Define: Complex Numbers

Define:Quaternions

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.

Adding and Subtracting Complex Numbers

Multiplying a Complex Number by a Scalar

Product of Complex Numbers

Square of Complex Numbers

Complex Conjugate

Absolute Value of a Complex Number

Quotient of Two Complex Numbers

Powers of i

The Complex Plane

Rotors

Quaternions

With this knowledge of the complex number system and the complex plane, we can extend this to 3-dimensional space by adding two imaginary numbers to our number system in addition to i.

...Hamilton also recognized that the i, j, and k imaginary numbers could be used to represent three cartesian unit vectors i, j, and k with the same properties of imaginary numbers, such that square(i) = square(j) = square(k) = -1..

• Quaternions as an Ordered Pair
• Adding and Subtracting Quaternions
• Quaternion Products
• A Real Quaternion
• Multiplying a Quaternion by a Scalar
• Pure Quaternions
• Additive Form of a Quaternion
• Unit Quaternion

Given an arbitrary vector v, we can express this vector in both its scalar magnitude and its direction.And we can also describe a unit quaternion that has a zero scalar and a unit vector.

• Binary Form of a Quaternion

We can now combine the definitions of the unit quaternion and the additive form of a quaternion, we can create a representation of quaternions which is similar to the notation used to describe complex numbers..

This gives us a way to represent the quaternion that is very similar to complex numbers..

• Quaternion Conjugate

The quaternion conjugate can be computed by negating the vector part of the quaternion..

• Quaternion Norm
• Quaternion Normalization

With the definition of a quaternion norm, we can use it to normalize a quaternion. A quaternion is normalized by dividing it by magnitude.

• Quaternion Inverse

...To compute the inverse of a quaternion, we take the conjugate of the quaternion and divide it by the square of the norm..

Inverse(q) = Conjugate(q) / Square(Magnitude(q))

To show this, we can take the fact that by definition of the inverse:

(q) Inverse(q) = [1, Vector(0)] = 1

And multiply both sides by the conjugate of the quaternion gives:

Conjugate(q) (q) Inverse(q) = Conjugate(q)

And by substitution we get:

Square(Magnitude(q)) Inverse(q) = Conjugate(q)

Inverse(q) = Conjugate(q) / Square(Magnitude(q))

And for unit-norm quaternions whose norm is 1, we can write:

Inverse(q) = Conjugate(q)

• Quaternion Dot Product

Similar to vector dot-product, we can also compute the dot product between two quaternions by multiplying the corresponding scalar parts and summing the results..

Quat(q1) = [S1, X1i + Y1j + Z1k]

Quat(q2) = [S2, X2i + Y2j + Z2k]

Quat(q1) Dot-Product Quat(q2) = S1S2 + X1X2 + Y1Y2 + Z1Z2

we can also use the quaternion dot-product to compute the angular difference between the quaternions.

And for unit-norm quaternions, we can simplify the equation:

Cosine( ) = S1S2 + X1X2 + Y1Y2 + Z1Z2

• Rotations
• Quaternion Interpolation
• SLERP(Spherical Linear Interpolation)
• SQUAD(Spherical and Quadrangle)

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quotient [ˈkwoʊʃənt]
1.the number that is the result of dividing one number by another

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