Complex numbers are expressions of form where are real numbers and is a new symbol.

Just as real numbers are points on a line, complex numbers are points on a plane

A complex number of the form is purely imaginary.

A complex number of the form is called real.

Complex numbers form an algebraic system. Basic vector addition and scalar multiplication are the same as usual on the plane

The new feature here is that we can multiply two complex numbers using the rule .

Just as the set of all real numbers is denoted , the set of all complex numbers is denoted .

Historical Origin of Complex Numbers

Most people think that complex numbers arose from attempts to solve quadratic equations, but acutally they first appeard in connection with cubic equations. Everyone knew that certain quadratic equations like,

had no solutions. The problem was certain for with cubic equations, for example

The equation was known to have three real roots, given by simple combination of the expression,

One of the roots is , it may not look like a real number, but it turns out to be one.

Note: The word complex does not mean complicated instead refers to a complex of real numbers.

Writing the complex numbers pictured from

Complex Number Concept

Is a real or a complex number?

Both, because can be identified with

Complex Number on a Complex Plane

Location: \({}\)

Angle: \({}\)

Operation on Complex Number

Real Part

Imaginary Part

Complex Conjugate

Note:

is a real number, it’s not .

Finding Complex Conjugate

To find the complex conjugate, negate the imaginary componnets. This is equivalent to reflecting the complex numbers across the real axis.

Complex Numbers can be

Added

Subtracted

Multiplied

Divided

Addition

Addition Visualization

Red arrow is the .

Multiplication

The only new operation is that we can now multiply two complex numbers using the rule,

Multiplication by a real number

Multiplication Visualization

Multiplication by a scalar number, enter the number to be multiplied with.

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