Download + i 1 - FTHS Wiki

5.7 Complex Numbers
12/4/2013
Quick Review
If a number doesn’t show an exponent, it is understood
that the number has an exponent of 1.
Ex: 8 = 81 ,
x = x1 ,
-5 = -51
Also, any number raised to the Zero power is equal to 1
Ex: 30 = 1 -40 = 1
Exponent Rule:
When multiplying powers with the same base,
you add the exponent.
x2 • x 3 = x 5
y • y7 = y8
The square of any real number x is never negative, so the equation x2 = -1 has
no real number solution.
To solve this x2 = -1 , mathematicians created an expanded system of numbers
i
using the IMAGINARY UNIT, .
i  1
2
i  1
Simplifying i given any powers
io  1
The pattern repeats after every 4.
So you can find i raised to any power by dividing the
exponent by 4 and see what the remainder is. Based
on that remainder, you can determine it’s value.
i1  i
i  1
2
i 3  i 2  i1  1  i  i
i  i  i  1  1  1
4
2
2
i 5  i 4  i1  1  i  i
i 6  i 4  i 2  1  1  1
i 7  i 4  i 3  1  i  i
i8  i 4  i 4  11  1
Do you see the pattern yet?
Ex : i 22
• Step 1. 22÷ 4 has a remainder of 2
• Step 2. i22 = i2
i 22  1
Ex : i 50
• Step 1. 51 ÷ 4 has a remainder of 3
• Step 2. i51 = i3
i 51  i
Checkpoint
1.
i 15
2.
i 20
3.
i 61
4.
i 122
Find the value of
Complex Number Is a number written in the
standard form a + bi
where a is the real part
and bi is the imaginary part.
Ex: 3 + 2i
Adding and
Add/Subtract the real parts,
Subtracting
then add/subtract the
Complex Numbers imaginary parts
Example 2
Add Complex Numbers
Write ( 3 + 2i + ( 1 – i as a complex number in
standard form.
(
(
SOLUTION
( 3 + 2i + ( 1 – i = 3 + 1 + 2i – 1i
(
=4+i
Group real and
imaginary terms.
Write in standard
form.
(
Example 3
Subtract Complex Numbers
Write ( 6 – 2i – ( 1 – 2i as a complex number in
standard form.
(
(
SOLUTION
( 6 – 2i – ( 1 – 2i =
6 – 1 – 2i + 2i
(
(
-1 + 2i
= 5 + 0i
=5
Write in standard
form.
Checkpoint
Add and Subtract Complex Numbers
Write the expression as a complex number in standard
form.
6. ( 4 – 2i + ( 1 + 3i
ANSWER
5 +i
(
ANSWER
5 + 3i
8. ( 4 + 6i – ( 2 + 3i
(
ANSWER
2 + 3i
9. ( – 2 + 4i – ( 2 + 7i
ANSWER
– 4 – 3i
(
(
(
7. ( 3 – i + ( 2 + 4i
(
(
(
Checkpoint
Add and Subtract Complex Numbers
Write the expression as a complex number in standard
form.
(
(
12. ( 2 – i – ( – 1 – 4i
(
11. ( 1 – 2i + ( 4 + 5i
ANSWER
5 + 3i
ANSWER
3 + 3i
(
Example 4
Multiply Complex Numbers
Write the expression as a complex number in standard
form.
remember :
b. (6 + 3i ( 4 – 3i
(
(
(
a. 2i ( – 1 + 3i
i 2  1
SOLUTION
a. 2i ( – 1 + 3i = – 2i + 6i 2
(
= – 6 – 2i
(
= – 2i + 6 ( – 1
Multiply using distributive
property.
Use i 2 = –1.
Write in standard form.
Example 4
Multiply Complex Numbers
b. (6 + 3i ( 4 – 3i = 24 – 18i + 12i – 9i 2 Multiply using FOIL.
(
= 24 – 6i – 9i 2
Simplify.
= 24 – 6i – 9 ( – 1
Use i 2 = –1.
(
= 33 – 6i
Write in standard
form.
(
Homework
WS 5.7 Do problems 13-38 only
Complex
Conjugates
Two complex numbers of the
form a + bi and a - bi
Their product is a real
number because
(3 + 2i)(3 – 2i) using FOIL
9 – 6i + 6i -4i2
9 – 4i2
i2 = -1
9 – 4(-1) = 9 + 4 = 13
Is used to write quotient of
2 complex numbers in
standard form (a + bi)
Example 5
Divide Complex Numbers
3 + 2i
as a complex number in standard form.
1 – 2i
a + bi
SOLUTION
Write
Multiply the numerator and the
denominator by 1 + 2i, the
complex conjugate of 1 – 2i.
3 + 2i
3 + 2i 1 + 2i
•
=
1 – 2i
1 – 2i 1 + 2i
3 + 6i + 2i + 4i 2
=
1 + 2i – 2i – 4i 2
3 + 8i + 4 ( – 1
=
1 – 4( – 1
(
(
=
– 1 + 8i
= –
Multiply using FOIL.
Simplify and use i 2 = – 1.
Simplify.
5
1 8
+ i
5 5
Write in standard form.
Checkpoint
Multiply and Divide Complex Numbers
Write the expression as a complex number in standard
form.
2+i
1– i
ANSWER
1
3
+ i
2
2
Properties of Square Root of Negative
Number
i  1
r 
1  r
r i r
Ex: −5 = 𝑖 5
i 2  1
Example 1
Solve a Quadratic Equation
Solve the equation.
a. 7x 2 = – 49
b. 3x 2 – 5 = – 29
SOLUTION
a. 7x 2 = – 49
x 2 = –7
Divide each side by 7.
x = +
– –7
Take the square root of each side.
x = +
–i 7
Write in terms of i.
Example 1
Solve a Quadratic Equation
b. 3x 2 – 5 = – 29
3x 2 = – 24
x 2 = –8
Write original equation.
Add 5 to each side.
Divide each side by 3.
x = +
– –8
Take the square root of each side.
x = +
–i 8
Write in terms of i.
8 4 22 2
x = +
– 2i 2
Simplify the radical.
Checkpoint
Solve a Quadratic Equation
Solve the equation.
1. x 2 = – 3
ANSWER
i 3, – i 3
2. x 2 = – 20
ANSWER
2 i 5, – 2 i 5
3. x 2 + 3 = – 2
ANSWER
i 5, – i 5
Graphing Complex Number
Imaginary axis
Real axis
Ex: Graph 3 – 2i
To plot, start at the origin,
move 3 units to the right and
2 units down
3
2
3 – 2i
Ex: Name the complex number represented
by the points.
Answers:
D
A is 1 + i
B
B is 0 + 2i = 2i
C is -2 – i
A
D is -2 + 3i
C
i r   i r  i r   i  i 
2
Since
i  i  i  1
2
i r   1r
2
and
or  r
r r
r  r r