Download Unit 13 Quadratics B Day 1 - Complex Numbers

Unit 13 Quadratics B Day 1 ­ Complex Numbers
Grab a new packet (if you didn't
Friday) and fill in the Unit 13 dates
Mon 4/3
Tues 4/4
Wed 4/5
Thurs 4/6
Mon 4/10
Tues 4/11
Wed 4/12
Thurs 4/13
Unit 13 Lesson 1 Complex Numbers
~I can define the imaginary number, i.
~I can add, subtract and multiply complex
numbers.
~I can graph complex numbers
~I can divide complex numbers and define the
conjugate.
~I can define the imaginary number, i.
~I can simplify imaginary numbers
Unit 13 Quadratics B Day 1 ­ Complex Numbers
Remember the factoring unit???
Factor to find roots, then sketch
2
x - 4x - 5 = 0
Roots: ___ ___
What are the solutions to this equation?
2
x +1=0
Solve
Unit 13 Quadratics B Day 1 ­ Complex Numbers
To express the square root of a negative number,
we use an imaginary unit called i .
i is defined by i 2 = -1 or
i =√ ­1
Numbers that include the real numbers as well
as the square roots of negative numbers are
called complex numbers.
3 + 2i
9 - 4√-1
17
i
A real number can be an integer, fraction, or
irrational, and can be found on a number line.
Real numbers have no imaginary parts.
An imaginary number only has imaginary
components and is written in the form bi .
A complex number might have a real part, an
imaginary part or both and can be written
in the form a + bi .
imaginary
real
4 + 2i
complex
Unit 13 Quadratics B Day 1 ­ Complex Numbers
Complex Numbers
3 + 4i, 2i, 5/4, -3 - i√2
Real Numbers
5/4, √3, -15, 2 + √7
Rational
5/6, -15, 6/11, 1.2
Imaginary Numbers
2i, -i, i√2, 1/5i
Irrational
√3, 2 + √7, e, π
Integers
..., -1, 0, 1, ...
Addition and Subtraction
Addition and subtraction of complex
numbers is similar to combining like terms.
Unit 13 Quadratics B Day 1 ­ Complex Numbers
Multiplying Complex Numbers
To multiply complex numbers, use FOIL or
the box method.
2
Remember that i = __ thus i = ___
3(2 - 4i)
(1 + 2i)(1 + 3i)
2i(5 - i)
(4 + 3i)(4 - 3i)
Multiply the following Observe what happens.
(2 - 4i) (2 + 4i)
(7 + 2i)(7 - 2i)
(-4 + 4i)(-4 - 4i)
Unit 13 Quadratics B Day 1 ­ Complex Numbers
Complex Conjugates
All complex numbers have a conjugate.
a + bi
has the conjugate
a - bi
What is the conjugate of:
-3 + 2i
2i
2 - 4i
-10i
5+i
-7i
*
Division
To divide two complex numbers in the form a + bi , you need
to eliminate the imaginary part of the denominator. An
answer in the form a +bi .
a.
c.
7 + 2i
-2i
2-i
8 - 6i
b.
d.
2 - 5i
3 + 4i
2 - 4i
2 + 4i
Unit 13 Quadratics B Day 1 ­ Complex Numbers
You cannot graph a complex number, such as
3 + 4i, on a real number line, but you CAN
graph it on a complex plane, where the
horizontal axis is thereal axis and the
vertical axis is the imaginary axis.
In the graph at the right,
3 + 4i is located at the
point with coordinates
(3,4). Any complex
number a + bi has (a,b)
as its coordinates on a
complex plane.
Graph on the Complex Plane
2 + 3i
-i
-4 - 2i
-5
Unit 13 Quadratics B Day 1 ­ Complex Numbers
*
We can graph powers of
imaginary numbers as well....
i =
i1
i
2
i =
i2
­1
i0
1
(0,0)
3
i =
­i i3
4
i =
*
Rewrite each power of i as i, -1, -i, or 1.
21
4
(i) =
(i) =
(i)
34
=
(i)
63
i
4
2
=
i =1
i = -1
3
i = -i
2
(i)
(i) =
(i)
104
=
47
6
=
(i) =
Unit 13 Quadratics B Day 1 ­ Complex Numbers
Let's talk to our neighbors...
Tell them two things you learned today and
one thing that you are still puzzled about.
Homework: Complex Numbers Worksheet
HOW ARE YOU DOING???
~I can define the imaginary number, i.
~I can add, subtract and multiply complex numbers.
~I can graph complex numbers
~I can divide complex numbers and define the
conjugate.
~I can define the imaginary number, i.
~I can simplify imaginary numbers