📐Geometric Algebra Unit 3 – Complex Numbers and Quaternions

What's the Deal with Complex Numbers?

• Complex numbers consist of a real part and an imaginary part, in the form $a + bi$
• The imaginary unit $i$ is defined as $i^2 = -1$
  • This allows for solutions to equations that would be impossible with only real numbers
• Complex numbers can represent points on a 2D plane called the complex plane
  • The real part is plotted on the horizontal axis and the imaginary part on the vertical axis
• Operations such as addition, subtraction, multiplication, and division can be performed on complex numbers
• Complex numbers have various applications in fields such as electrical engineering (AC circuits), quantum mechanics, and computer graphics
• The set of complex numbers is denoted as $\mathbb{C}$
• Complex numbers are fundamental in understanding more advanced concepts in mathematics and physics

Imaginary Numbers: Not So Imaginary After All

• Imaginary numbers are based on the imaginary unit $i$, where $i^2 = -1$
• Despite the name, imaginary numbers have real-world applications and are essential in many fields
• The term "imaginary" was coined by René Descartes in the 17th century, but it can be misleading
• Imaginary numbers allow us to solve equations that have no real solutions, such as $x^2 = -1$
• When combined with real numbers, imaginary numbers form complex numbers in the form $a + bi$
• Imaginary numbers are crucial in describing phenomena such as electromagnetic waves and quantum mechanics
• The use of imaginary numbers enables the representation of periodic functions using Euler's formula: $e^{ix} = \cos(x) + i\sin(x)$

Complex Plane: Where Numbers Get a New Dimension

• The complex plane is a 2D representation of complex numbers
  • The horizontal axis represents the real part, and the vertical axis represents the imaginary part
• Each complex number $a + bi$ corresponds to a unique point $(a, b)$ on the complex plane
• The complex plane allows for geometric interpretations of complex number operations
  • Addition of complex numbers corresponds to vector addition on the complex plane
• The distance of a complex number from the origin is called its magnitude or absolute value, denoted as $|a + bi| = \sqrt{a^2 + b^2}$
• The angle formed by the line connecting a complex number to the origin and the positive real axis is called the argument or phase
• The complex plane is divided into four quadrants, similar to the Cartesian coordinate system
• Plotting complex numbers on the complex plane helps visualize their relationships and properties

Quaternions: When Three Dimensions Aren't Enough

• Quaternions are an extension of complex numbers, discovered by William Rowan Hamilton in 1843
• A quaternion consists of a real part and three imaginary parts: $q = a + bi + cj + dk$
  • The imaginary units $i$, $j$, and $k$ have the properties: $i^2 = j^2 = k^2 = ijk = -1$
• Quaternions are used to represent rotations in 3D space more efficiently than Euler angles or rotation matrices
  • They avoid the problem of gimbal lock, which occurs when two rotation axes align
• Quaternions have applications in computer graphics, virtual reality, robotics, and aerospace engineering
• Quaternion multiplication is non-commutative, meaning $q_1q_2 \neq q_2q_1$ in general
• Unit quaternions, also called versors, have a magnitude of 1 and are used to represent rotations
• Quaternions can be interpolated using techniques such as spherical linear interpolation (SLERP) for smooth animations

Operations with Complex Numbers and Quaternions

• Addition and subtraction of complex numbers and quaternions are performed component-wise
  • For complex numbers: $(a + bi) \pm (c + di) = (a \pm c) + (b \pm d)i$
  • For quaternions: $(a + bi + cj + dk) \pm (e + fi + gj + hk) = (a \pm e) + (b \pm f)i + (c \pm g)j + (d \pm h)k$
• Multiplication of complex numbers follows the distributive property and the rule $i^2 = -1$
  • $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$
• Multiplication of quaternions is more involved due to the non-commutative property of the imaginary units
  • The product of two quaternions $q_1 = a + bi + cj + dk$ and $q_2 = e + fi + gj + hk$ is given by:
    • $q_1q_2 = (ae - bf - cg - dh) + (af + be + ch - dg)i + (ag - bh + ce + df)j + (ah + bg - cf + de)k$
• The conjugate of a complex number $a + bi$ is $a - bi$, while the conjugate of a quaternion $a + bi + cj + dk$ is $a - bi - cj - dk$
• Division of complex numbers and quaternions involves multiplying by the reciprocal of the divisor, which is obtained using the conjugate

Real-World Applications: Why This Stuff Matters

• Complex numbers have numerous applications in various fields:
  • Electrical engineering: AC circuit analysis, impedance, and phasors
  • Signal processing: Fourier transforms, filters, and spectral analysis
  • Quantum mechanics: Wave functions, Schrödinger equation, and quantum states
  • Chaos theory: Mandelbrot and Julia sets, fractals, and dynamical systems
• Quaternions are widely used in:
  • Computer graphics: 3D rotations, animations, and character orientations
  • Virtual reality and gaming: Smooth and efficient object rotations, camera control
  • Robotics: Orientation representation, motion planning, and control algorithms
  • Aerospace engineering: Aircraft and spacecraft attitude control, gyroscope data fusion
• Understanding complex numbers and quaternions is essential for advanced mathematics, physics, and engineering courses
• Many modern technologies and innovations rely on the principles and applications of complex numbers and quaternions

Key Formulas and Identities to Remember

• Euler's formula: $e^{ix} = \cos(x) + i\sin(x)$
• De Moivre's formula: $(\cos(\theta) + i\sin(\theta))^n = \cos(n\theta) + i\sin(n\theta)$
• Complex number magnitude: $|a + bi| = \sqrt{a^2 + b^2}$
• Complex number conjugate: $(a + bi)^* = a - bi$
• Quaternion multiplication: $i^2 = j^2 = k^2 = ijk = -1$
• Quaternion conjugate: $(a + bi + cj + dk)^* = a - bi - cj - dk$
• Quaternion magnitude: $|a + bi + cj + dk| = \sqrt{a^2 + b^2 + c^2 + d^2}$
• Quaternion inverse: $q^{-1} = \frac{q^*}{|q|^2}$

Common Pitfalls and How to Avoid Them

• Forgetting the imaginary unit: Always include the imaginary unit $i$ when working with imaginary numbers
• Mishandling the sign of $i^2$: Remember that $i^2 = -1$, not $1$
• Confusing the order of quaternion multiplication: Quaternion multiplication is non-commutative, so the order matters
• Mixing up the components of complex numbers or quaternions: Keep track of the real and imaginary parts separately
• Neglecting the conjugate when dividing complex numbers or quaternions: Use the conjugate to find the reciprocal of the divisor
• Misinterpreting the geometric meaning of complex numbers or quaternions: Visualize them on the complex plane or as 3D rotations, respectively
• Overlooking the applications: Understand how complex numbers and quaternions are used in various fields to appreciate their importance
• Not practicing enough: Solve a variety of problems to reinforce your understanding and avoid common mistakes