1. Who, what, where, why and how.
Complex Numbers1
Who, what, where, why and how.

2. Quadratic Equations
Girolamo Cardano, a contemporary of Leonardo DaVinci.
‘divide 10 into two parts such that the product is 40.’
x and 10 – x
x(10 – x) = 40

3. x2 - 10x + 40=0 ‘truly sophisticated’
‘as subtle as it would be useless’

4. Roots of equations x2 - x - 6 = 0
What is meant by the roots of an equation?
An equation with roots -2 and 3
(x - 3)(x + 2) = 0
x2 - x - 6 = 0
The roots of x2 + 3x + 5 = 0

5. Imaginary Numbers

6. Alternating systems

7. Side view

8. Alternating systems

9. Powers of j

10. Negative powers

11. Complex Numbers Have a real and imaginary part For 3 + 4j
Real part Re(3 + 4j) = 3
Imaginary part Im(3 + 4j) = 4
If z = 5 - 2j
Re(z) = 5, Im(z) = -2

12. The conjugate of a complex number
If z = j
z* = j
Only the imaginary part is in the opposite direction

13. Addition and subtraction