Complex Number Question Urgent (1 Viewer)

[theprodigy]

far out.. this questions got me wrappedround.. and i think its really simple too


If Z is a complex number such that Z + 1/z is real, prove that either Im(z) = 0 or |z| =1

 

[jules.09]

Make z = x+iy

Then sub into z + 1/z.

Then multiply 1/(x+iy) by its conjugate (x-iy).

Then split into real and imaginary parts. It's been stated that it's real? So you make the imaginary part = 0.

i.e. (x+iy) + (x-iy)/(x^2 + y^2)

x/(x^2 + y^2) + x + i[y - y/(x^2 + y^2)]

y - y/(x^2 + y^2) = 0.

Which means that x^2 + y^2 = 1. When you get the square root of both sides, you end up with |z| = 1.

 

[cutemouse]

Notice that there's a capital 'Z', which is different from lower case 'z'.

So I don't think your answer is correct.

 

[alakazimmy]

far out.. this questions got me wrappedround.. and i think its really simple too


If Z is a complex number such that Z + 1/z is real, prove that either Im(z) = 0 or |z| =1

Okay, we have Z is complex, Z + 1/z is real.

That means Im(Z + 1/z) = 0

The imaginary part of a complex number, say x, is (x-conjugate(x))/2i
For x = a + bi.
x - conjugate(x) = a + bi - (a - bi) = 2bi, and 2bi/2i = b, which is the imaginary part of x.
Apply this, by letting x = Z + 1/z

So Im(Z + 1/z) = ((Z + 1/z) - conjugate(Z + 1/z))/2i = 0. (We can ignore the denominator of 2i)
(Z + 1/z) - conjugate(Z) - conjugate(1/z) = 0
(Z - conjugate (Z)) - (1/z - conjuagte(1/z)) = 0
2i*Im(Z) - (2i*Im(z)/|z|²)=0

This question has infinite solutions, as they are 2 variables, and only one equation restricting them. I think both the z's are the same case.

Here's a scenario which contradicts the question.
Assume Im(z) =0. Then 1/z is obviously real. But Z is complex. Hence Z+1/z cannot be real.
Assume |z|=1, 1/z = conjugate(z)/|z|² = conjugate(z), which is complex.

 

[alakazimmy]

Okay, for both z's the same case, the question seems to be correct.

Im(x) = (x - conjugate(x))/2i

Im(z + 1/z) = 0
So, ((z + 1/z) - conjugate(z + 1/z))=0
(z - conjugate(z)) + (1/z - conjugate(1/z))=0
2i*Im(z) + (conjugate(z) - z)/|z|² = 0
2i*Im(z) - 2i*Im(z)/|z|² = 0
2i*Im(z)*(1 - 1/|z|²) = 0

Either Im(z) = 0, or (1 - 1/|z|²)
Therefore, Im(z) = 0, or |z|=1

Done