Complex Numbers Roots of Unity By multiplying two roots, one is the conjugate of the other,  we get one.
Does someone know why and proof that.
Many thanks
 A: Since $|z|=1$, and $z\overline z=|z|^2=1$, we get that $\overline z=1/z$.
A: If they are conjugates, their product is a nonnegative real number.
If they are roots of unity, their moduli are also unity.
So their product is a nonnegative real number of unit modulus, i.e., $1$.
A: If $z$ is a root of unity, then $z^n=1$ for some $n$.
Then $1=|1|=|z^n|=|z|^n$, which implies $|z|=1$ because $|z|\ge0$.
Finally, $z\bar z=|z|^2=1$.
A: Here's a proof from a different angle, on the principle that it's always good to see things from multiple angles (as it were).  Let $z = a+bi$, then $\overline{z} = a-bi$, and
$$
z\overline{z} = (a+bi)(a-bi) = a^2-(bi)^2 = a^2-b^2(i^2) = a^2+b^2
$$
But since $z$ is a root of unity, it must have modulus $1$; that is, in the complex plane, it must lie a distance $1$ from the origin.  By the Pythagorean theorem, $a^2+b^2$ is that distance for $z$; therefore $z\overline{z} = a^2+b^2 = 1$.
A: A root of unity is a complex number of modulus equal to $1$, so you can write it as $e^{i\theta}$; its conjugate is then $e^{-i\theta}$ from which you get $e^{i\theta}e^{-i\theta}=e^{i(\theta-\theta)}=e^0=1$.
