Introduction to Complex Numbers

Eddie Woo via YouTube

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Provider

YouTube
Pricing

Free Video
Languages

English
Duration & workload

13 hours 47 minutes
Sessions

On-Demand
Level

Intermediate

Found in

Explore the fascinating world of complex numbers through this comprehensive video series that builds from fundamental concepts to advanced applications. Begin with the historical backstory and necessity of complex numbers in algebra, then master basic arithmetic operations including addition, multiplication, conjugates, and division. Discover the geometric interpretation of complex numbers through the complex plane and learn to visualize operations as vector manipulations. Progress to polar (mod-arg) form representation, understanding how to convert between rectangular and polar forms while exploring the relationships between moduli and arguments in products and quotients. Delve into polynomial applications including the Complex Conjugate Root Theorem, linear factorization, and finding square roots of complex numbers. Investigate advanced topics such as Euler's formula and identity (e^iπ = -1), often called "the most beautiful identity in mathematics," through both formal proofs and intuitive explanations using Taylor series and polynomial interpolation. Apply geometric principles to solve complex number problems, work with distance calculations, and explore the parallelogram law for vector addition. Master techniques for manipulating complex expressions to achieve purely real results and tackle challenging problems involving moduli constraints. The series culminates with deep explorations of exponential form, trigonometric identities, and the profound connections between complex analysis, geometry, and calculus that make complex numbers essential tools in advanced mathematics.

Syllabus

Someone asked me a question about Euler's Identity (e^iπ = --1)...
Polynomials w/ Complex Roots (interesting exam question)
Exam Problem: Cubic Polynomial w/ 1 Real Root
Introduction to Complex Numbers (1 of 2: The Backstory)
Complex Arithmetic (1 of 2: Addition & Multiplication)
Complex Arithmetic (2 of 2: Conjugates & Division)
Introduction to Complex Numbers (2 of 2: Why Algebra Requires Complex Numbers)
Who cares about complex numbers??
Linear Factorisation of Polynomials (1 of 2: Working in the Complex Field)
Square Roots of Complex Numbers (1 of 2: Establishing their nature)
Square Roots of Complex Numbers (2 of 2: Introductory example)
Linear Factorisation of Polynomials (2 of 2: Introductory example)
Complex Numbers - Mod-Arg Form (1 of 5: Introduction)
Complex Numbers - Mod-Arg Form (2 of 5: Visualising Modulus & Argument)
Complex Numbers - Mod-Arg Form (3 of 5: Calculating the Modulus)
Complex Numbers - Mod-Arg Form (4 of 5: Conversion Example 1)
Complex Numbers - Mod-Arg Form (5 of 5: Conversion Example 2)
Multiplying Complex Numbers in Mod-Arg Form (1 of 2: Reconsidering powers of i)
Multiplying Complex Numbers in Mod-Arg Form (2 of 2: Generalising the pattern)
Relationships Between Moduli & Arguments in Products of Complex Numbers
Powers of a Complex Number (example question)
Understanding Complex Quotients & Conjugates in Mod-Arg Form
Manipulating Complex Numbers for Purely Real Results
Complex Numbers Question (Finding the greatest value of |z| if |z-4/z|=2)
Complex Conjugate Root Theorem (Formal Proof)
Complex Conjugate Root Theorem (1 of 2: Using the Conjugate Root Theorem to solve Polynomials)
Complex Conjugate Root Theorem (2 of 2: Alternatively solving with Long Division)
Extension II Assessment Review (1 of 5: Multiple Choice Questions Section)
Extension II Assessment Review (2 of 5: Complex Conjugate Root Theorem, DMT & Geometry)
The Most Beautiful Identity (1 of 8: Introducing Complex Numbers)
The Most Beautiful Identity (2 of 8: Same number, different clothes)
The Most Beautiful Identity (3 of 8: The Complex Plane)
The Most Beautiful Identity (4 of 8: Polar Form)
The Most Beautiful Identity (5 of 8: Polynomial Interpolation)
The Most Beautiful Identity (6 of 8: Taylor Series)
The Most Beautiful Identity (7 of 8: Revisiting Polar Form)
The Most Beautiful Identity (8 of 8: Conclusion)
Square Roots of Complex Numbers
Why √a√b isn't always equal to √ab
Geometry of Complex Numbers (1 of 6: Radians)
Geometry of Complex Numbers (2 of 6: Real vs. Complex)
Geometry of Complex Numbers (3 of 6: Real Arithmetic)
Geometry of Complex Numbers (4 of 6: The Complex Plane)
Geometry of Complex Numbers (5 of 6: Polar Form)
Geometry of Complex Numbers (6 of 6: Conversion Between Forms)
Polar Form (1 of 2: Using Complex Number examples to justify polar form's use)
Polar Form (2 of 2: Generalising to prove the multiplication identity of polar form)
Parallelogram Law (Geometrically representing the addition of complex numbers with vectors)
Solutions of (1+i)z² - z - i = 0
Graphing with Complex Numbers (1 of 3: Initial algebraic expansion)
Graphing with Complex Numbers (2 of 3: Determining the region)
Graphing with Complex Numbers (3 of 3: Is |z₁z₂| equal to |z₁| × |z₂|?)
Introducing Complex Numbers (1 of 3: History of numbers)
Introducing Complex Numbers (2 of 3: Revealing the invisible)
Introducing Complex Numbers (3 of 3: Defining fundamentals)
Complex Arithmetic (1 of 3: Basic operations)
Complex Arithmetic (2 of 3: Trigonometric identity)
Complex Arithmetic (3 of 3: Identity from ℝ component)
Further Complex Arithmetic (1 of 2: Basic questions and notation)
Further Complex Arithmetic (2 of 2: Equating components)
Introducing the Complex Plane
Exploring the Complex Plane (1 of 2: Visualising addition & subtraction)
Exploring the Complex Plane (2 of 2: Visualising multiplication)
Basics of Complex Geometry (example questions)
Introducing Polar Form (1 of 3: An alternative coordinate system)
Introducing Polar Form (2 of 3: Relationship to rectangular form)
Introducing Polar Form (3 of 3: Example conversion)
Working in Polar Form (1 of 2: Conjugate & negative)
Working in Polar Form (2 of 2: Square & reciprocal)
Distance between complex numbers
Exact value of cos(π÷12)
Quotient of Complex Numbers (1 of 2: Evaluating modulus)
Quotient of Complex Numbers (2 of 2: Evaluating argument)
Working with Moduli and Arguments (Proof Question)
Argument of the Complex Conjugate
Square Roots of a Complex Number (3 of 3: Solving in rectangular and exponential form)
Square Roots of a Complex Number (2 of 3: Principal square root)
Square Roots of a Complex Number (1 of 3: Review questions)
Proving Euler's Formula (4 of 4: Evaluating constants)
Proving Euler's Formula (3 of 4: Equating terms)
Proving Euler's Formula (2 of 4: Differentiating both sides)
Proving Euler's Formula (1 of 4: Fields)
What does an imaginary power mean?
Informal Proof of Euler's Formula (2 of 2: Trigonometric calculus)
Informal Proof of Euler's Formula (1 of 2: Exponential calculus)