Download 7.5 part 1: Complex Numbers This is the graph of the equation y = x2

7.5 part 1: Complex Numbers
This is the graph of the equation y = x2 + 4x + 5. Using the quadratic formula, we find the roots (zeros, x-intercepts) of
the graph are
Complex Numbers
Imaginary Unit
Conjugate Pair
Complex Conjugates
Group 1: The Classification of Numbers.
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Natural numbers are the counting number.
Integers are any natural number, 0, and the
negative counting numbers.
Rational numbers are any numbers that can
be written as a faction, including integers.
Every fraction can be written either as a:
o Terminating decimal
o Repeating decimal
Irrational numbers are any numbers that
cannot be written as a fraction. Their
decimals are:
o Non-terminating
o Non-repeating
Real numbers are any rational numbers or
irrational numbers.
An imaginary number is any number that is
the square root of a negative or has “i.”
Complex numbers are any real or imaginary
number that can be written in the form a +
bi.
Example: State all the classifications of the number -16/2.
The number is a negative, so it cannot be a natural.
-16/2 = -8. It is a whole counting number, so the lowest classification is an integer.
Answer: -16/2 is an integer, rational, real, and complex number.
Example2: State all the classifications of the number 7.26.
The number is not a counting number, so it cannot be an integer.
The number is a terminating decimal, which can be converted into the fraction 7
26
, so the lowest classification
100
is a rational number.
Answer: 7.26 is a rational, real, and complex number.
State all the classifications of each number.
a. 0
e.
3 7
b. 4.453423…
c. 3+4i
f.  4
g. -5.4
d. 4.453476
h.
5
3
i.
4
Group 2: Simplifying Complex Numbers
Simplifying complex numbers is the same as simplifying radicals. Our goal is to write the numbers in the form
a + bi, where a is the real number, and bi is the imaginary part.
Example: Simplify
 50  36 .
The
36 is the real number part.
The
 50 is the imaginary part.
So
36 simplifies to 6.
 50  25 2  1  5 2i  5i 2 .
 50  36 simplifies to 6  5i 2 .
Example2: Simplify  80 .
The negative is not under the radical, so this is a real number.
 80   16 5  4 5 .
Simplifying complex Numbers
a.
 25
b.
 19
c.
 45
d.
4
e.
7
f.
 12
g.
9 6
h.   18  7
i. 24   75
Group 3: Graphing Complex Numbers
The x-axis is now the real number, the y-axis represents the imaginary.
Example: State the real number represented on the complex plane on the right.
A = 5 – 4i
B = -3i
C = -3 – i
D = -6
E = -2 + 4i
F=1
G=1+i
**Notice how these are numbers in the form a + bi, not ordered pairs.
Graph each number on the complex plane below.
a. 3+4i
b. -2-i
c. 3
d. -1+  25
e. 8i
f.   4  1
Group 4: Determining Powers of i.
Powers of i are cyclic, meaning that there is a pattern. Note:
i1 = i
1  i
i2 =
i2 = -1
i1 =i
1  i
i3 = i2i = (-1)(i)
i4 = i2i2 = (-1)(-1)
i3 = -i
i4 = 1
i2 =___ i3 =___ i4 =___
Now, notice the following.
i5 = i4i1 = (1)(i)
i6 =i4i2 = (1)(-1)
i7=i4i3=(1)(-i)
i8=(i4)2 = (1)2
i5 = i
i6 = -1
i7=-i
i8 = 1
The same pattern repeats with every group of 4.
Because i4 = 1, then every power that is a multiple of 4 will also equal 1.
So to determine the value of a power of i
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Write the power as a multiple of 4 and then the remainder.
The multiple of 4 will equal 1, and then determine the value of the remainder.
Example: Simplify i43.
The closest multiple of 4 to 43 is 40. So, i43 = i40i3.
i40 = (i4)10 = (1)10 = 1
i43 = (1)(i3) = i3 = -i
Answer: i43 = -i
Simplify each power of i.
a. i24 =
b. i35 =
c. i50 =
d. i13 =
e. i102 =
f. i37 =
Sharing: Complex Numbers
With your home groups, share the most important points to remember from your expert group.
Classification of Numbers
Simplifying Complex Numbers
Graphing Complex Numbers
Powers of i
Home Group Practice
With your group, complete the following practice problems together.
1.
Classify each number in as many ways as possible.
a.
2.
b. -6
d. 4 – 19i
c. 0
Simplify each complex number.
a.
3.
9.489
36
e. 3
 30
b. 5   98
Plot the following complex numbers on the complex plane below.
a. -5
b. -3 + 2i
c.
 49  1
d.   9
e. 5i + 8
4.
Simplify each power of i.
a. i27
b. i96
c. i34
d. i62
e. i13
Back to Solving Quadratic Equations
There is no more “no solution” answers anymore. EVERY PROBLEM will have a solution!!
Solve the quadratic equations. Leave answers in simplest form.
a. x2 = -36
b. x2 = -28
c. -(x – 3)2 = 25
d. (2x + 7)2 – 15 = -28
e. 4(x – 11)2 + 27 = 3
f. -5(5x – 1)2 = 18
Now use the quadratic formula to solve these equations.
a.
b.
Name: _______________________
Date: ___________
Homework: 7.5 part 1 Complex Numbers
Simplify.
1.
 108
5. i83
2.
 144
3.
6. i57
 32  15
4.  54  28
7. i66
8. i21
List all the classifications of each number.
9. 28
13.  100  45
10.  72
11.
14. 8.57389…
Plot each number on the complex plane to the left.
17. 5 – 3i
18. i
19. -8
20. 2 + 5i
21.  4   64
 72
15. 8.235
12.
 100
16.
58
47
Solve each quadratic equation. Leave answers in simplest form.
c.
a.
d.
b.
c.