Why is the angle of $i^2 = \pi$? On the complex plane , the angle of $i =  \pi / 2$ and the angle of  $i^2 = \pi$ .
I understand that by definition $i^2 = -1$ but do not understand how to arrive at angle $\pi$ from $\pi  / 2$ when we square $i$  on the complex plane. Can anybody help?  
 A: You should visualize the complex numbers in the complex plane, that is a number $z = \alpha + \beta i$ has the coordinate $(\alpha,\beta)$ in the complex plane. The angle is then measured from the first axis.
This means in particular that $i = 0+1i$ has the coordinate $(0,1)$, and $i^2 = -1+0i$ has the coordinate $(-1,0)$. These have angles $\frac\pi2$ and $\pi$, respectively.

A: When you multiply complex numbers, their arguments (the angles) add. It is their modules that are  multiplied.
A: Multiplying any complex number by $i$ will rotate it 90 degrees counterclockwise. Multiplying any number by $-1$ will rotate it 180 degrees counterclockwise (because this is not different than multiplying by $i$ twice!).
A: Any complex number $z$ with magnitude $r$ and angle $\theta$ can be represented in polar form as $z = r(\cos(\theta) + i\sin(\theta))$.
To see what happens when you multiply complex numbers, suppose $z = r(\cos(\theta) + i\sin(\theta))$ and $w = s(\cos(\zeta) + i\sin(\zeta))$.
Then $zw = rs[(\cos(\theta) + i\sin(\theta))(\cos(\zeta) + i\sin(\zeta))]
= rs[(\cos(\theta)\cos(\zeta) - \sin(\theta)\sin(\zeta)) + i(\sin(\theta)\cos(\zeta) + \cos(\theta)\sin(\zeta))]
= rs[\cos(\theta + \zeta) + i\sin(\theta + \zeta)]$
So the product of two complex numbers $zw$ has an angle $\theta + \zeta$ equal to the sum of the angles of the individual complex numbers $z$ and $w$ that produce it.
In particular, the angle of $i^2$ is $\pi = \frac{\pi}{2} + \frac{\pi}{2}$.
A: Yes the angle of ${i}^{2}$ is $\pi$ bacause -1 is on the left side of the real axis.
A: $z = \alpha + i\beta$ 
$z(\theta) = x$
$iz(\theta) = x + 90^{\circ}$
$i(\theta) = \dfrac{\pi}{2}$
$i^2(\theta) = \dfrac{\pi}{2} + 90^{\circ} = \dfrac{\pi}{2} + \dfrac{\pi}{2} = \pi$
A: The number of rotations on your complex protractor is written to base $i$  as an exponent at top in terms of $\pi/2 $ radians or 90$^0.$
You have written 2, so it makes $\pi$.  
If you had written $ \frac {2}{\pi} $ that would make for one radian  as,...
$ i^{2/\pi} = 1. $
