|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 is the square root of -1
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.
(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
(2 + 3i ) (−5 + 4i ) =
(5 + 2i ) (2 − 7i ) =
(a + bi ) (a − bi ) =
( a + bi) and (a − bi ) are called
|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
This follows from the distance formula!
|The Complex Plane
c = 2 + 3i
c = 5 − 4i
c = −4 + 3i
c = −4 − 3i
|The Quadratic Formula
If ax2 + bx + c = 0
|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.