Complex Numbers

Question

If z is a complex number such that |z| = 1, then the value of (1 + z)/(1 + z̄) (where z̄ is the conjugate of z) lies on

RankGuru · Accepted Answer

1. **Modulus Constraint**: $|z| = 1$ implies $z \cdot \bar{z} = |z|^2 = 1 \implies \bar{z} = 1/z$.
2. **Substitution**: 
   - $\frac{1+z}{1+\bar{z}} = \frac{1+z}{1+1/z}$
   - $= \frac{1+z}{(z+1)/z} = \frac{z(1+z)}{1+z}$
3. **Result**: The expression simplifies exactly to $z$.
4. **Conclusion**: Since the expression equals $z$, and $|z| = 1$, the value lies on the **unit circle** centered at the origin.

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