Complex Numbers

Before we can fully understand quaterions, we must first understand where they came from. The root of quaternions is based on the concept of the complex number system.

In addition to the well-known number sets (Natural, Integer, Real, and Rational), the Complex Number system introduces a new set of numbers called imaginary numbers. Imaginary numbers were invented to solve certain equations that had no solutions such as:

To solve this expression, we must state that  which we know is not possible because the square of any number (positive or negative) is always positive.

Mathematicians generally can’t accept that an expression does not have a solution so a new term was invented called the imaginary number that can be used to solve such equations.

The imaginary number has the form:

Don’t try to actually understand this term as there is no logical reason why it exists. We just have to accept that  is just something that squares to .

The set of imaginary numbers can be represented by .

The set of complex numbers (represented by the symbol ) is the sum of a real number and an imaginary number and has the form:

It could also be stated that all Real numbers are complex numbers with  and all imaginary numbers are complex numbers with .

Adding and Subtracting Complex Numbers

Complex numbers can be added and subtracted by adding or subtracting the real, and imaginary parts.

Addition:

Subtraction:

Multiply a Complex Number by a Scalar

A complex number is multiplied by a scalar by multiplying each term of the complex number by the scalar:

Product of Complex Numbers

Complex numbers can also be multiplied by applying normal algebraic rules.

Square of Complex Numbers

A complex number can also be squared by multiplying by itself:

Complex Conjugate

The conjugate of a complex number is a complex number with the imaginary part negated and is denoted as either  or .

The product of a complex number and its conjugate gives a special result.

Absolute Value of a Complex Number

We can use the conjugate of a complex number to compute the absolute value (ornorm, or magnitude) of a complex number. The absolute value of a complex number is the square-root of the complex number multiplied by its conjugate and is denoted :

Quotient of Two Complex Numbers

To compute the quotient of two complex numbers, we multiply the numerator and denominator by the complex conjugate of the denominator.

Powers of i

If we state that  then it should be possible to raise i to other powers as well.

If we keep writing this sequence, we will see a pattern emerge .

A similar pattern emerges from the increasing negative powers.

You may have seen a similar pattern in mathematics before but in the form  which is generated by rotating a point 90 degrees counter-clockwise on a 2D Cartesian plane and the sequence  is generated by rotating a point 90 degrees clockwise on a 2D Cartesian plane.

Cartesian Plane

The Complex Plane

We can also map complex numbers in a 2D grid called the Complex Plane by mapping the Real part on the horizontal axis and the Imaginary part on the vertical axis.

Complex Plane

As shown in the previous sequence, we can say that if we multiply a complex number by i, we can rotate the complex number through the complex plane at 90 degree increments.

Let’s see if this is true. We’ll take an arbitrary point p in the complex plane:

and we multiply it by i gives q:

Multiplying q by i gives r:

And multiplying r by i gives s:

And multiplying s by i gives t:

Which is exactly what we started with (p). If we plot these complex numbers on the complex plane, we get the following result.

Complex Numbers on the Complex Plane

We can also rotate clock-wise in the complex plane by multiplying the complex number by -i.

Rotors

We can also perform arbitrary rotations in the complex plane by defining a complex number of the form:

Multiplying any complex number by the rotor q produces the general formula:

Which can also be written in matrix form:

Which is the method to rotate an arbitrary point in the complex plane counter-clockwise about the origin.

Refrences:

• http://3dgep.com/understanding-quaternions/