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Introduction to Complex Numbers: Form, Properties, and Calculations

Explore the basics of complex numbers, their representation, properties, and calculations. Learn about Cartesian and polar representations, Euler's formula, conjugates, and useful identities. Practice solving complex number problems with examples.

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Introduction to Complex Numbers: Form, Properties, and Calculations

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  1. Review of Complex Numbers

  2. Introduction to Complex Numbers • Complex numbers could be represented by the form • Where x and y are real numbers • Complex numbers are denoted: N = {x}+j{y}, where x is considered the REAL part and Y is considered the IMAGINARY part • If x = 0, N is considered an IMAGINARY NUMBER • If y = 0, N is considered a REAL number

  3. Properties of Complex Numbers • The sum of two complex numbers is a complex number: • (x1 + jy1) + (x2 + jy2) = (x1 + x2) + j(y1+ y2); • Example, Express the following complex numbers in the form x + iy, x, y real: • (−3 + i)(14 − 2i) • The product of two complex numbers is a complex number: • (x1 + jy1)(x2 + jy2) = x1(x2 + jy2) + (jy1)(x2 + iy2) • = x1x2 + x1(jy2) + (jy1)x2+ (jy1)(jy2) • = x1x2 + ix1y2 + iy1x2 + i2y1y2 • = (x1x2 + {−1}y1y2) + i(x1y2 + y1x2) • = (x1x2 − y1y2) + i(x1y2 + y1x2)

  4. Calculating with complex Numbers • Example 1: solve the system • (1 + i)z + (2 − i)w = 2 + 7i • 7z + (8 − 2i)w = 4 − 9i. • The determinant of the coefficient matrix is • = (1 + i)(8 − 2i) − 7(2 − i) • = (8 − 2i) + i(8 − 2i) − 14 + 7i • = −4 + 13i .

  5. Calculating with complex Numbers • Applying Cramer’s rule: Solve for w!

  6. Calculating with complex Numbers • Class exercise: solve the system: • (1 + i)z + (2 − i)w = −3i • (1 + 2i)z + (3 + i)w = 2 + 2i.

  7. Calculating with complex Numbers • Example 2: solve the system: z2= 1 + i. • Let z = x + iy. • >> (x + iy)2 = x2 − y2 + 2xyi = 1 + i, • >> x2− y2 = 1 and 2xy = 1. • >> x ≠0 and y = 1/(2x) • >> • >> 4x4− 4x2 − 1 = 0 • >> • >> • >> >>

  8. Calculating with complex Numbers • Class exercise: solve the system: z2= 1 + i√3

  9. Cartesian and polar representation of a complex number •  Every complex number z = x+iy can be represented by a point on the Cartesian plane known as complex plane by the ordered pair (x, y).

  10. Cartesian and polar representation of a complex number • The Cartesian coordinate pair (x, y) is also equivalent to the polar coordinate pair (r,θ), where r is the (nonnegative) length of the vector corresponding to (x, y), and θis the angle of the vector relative to positive real line. • x = r cosθ • y = r sin θ • Z = x + jy = r cosθ + j rsinθ = r (cosθ + j sin θ) • |z| = r = √(x2 + y2) • tanθ = (y/x) • θ = arctan(y/x)+ (0 or Π) (Πis added iffx is negative)

  11. The Euler Formula • ej θ = cosθ + j sin θ • Z = x+ jy = r cosθ + j rsinθ = r (cosθ + j sin θ) = r ej θ • R is the distance of the point z from the origin • 1/Z = 1/ r ej θ=( 1/ r) e-j θ

  12. Conjugate of a complex number • Let z = x + jy • The complex conjugate of z is the complex number defined by z* = x − jy. • Geometrically, the complex conjugate of z is obtained by reflecting z in the real axis • z* = x − jy = r e-j θ • z + z* = (x + jy) + (x – jy) = 2x = 2Re(z) • zz* = (x + jy) (x – jy) = x2+y2 = |z|2

  13. Some useful identities • 1e±j Π=-1 ; e±jnΠ=-1 for n odd integer • e±j2nΠ=1 for n integer • ej Π/2= j • E-j Π/2= -j

  14. Examples • Express the following numbers in polar form (also sketch the geometric representation): • 2+j3 • 1 – j3 • Use the MATLAB function cart2pol to convert the above numbers to polar form

  15. Examples • Express the following numbers in polar form (also sketch the geometric representation): • 2+j3 • r = |z| = √(22+32) = √13 • Θ = tan-1(3/2) = 56.30 • 2+j3 = √13ej56.3º

  16. Examples • Represent the following numbers in the complex plane and express them in Cartesian form: • 2 ej Π/3 • 4 e- j3Π/4 • Use the MATLAB function pol2cart to convert the above numbers from polar to Cartesian form

  17. Examples • Represent the following numbers in the complex plane and express them in Cartesian form: • 2 ej Π/3 • = 2cos(Π/3) + 2jsin(Π/3) • =2(1/2) +2 j(√3/2) • =1+j√3

  18. Examples • Determine z1z2 and z1/z2 for • z1 = 3 + j4 = 5ej53.1º • z2 = 2 + j3 = √13 ej56.3º • Solve this problem in both polar and Cartesian forms • Solve this problem using MATLAB

  19. Determine z1z2 and z1/z2 for • z1 = 3 + j4 = 5ej53.1º • z2 = 2 + j3 = √13 ej56.3º • Polar: • z1z2 = (3+j4)(2+j3) = (6-12)+j(8+9) = -6+j17 • z1/z2 = (3+j4)(2-j3) /(22+32) = (18/13) – j(1/13) • Cartesian: • z1z2 = (5ej53.1º )(√13 ej56.3º )= 5√13 ej(53.1º+ 56.3º ) =5√13 ej(109.4º) • z1/z2 = (5ej53.1º )/(√13 ej56.3º )=(5/√13) ej(53.1º- 56.3º ) =(5/√13) ej(-3.2º) Examples

  20. Examples • Consider X(ω), a complex function of a real variable ω: • X(ω) = (2 + j ω)/(3 + j4 ω) • Express X(ω) in Cartesian form, and find its real and imaginary parts. • Express X(ω) in polar form and find its magnitude and angle.

  21. Examples • Consider X(ω), a complex function of a real variable ω: • X(ω) = (2 + j ω)/(3 + j4 ω) • Express X(ω) in Cartesian form, and find its real and imaginary parts. • X(ω) = ((2 + j ω)(3 - j4 ω) )/(32 + 42 ω2) • (6+4ω2)/(9+16 ω2) - j5ω/9+ω2) • Express X(ω) in polar form and find its magnitude and angle. • X(ω) =[√(4 + ω2) ejarctan(w/2)]/ [√(9 + 16ω2) ejarctan(4w/3)] • √((4 + ω2)/ √(9 + 16ω2)) ej(arctan(w/2)-arctan(4w/3))

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