## Complex Numbers: Planar Rotations

The algebras of this chapter are called hyper-complex numbers, referring to the fact that they are extensions of the complex numbers. This section is a review of the important aspects of complex numbers. The most relevant of which is their ability to represent rotations in a plane.

The complex numbers are usually written as $z=a+bi$, where $a$ and $b$ are elements of any field (i.e. most generalizations of the rational numbers) and $i^2=-1$.

$(a+bi)+(c+di)=(a+c)+(b+d)i$
$(a+bi)(c+di)=(ac-bd)+(ab+bc)i$
$(a+bi)^*=a-bi$

This defines a field, and is algebraically complete over the real numbers. This means that any polynomial of degree n as n complex roots, counting multiplicity.

Further, the complex number have a norm over the original field, $N(z)=zz^*=a^2+b^2$ such that:

$N(x)N(y)=N(xy)$

Further, this value is positive definite (never negative). Usually it is common to discuss the square root of this quantity, called $abs(z)=|z|=\sqrt{N(z)}$.

The norm 1 complex numbers are closed over multiplication. Any complex number can be divided by its absolute value to determine a unit vector representing its direction. This leads to the polar notation of the complex numbers.

$r e^{i\theta}$

We also have $\tan(\theta)=\frac{b}{a}$, $a=r\cos\theta$ and $b=r\sin\theta$. The ancients used a trigonometry notation different from ours. Where we use sine and cosine they used r times these values. Further, angles were expressed by their tangents.

The angle addition formulas of trigonometry can now be seen to be the exactly the multiplication of the complex numbers. One could even picture ancient trigonometry being equivalent to the algebra of the complex numbers.

We can rotate a plane $(x,y)$ by the angle of $u=\cos\theta-i\sin\theta$, as follows:

$(x+yi)u=x'+y'i$

We also note that the principle square root of $u$ is the unit complex number of half the angle of $u$. Further, the complex numbers are commutative.

To pave the way for future discussion we present an alternate form of the above, using $v^2=u$.

$v(x+yi)v=x'+y'i$