# Operations on the Complex Plane

Related Topics:
Common Core (The Complex Number System)
Common Core for Mathematics

Examples, solutions, and lessons to help High School students learn how to represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.

For example, (-1 + √3 i)3 = 8 because (-1 + √3i) has modulus 2 and argument 120°.

### Suggested Learning Targets

• I can represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane.
• I can use formulas to multiply and divide complex numbers in polar form.
• I can use DeMoivre's Theorem to raise a complex number to a power.
Common Core: HSN-CN.B.5

### Operations on the Complex Plane

The Complex Plane
This video describes the Complex plane and uses four different Complex operations to show how the functions can be interpreted geometrically on the Complex plane. These operations include taking a conjugate, multiplying a complex number by i, absolute value, and addition of complex numbers. The Triangle Inequality Theorem is also discussed from the point of view of complex numbers.
Complex Numbers: Representation and Operations - Complex Plane
Methods of representing complex numbers: Argand diagram, a + bi, vectors, polar coordinates. Operations of addition, subtraction, and multiplication, FOIL method. Multiplying Complex Numbers
Here are two different ways to multiply complex numbers, one using rectangular notation and one using polar notation. They are equivalent so they both give the same answers. It also shows where the method that uses polar notation comes from. Complex Analysis Lecture 4—Geometry of complex numbers

### Multiply and Divide Complex Numbers

Complex Numbers: Multiplying and Dividing in Polar Form, Ex 1
It gives the formula for multiplication and division of two complex numbers that are in polar form. Complex Numbers: Multiplying and Dividing in Polar Form, Ex 2
It gives the formula for multiplication and division of two complex numbers that are in polar form. The Product and Quotient of Complex Numbers in Trigonometric Form
This video explains how to find the product and quotient of complex number is trigonometric form.

### Demoivre's Theorem

Raise a Complex Number in Polar Form to a Power - Demoivre's Theorem
This video explains how to use Demoivre's Theorem to raise a complex number in polar form to a power. DeMoivre's Theorem: Raising a Complex Number to a Power, Ex 1
This video gives DeMoivre's theorem and use it to raise a complex number to a power. Note that our number must be in polar form! DeMoivre's Theorem: Raising a Complex Number to a Power, Ex 2
This video gives DeMoivre's theorem and use it to raise a complex number to a power. Note that our number must be in polar form! DeMoivre's Theorem: Raising a Complex Number to a Power, Ex 3
This video gives DeMoivre's theorem and use it to raise a complex number to a power. Note that our number must be in polar form!

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