1. Complex Numbers

2. a + bi Is a result of adding together or combining a real number and an imaginary number a is the real part of the complex number. bi is the imaginary part of a complex number. both a and b are real coefficients and i is equal to the For example: 6 + 8 i

3. Complex Numbers and their graphs GRAPH the following: -5+4i 4+5i 2i -3 5-i -2-4i Complex numbers can be represented as an ordered pair. For example, (5, 6) represents the complex number 5+6i. This ordered pair can be graphed on the complex plane called an "Argand diagram." The real part of the complex number represents the horizontal coordinate. The coefficient of the imaginary part of the complex number represents the vertical coordinate. The origin corresponds to th point (0, 0) or 0+0i

4. Modulus What is the absolute value of -3? │-3│ or 3 Absolute value means the distance from 0 and -3 on the real number line. Well, there is an absolute value for a complex number as well, and we can find it using our new tool: the argand diagram!

5. Modulus By representing a complex number as a position vector drawn from the origin (0, 0) to the complex number coordinate (a, b), the absolute value of the complex number can be derived. This absolute value or MODULUS is equivalent to the vector line's length. Develop a formula for the absolute value of a complex number │a + bi│, in terms of a and b

6. A sports complex is a building that accommodates two or more sports. A complex number is any number having the a +bi. Explain in your own words why you think these numbers are called complex.

7. How would you graph i ^15 R Im

8. Well, first let's look at the cyclical number system of imaginary numbers R Im What is i ^0? 1 1 is a real number. The complex number for this value (a + b i ) is 1 + 0 i The ordered pair is (1, 0) on the complex plane

9. Well, first let's look at the cyclical number system of imaginary numbers R Im What is i ^1? i i is an imaginary number. The complex number for this value (a + b i ) is 0 + 1 i The ordered pair is (0, 1) on the complex plane

10. Well, first let's look at the cyclical number system of imaginary numbers R Im What is i ^2? -1 is a real number. The complex number for this value (a + b i ) is -1 + 0 i The ordered pair is (-1, 0) on the complex plane Are we starting to see a pattern here?

11. Well, first let's look at the cyclical number system of imaginary numbers R Im What is i ^3? -i -i is an imaginary number. The complex number for this value (a + b i ) is 0 - 1 i The ordered pair is (0, -1) on the complex plane

12. Well, first let's look at the cyclical number system of imaginary numbers R Im What is i ^4? 1 1 is a real number. The complex number for this value (a + b i ) is 1 + 0 i The ordered pair is (1, 0) on the complex plane And now we are back where we started...

13. How would you graph i ^15 R Im What is i ^15? (1)(1)(1)(i^3) i^3 is the imaginary number - i. The complex number for this value (a + b i ) is 0 - 1 i The ordered pair is (0, -1) on the complex plane

14. How would you graph i ^-7 R Im

15. Well, first let's look at the cyclical number system of imaginary numbers R Im What is i ^- 1? -i -i is an imaginary number. The complex number for this value (a + b i ) is 0 - 1 i The ordered pair is (0, -1) on the complex plane

16. Well, first let's look at the cyclical number system of imaginary numbers R Im What is i ^- 2? -1 is a real number. The complex number for this value (a + b i ) is -1 + 0 i The ordered pair is (-1, 0) on the complex plane

17. Well, first let's look at the cyclical number system of imaginary numbers R Im What is i ^- 3? i i is an imaginary number. The complex number for this value (a + b i ) is 0 + 1 i The ordered pair is (0, 1) on the complex plane Are we starting to see a pattern here?

18. How would you graph i ^-7 R Im What is i ^- 7? (1)i^-3 i^-3 is an imaginary number i. The complex number for this value (a + b i ) is 0 + 1 i The ordered pair is (0, 1) on the complex plane

19. Rules for graphing NEGATIVE EXPONENTSPOSITIVE EXPONENTS For each successive negative power of i, the point rotates clockwise by 90 degrees For each succesive positive power of i, the point rotates counter-clockwise by 90 degrees i^0 = 1 i^-1 = -ii^1 = i i^-2 = -1i^2 = -1 i^-3 = ii^3 = -i i^-4 = 1i^4 = 1