Download Algebra 2 - peacock

Algebra 2
Complex Numbers
Lesson 4-8 Part 1
Goals
Goal
• To identify, graph, and
perform operations with
complex numbers.
Rubric
Level 1 – Know the goals.
Level 2 – Fully understand the
goals.
Level 3 – Use the goals to
solve simple problems.
Level 4 – Use the goals to
solve more advanced problems.
Level 5 – Adapts and applies
the goals to different and more
complex problems.
Vocabulary
•
•
•
•
•
•
•
Imaginary Unit
Imaginary Number
Complex Number
Pure Imaginary Number
Complex Number Plane
Absolute Value of a Complex Number
Complex Conjugate
Essential Question
Big Idea: Solving Equations
1. What are complex numbers?
2. How do you perform basic operations on
complex numbers?
What is a imaginary
number?
• It is a tool to solve an equation.
– Invented to solve quadratic equations of the
form x 2  4 .
• It has been used to solve equations for the
last 200 years or so.
• “Imaginary” is just a name, imaginary do
indeed exist, they are numbers.
Imaginary Numbers
You can see in the graph of f(x) = x2 + 1 below that f has no
real zeros. If you solve the corresponding equation 0 = x2 + 1,
you find that x =
,which has no real solutions.
However, you can find solutions if you define
the square root of negative numbers, which is
why imaginary numbers were invented. The
imaginary unit i is defined
as
. You can use the imaginary unit to
write the square root of any negative number.
Powers of i
ii
i  1
2
We could continue but notice that
they repeat every group of 4.
For every i 4 it will = 1
i  i i  1(i)  i
To simplify higher powers of i
4
2 2
then, we'll group all the i
and
i  i i   1 1  1 see what is left.
5
4
8
33
4 8
i  i i  1i   i
i  i  i  1 i  i
6
4 2
i  i i  1 1  1
4 will go into 33 8 times with 1 left.
7
4 3
i  i i  1 i   i
20 3
83
4 20 3
i  i  i  1 i  i
8
4 4
i  i i  11  1
3
2
4ths
4 will go into 83 20 times with 3 left.
Powers of i
Helpful Hint
Notice the repeating pattern in each row of the table.
The pattern allows you to express any power of i as
one of four possible values: i, –1, –i, or 1.
Powers of i
• Always reduce powers of i to;
i, -1, 1, or –i.
• To reduce, divide the exponent by 4 and the
remainder determines the reduced value.
–
–
–
–
remainder
remainder
remainder
remainder
1⇒ i
2 ⇒ -1
3 ⇒ -i
0 (no remainder) ⇒1
• Negative exponent, the term goes in the
denominator and the exponent becomes positive.
Powers of i
For exponents larger than 4, divide the exponent by
4, then use the remainder as your exponent instead.
Example:
i ?
23
23
 5 with a remainder of 3
4
So, use i which  -i
3
i  i
23
Your Turn:
• Simplify each of the following powers.
1) i22
-1
2)
i36
1
3)
i13
i
4)
i47
-i
Example: Simplifying Square
Roots of Negative Numbers
Express the number in terms of i.
Factor out –1.
Product Property.
Simplify.
Multiply.
Express in terms of i.
Example: Simplifying Square
Roots of Negative Numbers
Express the number in terms of i.
Factor out –1.
Product Property.
Simplify.
4 6i  4i 6
Express in terms of i.
Your Turn:
Express the number in terms of i.
Factor out –1.
Product Property.
Product Property.
Simplify.
Express in terms of i.
Your Turn:
Express the number in terms of i.
Factor out –1.
Product Property.
Simplify.
Multiply.
Express in terms of i.
Your Turn:
Express the number in terms of i.
Factor out –1.
Product Property.
Simplify.
Multiply.
Express in terms of i.
The Complex-Number
System
• The complex-number system is used to find
zeros of functions that are not real numbers.
• When looking at a graph of a function, if
the graph does not cross the x-axis, it has no
real-number zeros.
Complex Numbers
A complex number is a number that
can be written in the form a + bi,
where a and b are real numbers and
i=
. The set of real numbers is a
subset of the set of complex numbers
C.
Every complex number has a real part a and an imaginary part b.
The Complex-Number
System
Complex Numbers C
Real Numbers R
Integers Z
Whole numbers W
Natural
Numbers N
Rational Numbers Q
R
Irrational
Numbers
Q -bar
Imaginary
Numbers i
Standard Form of
Complex Numbers
Complex numbers can be written in the form
a + bi (called standard form), with both a and
b as real numbers.
a is a real number and bi would be an
imaginary number.
If b = 0, a + bi is a pure real number (2, -56,
34 etc.).
If a = 0, a + bi is an pure imaginary number
(2i, -56i, 34i etc.).
Standard Form of
Complex Numbers
• Since all numbers belong to the Complex number
system, C, all numbers can be classified as complex.
Real numbers, R, and Imaginary numbers, i, are subsets
of Complex Numbers.
Complex Numbers
a + bi
Real Numbers
a + 0i
Imaginary Numbers
0 + bi
Standard Form of
Complex Numbers
Zero is the only number that is both real and imaginary.
Standard Form of Complex
Numbers
Example:
Write each of the following in the form of a
complex number in standard form a + bi.
6 = 6 + 0i
8i = 0 + 8i
 24 
4  6  1  2 i 6  0  2 i 6
6   25  6  25   1  6 + 5i
Your turn:
• Write each of the following in the form of a
complex number in standard form a + bi.
•
9  5
•
 24
5  3i
0  2i 6
Equity of Complex
Numbers
Two complex numbers are equal if and only if their
real parts are equal and their imaginary parts are equal.
Equity of Complex Numbers
a+bi =c+di if and only if a=c and b=d.
Example:
Find the values of x and y that make the equation 4x + 10i = 2 – (4y)i
true .
Real parts
4x + 10i = 2 – (4y)i
Imaginary parts
4x = 2
Equate the
real parts.
Solve for x.
10 = –4y Equate the
imaginary parts.
Solve for y.
Your Turn:
Find the values of x and y that make each equation true.
2x – 6i = –8 + (20y)i
Real parts
2x – 6i = –8 + (20y)i
Imaginary parts
2x = –8
Equate the
real parts.
x = –4
Solve for x.
–6 = 20y
Equate the
imaginary parts.
Solve for y.
Your Turn:
Find the values of x and y that make each equation true.
–8 + (6y)i = 5x – i 6
Real parts
–8 + (6y)i = 5x – i 6
Imaginary parts
–8 = 5x
Equate the real
parts.
Solve for x.
Equate the imaginary
parts.
Solve for y.
Complex Plane
Just as you can represent real
numbers graphically as points
on a number line, you can
represent complex numbers in
a special coordinate plane.
The complex plane is a set of coordinate axes in which the
horizontal axis represents real numbers and the vertical axis
represents imaginary numbers.
Complex Plane
Helpful Hint
The real axis corresponds to the x-axis, and the
imaginary axis corresponds to the y-axis. Think
of a + bi as x + yi.
Example:
Graph each complex number.
A. 2 – 3i
–1+ 4i
•
B. –1 + 4i
C. 4 + i
D. –i
4+i
•
•
–i
•
2 – 3i
Your Turn:
Graph each complex number.
a. 3 + 0i
b. 2i
c. –2 – i
2i
•
–2 – i
d. 3 + 2i
•
3 + 2i
•
•
3 + 0i
Absolute Value of a
Complex Number
Recall that absolute value of a real number is its distance from 0
on the number line. Similarly, the absolute value of a complex
number is its distance from the origin in the complex plane.
Example:
Find each absolute value.
A. |3 + 5i|
B. |–13|
C. |–7i|
|–13 + 0i|
|0 +(–7)i|
13
7
Your Turn:
Find each absolute value.
a. |1 – 2i|
b.
c. |23i|
|0 + 23i|
23
Adding and Subtracting
Complex Numbers
Sum or Difference of Complex Numbers
If a + bi and c + di are complex numbers, then
their sum is
(a + bi) + (c + di) = (a + c) + (b + d)i
Their difference is
(a + bi) – (c + di) = (a – c) + (b – d)i
Combine like Terms: add/sub. real parts and
add/sub. Imaginary parts.
Adding and Subtracting
Complex Numbers
Adding and subtracting complex numbers is similar to
adding and subtracting variable expressions with like
terms. Simply combine the real parts, and combine the
imaginary parts.
The set of complex numbers has all the properties
of the set of real numbers. So you can use the
Commutative, Associative, and Distributive
Properties to simplify complex number
expressions.
Helpful Hint
Complex numbers also have additive inverses.
The additive inverse of a + bi is –(a + bi), or
–a – bi.
Procedure:
1. Pretend that “i” is a variable and that a
complex number is a binomial
2. Add and subtract as you would binomials
Example:
(2 + i) – (-5 + 7i) + (4 – 3i)
2 + i + 5 – 7i + 4 – 3i
11 – 9i
Example:
Add or subtract. Write the result in the form a + bi.
(4 + 2i) + (–6 – 7i)
(4 – 6) + (2i – 7i)
–2 – 5i
Add real parts and imaginary
parts.
Example:
Add or subtract. Write the result in the form a + bi.
(5 –2i) – (–2 –3i)
(5 – 2i) + 2 + 3i
Distribute.
(5 + 2) + (–2i + 3i)
Add real parts and imaginary
parts.
7+i
Example:
Add or subtract. Write the result in the form a + bi.
(1 – 3i) + (–1 + 3i)
(1 – 1) + (–3i + 3i)
0
Add real parts and imaginary
parts.
Your Turn:
Add or subtract. Write the result in the form a + bi.
(–3 + 5i) + (–6i)
(–3) + (5i – 6i)
–3 – i
Add real parts and imaginary
parts.
Your Turn:
Add or subtract. Write the result in the form a + bi.
2i – (3 + 5i)
(2i) – 3 – 5i
(–3) + (2i – 5i)
–3 – 3i
Distribute.
Add real parts and imaginary
parts.
Your Turn:
Add or subtract. Write the result in the form a + bi.
(4 + 3i) + (4 – 3i)
(4 + 4) + (3i – 3i)
8
Add real parts and imaginary
parts.
Multiplying Complex
Numbers
The technique for multiplying complex numbers
varies depending on whether the numbers are written
as single term (either the real or imaginary
component is missing) or two terms.
You can multiply complex numbers by using the
Distributive Property and treating the imaginary
parts as like terms. Simplify by using the fact i2 = –1.
Multiplying Complex
Numbers
Note that the product rule for radicals does NOT
apply for imaginary numbers.
2
 16   25  4i  5i  20i  20(1)   20
 16   25 
16  25 
400  20
Before performing operations with imaginary
numbers, be sure to rewrite the terms or factors in
i-form first and then proceed with the operations.
Procedure:
1. Pretend that “i” is a variable and that a complex
number is a binomial
2. Multiply as you would binomials
3. Simplify by changing “i2” to “-1” and combining
like terms
Example:
(-4 + 3i)(5 – i)
-20 + 4i + 15i - 3i2
-20 + 4i + 15i + 3
-17 + 19i
Example:
Multiply. Write the result in the form a + bi.
–2i(2 – 4i)
–4i + 8i2
Distribute.
–4i + 8(–1)
Use i2 = –1.
–8 – 4i
Write in a + bi form.
Example:
Multiply. Write the result in the form a + bi.
(3 + 6i)(4 – i)
12 + 24i – 3i – 6i2
Multiply.
12 + 21i – 6(–1)
Use i2 = –1.
18 + 21i
Write in a + bi form.
Example:
Multiply. Write the result in the form a + bi.
(2 + 9i)(2 – 9i)
4 – 18i + 18i – 81i2
Multiply.
4 – 81(–1)
Use i2 = –1.
85
Write in a + bi form.
Example:
Multiply. Write the result in the form a + bi.
(–5i)(6i)
–30i2
Multiply.
–30(–1)
Use i2 = –1
30
Write in a + bi form.
Your Turn:
Multiply. Write the result in the form a + bi.
2i(3 – 5i)
6i – 10i2
Distribute.
6i – 10(–1)
Use i2 = –1.
10 + 6i
Write in a + bi form.
Your Turn:
Multiply. Write the result in the form a + bi.
(4 – 4i)(6 – i)
24 – 4i – 24i + 4i2
Distribute.
24 – 28i + 4(–1)
Use i2 = –1.
20 – 28i
Write in a + bi form.
Your Turn:
Multiply. Write the result in the form a + bi.
(3 + 2i)(3 – 2i)
9 + 6i – 6i – 4i2
Distribute.
9 – 4(–1)
Use i2 = –1.
13
Write in a + bi form.
Complex Conjugates
The complex numbers
and
are related.
These complex numbers are a complex conjugate pair. Their real
parts are equal and their imaginary parts are opposites. The
complex conjugate of any complex number a + bi is the
complex number a – bi.
If a quadratic equation with real coefficients has nonreal solutions,
those solutions are complex conjugates.
Helpful Hint
When given one complex solution, you can
always find the other by finding its conjugate.
Example:
Find each complex conjugate.
B. 6i
A. 8 + 5i
8 + 5i
8 – 5i
Write as a + bi.
Find a – bi.
0 + 6i
0 – 6i
–6i
Write as a + bi.
Find a – bi.
Simplify.
Your Turn:
Find each complex conjugate.
B.
A. 9 – i
9 + (–i)
Write as a + bi.
Write as a + bi.
9 – (–i)
9+i
Find a – bi.
Find a – bi.
Simplify.
C. –8i
0 + (–8)i
0 – (–8)i
8i
Write as a + bi.
Find a – bi.
Simplify.
Product of Complex
Conjugates
Complex Conjugates
The complex numbers (a + bi) and (a – bi) are
complex conjugates of each other, and
(a + bi)(a – bi) = a2 + b2
Complex Conjugate
Examples:
6 + 7i and 6  7i
8  3i and 8 + 3i
14i and 14i
• The product of a complex number and its conjugate is a
real number.
Example:
(4 + 9i)(4  9i) = 42  (9i)2
= 16  81i2
= 16  81(1)
= 97
Example:
(7i)(7i) =  49i2
= 49(1)
= 49
Your Turn:
• Multiply each complex number by its
complex conjugate.
1) 3 + i
10
2) 5 – 4i
41
Dividing Complex
Numbers
Recall that expressions in simplest form cannot have
square roots in the denominator. Because the imaginary
unit represents a square root, you must rationalize any
denominator that contains an imaginary unit. To do this,
multiply the numerator and denominator by the complex
conjugate of the denominator.
Helpful Hint
The complex conjugate of a complex number
a + bi is a – bi.
Procedure:
1. Write the division problem in fraction form.
2. The trick is to make the denominator real, this is
done using the complex conjugate.
3. Multiply the fraction by a special “1” where “1” is
the conjugate of the denominator over itself.
4. Simplify and write answer in standard form: a + bi.
Example:
Simplify.
Multiply by the conjugate.
Distribute.
Use i2 = –1.
Simplify.
Example:
Simplify.
Multiply by the conjugate.
Distribute.
Use i2 = –1.
Simplify.
Your Turn:
Simplify.
Multiply by the conjugate.
Distribute.
Use i2 = –1.
Simplify.
Your Turn:
Simplify.
Multiply by the conjugate.
Distribute.
Use i2 = –1.
Simplify.
Your Turn:
3  11i
1)
1  2i
5  i
7  4i
2)
2  5i
6 43
  i
29 29
Essential Question
Big Idea: Solving Equations
1. What are complex numbers?
–
The set of complex numbers includes real numbers and
imaginary numbers. The basis for the complex numbers is the
imaginary unit i, whose square is -1. Use a complex number of
the form a + bi to simplify the square root of a negative number.
2. How do perform basic operations on complex numbers?
–
To perform basic operations on complex numbers, treat the
imaginary parts like variable terms of binomials. Add, subtract,
or multiply as you would with algebraic expressions. To divide
complex numbers, multiply numerator and denominator by the
complex conjugate of the denominator and simplify.
Assignment
• Section 4-8 Part 1, Pg 273;# 1 – 6 all, 8 – 30
even.