1. For many calculators if you input "the square root of -1", you will get out "domain error" Input Output This was done with a TI-30XS calculator

2. Why do some calculators give a "domain error" for the square root of a negative?

3. ??

4. Complex Numbers Imaginary NumbersReal Numbers Rational Numbers Irrational Numbers Integers Whole Numbers Counting Numbers Because the domain programmed into the calculator is the Real Numbers not the Complex Numbers or Imaginary Numbers

5. These questions should be on your mind. 2. How do I process them? 1. Why is the square root of a negative useful?

6. Why is the square root of a negative useful?

7. Imaginary Numbers are useful in these fields of work: 1. Computer Graphics 2. Electric Engineering 3. Quantum Physics

8. We can learn about a 4 Cycle System that is a basis of study in these fields. 0 1 2 3 i clock

9. Let's begin with a definition By definition, "the square root of -1" is named i.

10. Let's combine that definition with these 2 facts. Fact #1 Any non-zero number to the "zero power" is 1 Fact #2 Any number to the "power of 1" is itself

11. So, the top and the right side of the "i clock" are explained. 0 1 2 3 i clock

12. Let's investigate the bottom position 0 1 2 3 i clock represent? What does this

13. But why is this true?

14. Let's investigate fact #3 first. Fact #3 A "square root" times itself cancels the radical, only

18. Now the top, right and bottom positions of the clock are explained 0 1 2 3 i clock

19. Let's investigate the position on the left 0 1 2 3 i clock represent? What does this

20. But why is this true?

23. and

24. A "square root" times itself cancels the radical, only?

26. What is this?

29. 0 1 2 3 i clock The "i clock"

30. What about a power of i that is 4 or greater?

31. Add to the clock like this.

32. Let's make sure we understand this

33. i clock to the 99th power