1. Imaginary Numbers
Unit 1 Lesson 1

2. Make copies of:
How do I simplify Powers of i version 2.docx

3. GPS Standards
MM2N1a- Write square roots of negative numbers in imaginary form.
MM2N1b- Write complex numbers in the form a + bi.

4. Essential Questions
How do I write square roots of negative numbers as imaginary numbers?
How do I simplify powers of i?

5. Why do we need imaginary numbers?

6. 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

7. 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

8. 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??

9. 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”

10. Imaginary Numbers
i is the imaginary number unit
i = √-1
i2 = -1

11. 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 =

12. Using imaginary numbers
Simplify the following
√-49
√-72
√50
√-500
√-22

13. Test Prep Example
Express in terms of i: -3√-64
-24i
-24√i
24i
24√i

14. Test Prep Example Simplify: -10 + √-16 2 -5 + 2i -5 – 4i 20 + 4i

15. i 2 = -1
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 ………

16. Simplifying Powers of i
Simplification
Simplest Form i
None needed i2
By definition
- 1 i3
i2 x i = -1 x i =
- i i4
( i2)2 = ( -1)2 = 1 i5
( i2)2 x i = ( -1)2 x i i6
( i2)3 = ( -1)3= i7
( i2)3 x i = ( -1)3 x i i8
( i2)4 = ( -1)4 =
Notice that there are only four possible answers any time you ever simplify a power of i.
An ODD power of i will always lead to either i or –i as the answer.
An EVEN power of i will always lead to either 1 or -1 as the answer.

17. 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

18. How Do I Simplify Powers of i graphic organizer
How do I simplify Powers of i version 2.dcx

19. Examples

20. Test Prep Example
Simplify i4 + i3 + i2 + i 1 -1 i

21. Test Prep Example
Simplify (i)237
A) -1
B) 1
C) i
D) -i

22. Lesson 1 Support Assignment
Pg. 4: #1-33

23. 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

24. Examples

25. Test Prep Example
(√-4)(√-4)
Simplify the expression -4 2i
2i2 4

26. Solving Equations in the Complex Numbers
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.

27. Examples