How Complex Numbers fit with other Numbers

Example 1

Example 1 Video

Example 2

Example 2 Video

Example 3

Example 3 Video

Geometric Explanation of i

Real World Uses of Complex Numbers

The Complex Plane

Powers of i

Example 4

Example 4 Video

Example 5

Example 5 Video

Exercise 2.1

Exercise 2.1 Recorded Solutions

Forms of a Complex Number

Example 1

Example 1 Video

Example 2

Example 2 Video

Example 3

Example 3 Video

Conjugate of a Complex Number

Example 4

Example 4 Video

Example 5

Example 5 Video

Example 6

Example 6 Video

Example 7

Example 7 Video

Exercise 2.2

Exercise 2.2 Recorded Solutions

Modulus, Argument and Mod-arg Form

Example 1

Example 1 Video

Example 2

Example 2 Video

Multiplying Complex Numbers in Mod-Arg Form

Example 3

Example 3 Video

Division of Complex Numbers in Mod-Arg Form

Example 4

Example 4 Video

Conjugate in Mod-Arg Form

Example 5

Example 5 Video

Exercise 2.3

Exercise 2.3 Recorded Solutions

Exponential Form

Euler's Formula

Understanding Exponential Form

Proving Euler’s Formula

Proof 1

Proof 1 Video

Euler’s Identity

Proof 2

Proof 2 Video

Example 3

Example 3 Video

Calculations in Exponential Form

Example 4

Example 4 Video

Example 5

Example 5 Video

Exponential Form

Example 6

Example 6 Video

Compare and Convert Between Forms

Example 7

Example 7 Video

Exercise 2.4

Exercise 2.4 Recorded Solutions

Method 1 - Simultaneous Equations

Example 1

Example 1 Video

Method 1a - Simultaneous Equations (Shortcut Method)

Example 2

Example 2 Video

Example 3

Example 3 Video

Method 2 - Formula

Example 4

Example 4 Video

Example 5

Example 5 Video

Exercise 2.5

Exercise 2.5 Recorded Solutions

Product of Conjugates

Example 1

Example 1 Video

Reciprocal of z and its conjugate

Example 2

Example 2 Video

Sum of Conjugates

Example 3

Example 3 Video

Example 4

Example 4 Video

Example 5

Example 5 Video

Difference of Conjugates

Example 6

Example 6 Video

Example 7

Example 7 Video

Operations with Conjugates

Example 8

Example 8 Video

Conjugate Root Theorem

Example 9

Example 9 Video

Exercise 2.6

Exercise 2.6 Recorded Solutions

Complex Numbers as Vectors

Triangle and Parallelogram Laws (Adding Vectors)

Subtracting Vectors

Polygon Law (Adding Vectors)

Example 1

Example 1 Video

Rotation and Dilation

Parallel Vectors

Example 2

Example 2 Video

Perpendicular Vectors

Example 3

Example 3 Video

Rotating by alpha

Example 4

Example 4 Video

Midpoint

Example 5

Example 5 Video

Triangle Inequality

Example 6

Example 6 Video

Exercise 2.7

Exercise 2.7 Recorded Solutions

Curves and Regions

Circles

Example 1

Example 1 Video

Perpendicular Bisector

Example 2

Example 2 Video

Rays and Sectors

Example 3

Example 3 Video

Example 4 (Horizontal and Vertical Lines)

Example 4 Video

Example 5 (Algebraic Method)

Example 5 Video

Arcs

Example 6

Example 6 Video

Exercise 2.8

Exercise 2.8 Recorded Solutions

De Moivre’s Theorem

Example 1

Example 1 Video

Example 2

Example 2 Video

Trigonometrical Applications of De Moivre’s Theorem

Example 3

Example 3 Video

Example 4

Example 4 Video

Solving Equations with Trigonometric Identities

Example 5

Example 5 Video

Exercise 2.9

Exercise 2.9 Recorded Solutions

Complex Roots of Unity

Example 1

Example 1 Video

Example 2

Example 2 Video

Some points to note before we continue

Example 3

Example 3 Video

Complex Roots of -1

Example 4

Example 4 Video

Roots of Other Complex Numbers

Example 5

Example 5 Video

Example 6

Example 6 Video

Exercise 2.10

Exercise 2.10 Recorded Solutions