Complex Numbers

Introduction

Although not covered in the notes on Moodle, this topic may be of interest for those who want to know more about Quaternions. The original Quaternion, which is not the same as the one we are using for 3D rotations, is a mathematical construct using complex numbers. Complex numbers are a combination of real and imaginary numbers. You may have learned that $i=\sqrt{-1}$ . So a complex number could be represented as $a+bi$ .

Complex Number Math

Given two complex numbers $a+bi$ and $c+di$ then $(a+bi)(c+di)=ac+adi+cbi+bdi^2=(ac-bd)(ad+bc)i$ .

However for the quaternion to work two extra imginary numbers were needed. These were defined as $j^2=-1$ and $j^2=-1$ . Using all three imginary numbers we have $ijk=-1$ .

Imaginary Number Math

Given that $ijk=-1$ if both sides of this expression are multipliedn by i we get:
$iijk=-i$
$i^2jk=-i$
$-jk=-i$ or
$jk=i$

The same math can be applied to different combinations of multiplying each side of the original expression, $ijk=-1$ , to get the results shown in the table below:

	i	j	k
i	-1	k	-j
j	-k	-1	i
k	j	-i	-1

Quaternions with Imaginary Numbers

As the origianl Quaternion was a complex number, this section uses that terminology.

Given $q_1=a+bi+cj+dk$ and
$q_2=e+fi+gj+hk$

Then
$q_1q_2=(a+bi+cj+dk)(e+fi+gj+hk)$
$q_1q_2=ae+afi+agj+ahk+bei+bfi^2+bgij+bhik+cej+cfji+cgj^2+chjk+dek+dfki+dgkj+dhk^2$

Substituting using the values of multiplying two imaginary numbers from the table above gives:
$q_1q_2=ae+afi+agi+ahk+bei-bf+bgk-bhj+cej-cfk-cg+chi+dek+dfj-dgi-dh$

Collecting all the like terms to get:
$q_1q_2=[ae+bf-cg-dh,(af+be+ch-dg)i,(ag-bh+ce+df)j,(ah+bg-cf+de)k]$

If the order of the multiplication is reveresed it can be shown that $q_1q_2\neq{q_2q_1}$ . However, the following is true:
$(q_1q_2)q_3=q_1(q_2q_3)$

Rlationship to 3D Rotation Quaternions

Given $q_1=(a,bi,cj,dk)$ then a is the scalar (real) portion of the quaternion or $a\approx{w}$ . Then it can be said that the vector (imaginary) component $(bi,cj,dk)\approx{(x,y,z)}$ .

Exercises & Assignments

There are no exercises or assignments associated with this Addendum.