1. What are they? Why do we need them? 5i
Complex Numbers
What are they?
Why do we need them? 5i

2. Complex Numbers Why do we need new numbers?
The hardest thing about working with complex numbers is understanding why you might want to. Before introducing complex numbers, let's backup and look at simpler examples of the need to deal with new numbers.

3. subtraction whole number integers negative
If you are like most people, initially number meant _____________, 0,1,2,3,...
Whole numbers make sense. They provide a way to answer questions of the form "How many ... ?" You also learned about the operations of addition and ____________, and you found that while subtraction is a perfectly good operation, some subtraction problems, like 3 - 5, don't have answers if we only work with whole numbers. Then you find that if you are willing to work with ________, ...,-2, -1, 0, 1, 2, ..., then all subtraction problems do have answers! Furthermore, by considering examples such as temperature scales, you see that ________ numbers often make sense.
whole number
subtraction
integers
negative

4. solutions of equations
Now that we have fixed subtraction, we will deal with division. Some, in fact most, division problems do not have answers that are ________. For example, 3 ÷ 2 is not an integer. We need new numbers! Now we have ________________ (fractions). There is more to this story. There are problems with square roots and other operations, but we will not get into that here. The point is that you have had to expand your idea of “number” on several occasions, and now we are going to do that again.
The "problem" that leads to complex numbers concerns _______________________.
integers
rational numbers
solutions of equations

5. x-intercepts (-1,0) and (1,0). Equation 1:
Equation 1 has two solutions, x = -1 and x = 1. We know that solving an equation in x is equivalent to finding the ____________
of a graph; and,
the graph of
crosses the x-axis at
_______________
x-intercepts
(-1,0) and (1,0).

6. x-intercepts no Equation 2:
Equation 2 has no solutions, and we can see this by looking at the graph of  
Since the graph has no ______________, the equation has ____ solutions. When we define complex numbers, equation 2 will have two solutions.
x-intercepts no

7. The symbol i represents ____________________________.
Graph the complex numbers:
1.  3 + 4i       (3,4)
2.  2 - 3i       (2,-3)   
3.  i     (-4,2)
4.  3 (which is really 3 + 0i)   (3,0)
5.  4i  (which is really 0 + 4i)  (0,4)
The complex number is represented by the point, or by the vector from the origin to the point.
                                                      
The symbol i represents ____________________________.
Here’s how it looks using math symbols:
Another way to think of it is this:
____________________________.
Write that equation down 3 times:
the number whose square is
the square root of is i

8. Solve this equation: Try this equation: Graph the complex numbers:
4.  3 (which is really 3 + 0i)   (3,0)
5.  4i  (which is really 0 + 4i)  (0,4)
The complex number is represented by the point, or by the vector from the origin to the point.
                                                      
Solve this equation:
Try this equation:

9. Simplify each number by using the imaginary number i.

10. Imaginary
Rational

11. A complex number is written in the form a + b i, where a and b are real numbers.
Write the complex number in the form
a + b i.
Try these:

12. To add or subtract complex numbers, combine the real parts and combine the imaginary parts.
Try these:

13. To recap:
1.The imaginary number, i, is defined as________.
2. Complex numbers are written as ____________.
3. To simplify the sq. root of a negative number, use ______ and complete the process as you would for a positive number.
4. To combine complex numbers, combine the ____ parts and combine the ________________ parts.
5. To solve an equation such as x2= --36 , isolate x2 and follow the step listed above in #3.

14. Just imagine the possibilities!!!!!!!!!!!
Use your imagination!!!
Tomorrow we will see how imaginary numbers look on a complex plane, how to multiply them, and how to graph them using an imaginary axis.
Just imagine the possibilities!!!!!!!!!!!
Work in your group and begin your homework!