Prove Re(z+1)/(z-1) = 0 for Complex Number Magnitude 1

Published on September 7, 2026

Algebra

Learn how to prove that the real part of the complex expression (z+1)/(z-1) is equal to zero when |z|=1 and z is not equal to 1, using complex conjugates and magnitude properties.

Question

Let z be a complex number such that |z| = 1 and z ≠ 1. Prove that the real part of the expression \(\frac{z+1}{z-1}\) is equal to 0 (meaning the result is purely imaginary).

Answer

To prove that

for

and

, we can use the property that a complex number

is purely imaginary if and only if

Let

. We compute its complex conjugate

Since

, we have

, which implies

. Substituting this into the expression for

Now, we add

and

Since

, we can rewrite the second term:

Because

, it follows that

. Thus, the expression is purely imaginary.

Theorems

• Complex Conjugate Property for Purely Imaginary Numbers

A complex number

is purely imaginary (i.e., its real part is zero) if and only if

• Magnitude of Unit Complex Number

For any complex number

with magnitude

, the conjugate of

is the multiplicative inverse of

, expressed as

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