2. Complex Numbers S. Awad, Ph.D. M. Corless, M.S.E.E. E.C.E. Department University of Michigan-Dearborn Math Review with Matlab: Complex Math

3. Complex Numbers:Complex Math 2 Complex Number Math n Rectangular Addition Rectangular Addition n Rectangular Subtraction Rectangular Subtraction n Polar Multiplication Polar Multiplication n Rectangular Multiplication Rectangular Multiplication n Polar Division Polar Division n Complex Conjugate Complex Conjugate n Rectangular Division Rectangular Division

4. Complex Numbers:Complex Math 3 Complex Addition n The addition of two complex number z 1 and z 2 gives another complex number n Addition of complex numbers is most easily done in Rectangular Form

5. Complex Numbers:Complex Math 4 Addition Example n As an example, the following two complex numbers can be added mathematically and graphically

6. Complex Numbers:Complex Math 5 » Z1=2+3i; » Z2=4+i; » Z3=Z1+Z2 Z3 = 6.0000 + 4.0000i Matlab Addition n This result can be verified in Matlab

7. Complex Numbers:Complex Math 6 Complex Subtraction n Similarly, subtraction of two complex number z 1 and z 2 gives another complex number n Subtraction of complex numbers is most easily done in Rectangular Form

8. Complex Numbers:Complex Math 7 Subtraction Example n As an example, the following two complex numbers can be subtracted graphically and mathematically n Subtracting z 2 is the same as adding -z 2

9. Complex Numbers:Complex Math 8 » Z1=3+3i; » Z2=2+i; » Z3=Z1-Z2 Z3 = 1.0000 + 2.0000i Matlab Subtraction n This result can be verified in Matlab

10. Complex Numbers:Complex Math 9 Polar Multiplication n Multiplication of complex numbers is most easily done in polar form since:

11. Complex Numbers:Complex Math 10 Polar Multiplication n Similarly, the shorthand angle notation can be used to express polar multiplication

12. Complex Numbers:Complex Math 11 Rectangular Multiplication n Multiplication of complex numbers can also be done in Rectangular Form by directly multiplying z 1 and z 2

13. Complex Numbers:Complex Math 12 Multiplication Example n Multiply the two complex numbers first using the direct rectangular form n Then verify the results using the polar version of multiplication.

14. Complex Numbers:Complex Math 13 Direct Multiplication n Direct multiplication in the rectangular form yields:

15. Complex Numbers:Complex Math 14 Polar Multiplication n z 1 and z 2 must first be converted to polar form

16. Complex Numbers:Complex Math 15 Polar Multiplication n Verify that this is same result as rectangular multiplication

17. Complex Numbers:Complex Math 16 » z1=3+2i; z2=1-4i; » mult=z1*z2 mult = 11.0000 -10.0000i » r = abs(mult) r = 14.8661 » theta=angle(mult) theta = -0.7378 Matlab Verification n Verify the multiplication of z 1 and z 2 using Matlab

18. Complex Numbers:Complex Math 17 Polar Division n Division of complex numbers is most easily done in Polar Form

19. Complex Numbers:Complex Math 18 Polar Division n Similarly, the shorthand angle notation can be used to express polar multiplication

20. Complex Numbers:Complex Math 19 Polar Division Example n Divide the complex number z 1 by z 2 by hand, then use Matlab to verify the result

21. Complex Numbers:Complex Math 20 » z1=10*exp(i*60*(pi/180)); » z2=5*exp(i*30*(pi/180)); Matlab Division Convert to Radians Convert to Degrees » div=z1/z2 div = 1.7321 + 1.0000i » Mag=abs(div) Mag = 2 » Theta=angle(div)*180/pi Theta = 30.0000

22. Complex Numbers:Complex Math 21 Complex Conjugate n The Complex Conjugate of a complex number is found by changing the sign of the imaginary portion n Complex Conjugate is denoted as z* n This is equivalent to negating the angle n Corresponds to a reflection of z in the real axis of an Argand diagram

23. Complex Numbers:Complex Math 22 Conjugate Example Plot the complex number z = 4 + i2 and it’s complex conjugate z*

24. Complex Numbers:Complex Math 23 Matlab Conjugate The conj command returns the complex conjugate of a complex number » z=4+2i; » zconj=conj(z) zconj = 4.0000 - 2.0000i » feather(z); » hold on » feather(zconj,'r') » xlabel('Real'); » ylabel('Imaginary');

25. Complex Numbers:Complex Math 24 Useful Complex Conjugate Relationships Addition Subtraction Multiplication

26. Complex Numbers:Complex Math 25 Rectangular Division n Division of complex numbers can also be done in Rectangular Form by use of the Complex Conjugate n The result is the multiplication of z 1 by the conjugate of z 2 divided by the magnitude of z 2 squared

27. Complex Numbers:Complex Math 26 Rectangular Division n Explicitly worked out, the division is:

28. Complex Numbers:Complex Math 27 Division Example n Divide z 1 by z 2 using the complex conjugate method

29. Complex Numbers:Complex Math 28 Matlab Verification » z1=4+j; » z2=2-3j; » z3 = z1/z2 z3 = 0.3846 + 1.0769i n The previous result is easily verified using Matlab

30. Complex Numbers:Complex Math 29 Summary n Complex addition and subtraction are most easily done using the rectangular form n Complex multiplication and division are most easily done using the exponential polar form n The complex conjugate can be used as a tool for implementing division using the rectangular form of complex numbers