1. Complex Numbers 2-4

2. Imaginary Numbers Designed so negative numbers can have square roots.
Imaginary numbers consist of all numbers bi, where b is a real number and i is the imaginary unit, with the property that i 2 = -1.
Example

3. Practice

4. Powers of i
i 1 = i
i 2 = -1
i 3 = -i
i 4 = 1

5. Higher Powers of i 2  i 2 = -1 3  i 3 = -i 0  i 0 = i 4 = 1
Divide the exponent by 4, then determine the remainder.
If the remainder is
1  i 1 = i
2  i 2 = -1
3  i 3 = -i
0  i 0 = i 4 = 1

6. Practice i 20 = i 27 = i 71 - i 49 = i 4444484044844444441 = i -2 =

7. Negative Exponents

8. Multiplying Imaginary Numbers

9. Practice
24i
-18 6
-10

10. Complex Numbers
Complex numbers are the sum of a real number and an imaginary number.
They are written in the form a + bi where a is a real number and bi is an imaginary number.
Complex numbers include real and imaginary numbers since 3 = 3 + 0i or 4i = 0 + 4i
Imaginary numbers follow the properties we have learned (commutative, associative, distributive…)

11. Imaginary Numbers b≠0 Complex Numbers a + bi Real Numbers b=0 PURE

12. The Big Picture where does everything fit
복소수 허수
순허수
Pure

13. Addition and Subtraction
Combine the real parts and combine the imaginary parts.
(6 + 3i) + (8-2i) = (6+8) + (3 – 2)i = 14 + i
(6 + 3i) - (8-2i) = (6-8) + (3 – -2)i = i
(6 - 3i) + (8-2i) = (6+8) + (-3 – 2)i = i

14. Practice 3i – (5 – 2i)= (-2 + 8i) – (7+3i)= 4 – 10i + 3i – 2 = 2 – 7i

15. Graphing Complex Numbers

17. Finding Absolute Values
The absolute value of a complex number | a + bi | is its distance from the origin. So we use the distance formula or simplified as
we ignore the i for the formula

18. Finding Absolute Values
Practice - Find
|6 – 4i |
|-2 + 5i |
|4i|
3) 4

19. Multiplying Complex Numbers
Multiply like real numbers and treat i like a variable but i2 = -1
(3 + 2i)(4-7i) = 12 – 21i +8i -14i2 =
12 – 13i -14(-1) = 12-13i +14 =
26 – 13i

20. Practice 6 – 3i +4i – 2i2 = 6+i+2= 8+ i 4 + 2i -2i – i2 = 4 – (-1) = 5
4 – (-1) = 5
18 – 12i -15i -10= i

21. Division

22. Complex Conjugates

23. Practice

24. Equations with complex numbers
Two complex numbers are equal if their real part is equal and their imaginary part is equal.
If a+bi = c+di then a=c and b=d
5x+1 + (3+2y)i = 2x-2 + (y-6)i
real part 5x+1 = 2x-2, 3x = -3, x=-1
imaginary part 3+2y = y-6, y=-9

25. Conjugates 켤레복소수
In algebra, a conjugate is a binomial formed by negating the second term of a binomial. The conjugate of x + y is x − y, where x and y are real numbers. If y is imaginary, the process is termed complex conjugation: the complex conjugate of a + bi is a − bi, where a and b are real.

26. Complex Conjugates

27. Practice

28. Equations with complex numbers
Two complex numbers are equal if their real part is equal and their imaginary part is equal.
If a+bi = c+di then a=c and b=d
5x+1 + (3+2y)i = 2x-2 + (y-6)i
real part 5x+1 = 2x-2, 3x = -3, x=-1
imaginary part 3+2y = y-6, y=-9

29. Graphing Points
You have the real axis (x-axis) and the imaginary axis (y-axis).
Plot point (a, b)
You can plot inequalities by shading areas of the graph.
How would we graph
{a + bi | a≤3 and b ≤2} ?

30. Graph {a + bi | a≤3 and b ≤2}

32. What happens when we multiply a complex number, a + bi by i ?

33. More about Complex Numbers
To solve an equation involving complex numbers, equate the real parts and equate the imaginary parts.

34. Product of a Complex Number and its Conjugate
What happens when you multiply a complex number by it’s conjugate?
(3+4i )(3-4i ) =
So (a + bi )(a – bi ) =
9-(-16) = 25
a2 – (b2)(i2) = a2 – (-1)(b2)
= a2 + (b2)

35. Dividing and Reciprocals
When you divide by a complex number (a fraction with a complex number as a denominator) you multiply both the numerator and denominator by 1 by multiplying both by the conjugate of the denominator,

36. Example

37. Finding the Reciprocal
To find the reciprocal of a complex number, you divide 1 by that complex number.
The reciprocal of 3 + 2i is
But now you need to rationalize the denominator by multiplying the numerator and denominator by the conjugate.
So find the reciprocal of 3 + 2i