y² and that is a real number ! am i right ?

 

[DrClaude]

$a^2+b^2$ will always be real. That doesn't tell you anything about $z$.

 

[alijan kk]

$a^2+b^2$ will always be real. That doesn't tell you anything about $z$.

options are:
z is purely imaginary
z is any complex number
z is real
none of these

 

[alijan kk]

options are:
z is purely imaginary
z is any complex number
z is real
none of these

if it doesn't tell about z ,, then should the answer be (none of these)

 

[mjc123]

It does tell you something about z, but not one of those options. You got it in post #8 - a unit circle. More specifically, z is any complex number represented by a point in the complex plane that lies on the circumference of a unit circle centered on the origin.
Are you familiar with the polar notation for complex numbers: z = re^iθ, where r² = a²+b² and tanθ = b/a?
Then z* = re^-iθ and 1/z* = (1/r)e^iθ
So z = 1/z* for any number for which r = 1, for any value of θ.