Where are the points located in the complex plane for which…

Where are the points located in the complex plane for which $z\bar z = 1$?

Step 1 · Apply $z\bar z = |z|^2$

$$ z\bar z = 1 \quad\Longleftrightarrow\quad |z|^2 = 1 \quad\Longleftrightarrow\quad |z| = 1 $$

Step 2 · Geometric interpretation

$|z|=1$ says every such point is at distance exactly $1$ from the origin — the defining property of a circle of radius $1$ centered at the origin, i.e. the unit circle.

Step 3 · Cartesian equation

With $z=x+iy$: $$ x^2+y^2 = 1 $$ the familiar equation of a circle centered at $(0,0)$ with radius $1$.

Step 4 · Polar (parametric) form

Every point on the unit circle can be written as:

$$ z = e^{i\theta} = \cos\theta+i\sin\theta, \qquad \theta \in [0,2\pi) $$

and as $\theta$ sweeps through $[0,2\pi)$, $z(\theta)$ traces the entire circle exactly once.

Step 5 · Check with examples

$z=1$: $z\bar z = 1\cdot1=1$. ✓

$z=i$: $z\bar z = i\cdot(-i) = -i^2 = 1$. ✓

$z=\dfrac1{\sqrt2}+i\dfrac1{\sqrt2}$: $|z|^2 = \dfrac12+\dfrac12=1$. ✓

$z=\dfrac{\sqrt3}{2}-i\dfrac12$: $|z|^2 = \dfrac34+\dfrac14=1$. ✓

Step 6 · Special properties on the unit circle

Reciprocal equals conjugate: if $|z|=1$, then $\dfrac1z = \dfrac{\bar z}{z\bar z} = \dfrac{\bar z}{1} = \bar z$.

Closed under multiplication: if $|z_1|=|z_2|=1$, then $|z_1z_2|=|z_1||z_2|=1$, so the product stays on the unit circle.

Closed under conjugation: if $|z|=1$, then $|\bar z|=|z|=1$.