Download Complex Numbers

Imaginary Numbers
Unit 1 Lesson 1
Make copies of:
• How do I simplify Powers of i version 2.docx
GPS Standards
• MM2N1a- Write square roots of negative
numbers in imaginary form.
• MM2N1b- Write complex numbers in the form
a + bi.
Essential Questions
• How do I write square roots of negative
numbers as imaginary numbers?
• How do I simplify powers of i?
Why do we need imaginary numbers?
Why do we need imaginary
numbers?
Think back to when you first learned about
numbers…
Number probably meant 0,1,2,3,…. (these
are the whole numbers)
Then you came upon a problem like 3 – 5
So we had to expand number to include all
the negative numbers ….-3,-2,-1,0,1,2,3,...
That was the set of integers, which are also
numbers
Let’s look at division…try 3 divided by 5
So now our definition of numbers needs to
include fractions…this is the set of rational
numbers
How about trying to take a square root of a
number like 2?
This means numbers has to include
radicals….these are irrational numbers
So..why do we need imaginary
numbers?
Let’s look at an equation:
X2 + 1 = 0
Isolate x term
X2 = -1
Take the square root of both sides…
Can you take the square root of a negative
number??
Let’s investigate…
(-4)2 = 16 and 42 = 16
Is there any time that you can
square something and get a
positive answer?
So…how do we take the
square root of a negative
number?
We need a new type of
“number”
Imaginary Numbers
i is the imaginary number unit
i = √-1
i2 = -1
Simplifying Square Roots of Negative
Numbers
•√–9
does not exist in the reals because
there is no number that can be squared to
give a negative answer. Therefore, you must
use i2 to replace the negative.
• √ – 9 = √9  √ –1 =
• √ – 20 = √20  √ –1 =
Using imaginary numbers
Simplify the following
√-49
√-72
√50
√-500
√-22
Test Prep Example
• Express in terms of i: -3√-64
A)
B)
C)
D)
-24i
-24√i
24i
24√i
Test Prep Example
• Simplify: -10 + √-16
2
A) -5 + 2i
B) -5 – 4i
C) 20 + 4i
D) 30 + 2i
i = -1
2
• i 2 = -1 is the basis of everything you
will ever do with complex numbers.
• Simplest form of a complex number
never allows a power of i greater than
the 1st power to be present, so ………
Simplifying Powers of i
i
i2
i3
i4
i5
i6
i7
i8
Simplification
None needed
By definition
i2 x i = -1 x i =
( i2)2 = ( -1)2 =
( i2)2 x i = ( -1)2 x i
( i2)3 = ( -1)3=
( i2)3 x i = ( -1)3 x i
( i2)4 = ( -1)4 =
Simplest Form
i
-1
-i
1
i
-1
-i
1
A Different Twist
Let’s look back at that pattern…
i33 =
Divide exponent by 4 (33÷4 = 8 R 1)
Our remainder will determine the answer based
on that pattern.
Remainder of 1 = i
Remainder of 2 = -1
Remainder of 3 = - i
No Remainder = 1
How Do I Simplify Powers of i graphic
organizer
• How do I simplify Powers of i version 2.dcx
Examples
Test Prep Example
• Simplify
i4 + i3 + i2 + i
A)
B)
C)
D)
0
1
-1
i
Test Prep Example
• Simplify
(i)237
A) -1
B) 1
C) i
D) -i
Lesson 1 Support Assignment
• Pg. 4: #1-33
Simplifying Square Roots of Negative
Numbers
•√–9
does not exist in the reals because
there is no number that can be squared to
give a negative answer. Therefore, you must
use i2 to replace the negative.
• √ – 9 = √9i2 = 3i
• √ – 20 = √20i2 = √45i2 = 2i√5
Examples
Test Prep Example
• (√-4)(√-4)
Simplify the expression
A) -4
B) 2i
C) 2i2
D) 4
Solving Equations in the Complex
Numbers
• x2 + 4 = 0
• Remember this equation that we used to
show why a sum of two squares never factors
in the reals?
• x2 = - 4  √x2 = √-4
• x =  √-4 =  √4i2 =  2i
• Complex solutions always come in conjugate
pairs.
Examples