Fundamentals of Engineering Analysis EGR 1302 - Introduction to Complex Numbers, Standard Form

1. Fundamentals of Engineering Analysis EGR 1302 - Introduction to Complex Numbers, Standard Form

2. T Exponential Decay Feedback Tout Sinusoidal Response Complex Numbers Definitions and Formats Complex Numbers mathematically represent actual physical systems Tin Tout SYSTEM

3. Complete the Square The Solution to the General Quadratic Equation Take the Square Root The General Quadratic Equation

4. 3rd Order 2nd Order If there is one real root there are two real roots, as above If If there are no real roots, as shown Solutions of the Quadratic Equation By solution, we mean “roots”, or where x=0

5. Complete the square: Take the square root: becomes The solution: Because does not exist, we call this an “imaginary” number, and we give it the symbol “ ”or “ ”. j i The Imaginary Number Consider:

6. Substitute into Checks! A general solution for is Complex Numbers

7. z1=z2 z2=x2+iy2 z1=x1+iy1 and Then if Given Complex Numbers Definitions The “Standard Form” z=x+iy Im(z)=y Re(z)=x z=x+iy and if y=0, then z=x, a real number i6=-1 i5=i i4=1 i3=-i i2=-1 x1 = x2 y1 = y2

8. z3= (x1 - x2) + i(y1 - y2) z1-z2=z3 Algebra of Complex Numbers z2=x2+iy2 z1=x1+iy1 Definitions: Given z3= (x1 + x2) + i(y1 + y2) z1+z2=z3 z1 * z2=z3 z3= (x1 + iy1)(x2 + iy2) = x1x2+ix1y2+ix2y1+i2y1y2 z3= x1x2+i2y1y2 +i(x1y2+x2y1) Re(z3)= x1x2-y1y2 Im(z3)= x2y1-x1y2

9. Dividing Complex Numbers To divide, must eliminate the “i” from the denominator We do this with the “Complex Conjugate” - by CHANGING THE SIGN OF i

10. Reciprocals of Complex Numbers Multiply by the Complex Conjugate to put in Standard Form

11. Questions?