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Complex Numbers


 

Section 1.3

Complex Numbers
 

 

How can we solve the equation x 2= −1?
x = ±

It turns out that this is not a real number !
So we have to "make up " some new numbers!
We define the new number i as follows

i =
 
Complex Numbers

Let i2=-1
then i=

 i is the square root of -1


 

Examples



 

Complex Numbers

A complex number has the form a + bi

a is called the real part of the complex number

b is called the imaginary part of the complex number


 

The Complex Plane
-


 

Adding Complex Numbers

(a + bi ) + (c + di ) = (a + c) + (b + d ) i

To add two complex numbers, add the real
parts and add the imaginary parts, the result
is a complex number.

 

Examples

(2 + 3i ) + (−5 + 4i ) =

(5 + 2i ) − (2 − 7i ) =
 
Multiplying Complex Numbers

(a+bi ) (c+di ) = ac + adi + bci + bdi2
= (ac −bd ) + (ad + bc) i

This follows from the distributive property
and the fact that t2 = −1
 
Examples

(2 + 3i ) (−5 + 4i ) =
(5 + 2i ) (2 − 7i ) =
(a + bi ) (a − bi ) =

( a + bi) and (a − bi ) are called
complex conjugates

 

If  z=a + bi is a complex number,
then =  a -bi is the conjugate of  z.
The product of a complex number
and its conjugate is a real number.


 

Multiply the complex conjugates

(2 + 3i ) (2 − 3i ) =
(5 − 2i ) (5 + 2i ) =
(c + di ) (c − di ) =


 

Dividing Complex Numbers


 

Magnitude of Complex Numbers

If c = a + bi is a complex number,
The absolute value (magnitude) of is
the distance away from 0 0 in the
complex plane.



This follows from the distance formula!


 

The Complex Plane


 

Examples
Find

c = 2 + 3i

c = 5 − 4i

c = −4 + 3i

c = −4 − 3i
 
The Quadratic Formula

 If ax2 + bx + c = 0

then


 

Use the quadratic formula to solve

x2 + 4 = 0

3y2 + 2y +1= 0

2t2− t = −2
 
Analytical Solutions

Given a quadratic equation ax 2 + bx + c = 0
Use the quadratic formula

 If b 2− 4ac > 0 there are two real solutions
If b2 − 4ac = 0 there is one real solution
If  b2 − 4ac< 0there are no real solutions,
but two complex conjugate solutions.
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