12 Feb 2017

# Complex Numbers

Complex numbers are expressions of form $a+bi$ where $a,b$ are real numbers and $i$ 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 $0+ib$ is purely imaginary.
• A complex number of the form $a+0i$ is called real.

Complex numbers form an algebraic system. Basic vector addition and scalar multiplication are the same as usual on the plane $(\mathbb R^{2})$

The new feature here is that we can multiply two complex numbers using the rule $i^{2}-1$.

Just as the set of all real numbers is denoted $\mathbb R$, the set of all complex numbers is denoted $\mathbb C$.

# 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 $x^{3} - 6x - 2=0$

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

One of the roots is $A+B$, 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 $a+bi$

## Complex Number Concept

Is $9$ a real or a complex number?

Both, because $9$ can be identified with $9+0i$

## Complex Number on a Complex Plane

Location: ${}$
Angle: ${}$

## Operation on Complex Number

### Note:

$Im(a+bi) = b$ is a real number, it’s not $bi$.

# 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

• Subtracted
• Multiplied
• Divided

Red arrow is the $A+B$.

## Multiplication

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

### Multiplication Visualization

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

1. Multiplying $2+3i$ and $1-5i$

### … To be continued

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Mathematics Complex-Analysis