Going Round in Circles

Or : Multiplying by $i$.

The beginning of complex numbers is often attributed to Cardano (1501 – 1576).  Complex numbers provide a solution to the equation $x^2 = -1$.

They are an extension of the number system from $\Bbb R$ (the real numbers) to $\Bbb C$ (the complex numbers).

Complex numbers take the form :

$z = a + bi$,

where $a,b$ are real numbers and $i = \sqrt -1$.

The normal rules of arithmetic and algebra apply with the additional rule that :

$i^2 = -1$

When two complex numbers are multiplied together the real numbers, $a$ and $b$, act as scalars and the imaginary number, $i$, acts as a rotation about the origin, $0$, (counter-clockwise from the positive real axis). Multiplication by $i$ on its own leads to a rotation of $\pi/2$ radians in the complex plane.

$\begin{align}1\cdot i & =  i & = i\\ i\cdot i & = i^2 & = -1\\ -1\cdot i & = i^3 & = -i\\ -i\cdot i & =  -i^2 & = 1 \end{align}$

Starting at any point on the real axis, (Re), and repeatedly mutilplying by $i$ traces larger or smaller circles.

The multiplication of two complex numbers gives another complex number.

Let :

$z = a + bi$ and $w = c + di$

Then :

$\begin{align} zw & = (a + bi)(c + di)\\ & = ac + bdi^2 + adi + bci\\ & = ac-bd + (ad + bc)i \end{align}$

Taking our starting point as one again, this time we multiply by $1 + i$ repeatedly.

$\begin{align}1(1 + i) & = 1 + i\\ (1 + i)(1 + i) & = 2i\\ 2i(1 + i) & = -2 + 2i\\ (-2 + 2i)(1 + i) & = -4\\ -4(1 + i) & = -4-4i\\ (-4 – 4i)(1 + i) & = -8i\\ (-8i)(1 + i) & = 8-8i\\ (8 – 8i)(1 + i) & = 16 \end{align}$

We still have a rotational locus but now in the form of a spiral.

Each multiplication by $1 + i$ leads to a rotation by $\pi/4$ and an increase in distance from the origin by $\sqrt 2$.

This is because the point $1 + i$ is at a distance $\sqrt 2$ from the origin, $0$, (by Pythagorus) and the angle that this point makes to the positive real axis is arctan $(1/1) = \pi/4$, where arctan is the inverse of the tangent function.

We look at an example.

Let :

$z = 3 + 4i$ and $w = 5 + 2i$

Then :

$zw = 7 + 26i$,

by the above formula.

We can break it down to see what is happening.

$(3 + 4i)5 = 15 + 20i$

The point has been scaled up by a factor of $5$ and the angle is left unchanged.

$(3 + 4i)2i = -8 + 6i$

The point has been scaled up by a factor of $2$ and rotated by an angle of $\pi/2$.

The addition of these two parts gives us the point $7 + 26i$

In terms of distances and angles (the polar representation) :

$z = (r_1, \theta), w = (r_2, \phi)$ and $zw = (r_1r_2, \theta + \phi)$.

$z = 3 + 4i = (5, \theta)$, where $\theta =$arctan$(4/3) \approx 0.9273$ radians $\approx 53.13^\circ$

$w = 5 + 2i = (\sqrt29, \phi)$, where $\phi =$arctan$(2/5) \approx 0.3805$ radians $\approx 21.80^\circ$

$zw = 7 +26i = (5\sqrt29, \theta + \phi)$, where$\theta + \phi =$arctan$(26/7) \approx 1.3078$ radians $\approx 74.93^\circ$

 

Next time we will rotate in three different directions using the quaternions.
 

© OldTrout $2019$