**2.4 Complex Numbers**

I. A complex number is written in standard form:

*a*+ b

*i*

where

*a*ε set of real numbers

and

b

*i*is a pure imaginary number

Therefore

*a*+ bi is an imaginary number

√(-1) =

*i*

(√(-1))

^{2}=

*i*

^{2}= -1

(√(-1))

^{3}=

*i*

^{3}= -

*i*

(√(-1))

^{4}= (√(-1))

^{2}(√(-1))

^{2}= (-1)(-1) = 1

(√(-1))

^{0}= 1

Therefore you can see the pattern that:

*i*

^{0}= 1

*i*

^{1}= √(-1) =

*i*

i

i

^{2}= -1

*i*

^{3}= -

*i*

*i*

^{4}= 1

so

*i*

^{5}=

*i*

*i*

^{6}= -1

*i*

^{7}= -

*i*

*i*

^{8}= 1

so what would

*i*

^{63}= ?

*i*

^{60}

*i*

^{3}

*i*

^{60}= 1 and so therefore

*i*

^{60}

*i*

^{3}= (1)

*i*

^{3}= -

*i*

**B. Additive Identity**is zero

what do you add to a number to get that number? of course, zero so

*a*+ b

*i*+ ________ = 0

______ = -

*a*- b

*i*

**C. Adding and Subtracting complex Numbers:**

**Example 1:**(3 + 4i) + (7 + 2i)

= (3 + 7) + (4i + 2i) by grouping like terms

= 10 = 6i

**Example 2:**(3 + 4i) - (7 + 2i)

= (3 - 7) + (4i - 2i)

= (-4) + (2i)

= -4 + 2i

**D. Multiplying Complex Numbers**

**Example 1:**(3 + 4i)(7 + 2i)

= (3)(7) + (3)(2i) + (4i)(7) + (4i)(2i)

= 21 + 6i + 28i + 8i

^{2}

= 21 + 34i + 8(-1)

= 21 + 34i - 8

= 13 + 34i

**E. Dividing Complex Numbers - multiply by the complex conjugate**

Given "a + bi", the complex conjugate would be "a - bi"

Example 1: complex #1

but recall that "a + bi" has to be in standard form so...

29/53 + 22i/53 would be the answer!

**F. Applications:**

1. Fractal Geometry

1. Fractal Geometry

**Mandelbrot Set:**to draw this, consider: Sequence of numbers

c, c

^{2}+ c, (c

^{2}+ c)

^{2 }+ c, [(c

^{2}+ c)

^{2}+ c ]

^{2}+ c, ...

For some values, it is

**Bounded**, which means that all elements in the sequence are less than some fixed number N. Therefore, complex number "c" is in the Mandelbrot Set.

For other values, it is Unbounded, which means that the elements in the sequence become infinitely large. Therefore, complex number "c" is not in the Mandelbrot Set.

Check out this website:

http://mathworld.wolfram.com/MandelbrotSet.html

**Example 1:** Let's let c = -1

So the sequence would be:

-1, (-1)^{2} + (-1) = 0, (0)^{2} + (-1) = -1, (-1)^{2} + (-1) = 0

or -1, 0, -1, 0

so this is bounded!

**Example 2:** Let's let c = - *i*

-*i*, (-*i*)^{2} + (-*i*) = -1 + -*i*, (-1 -*i*)^{2} + (-*i*) = -3*i*, (-3*i*)^{2} + (-*i*) = -9 - *i*

or -*i*, -1 -* i*, -3*i*, - 9 -* i*

so this is not bounded.

2. Impedance - the opposition to current in an electrical circuit.

**Equation of 2 pathways:**

1/z = 1/z_{1} + 1/z_{2}

where

z_{1} is the impedance of pathway 1

z_{2} is the impedance of pathway 2