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Complex Numbers

Written by tutor Colin D.

How to Envision Complex Numbers Graphically: The Complex Plane

How to Envision Complex Numbers on a Graph
  • The complex number x + yi corresponds to the point with coordinates (x, y)
  • The x-axis is the real axis
  • The y-axis is the imaginary axis
  • Real numbers are associated with points on the x-axis
              For example: x = x + 0i <- -=""> (x,0)
  • Imaginary numbers are associated with points on the y-axis
              For example: yi = 0 + yi <- -=""> (0,y)

How to Find a Point (P) in the Complex Plane

  • Any point in the complex plane can be identified by the coordinate pair (r, θ)
  • r = distance from the origin to point P (i.e., line segment OP)
  • θ = angle from the positive x-axis (between Quadrants I and IV) to segment OP
  • All points on the terminal side can be expressed as (r cos θ, r sin θ)
              -Because cos θ = adjacent/hypotenuse, and hypotenuse = r, to solve for θ, one would proceed: cos θ = x/r.
               Solving this for x would result in x = r cos θ
              -Likewise, because sin θ = opposite/hypotenuse, solving for θ would result in sin θ = y/r.
               Solving this for y would result in y = r sin θ
  • Piecing it altogether:
              -If we have complex number x + yi
              -Then P has coordinates (x,y)
              -And x = r cos θ, y = r sin θ

Trigonometric (Polar) Form

Trigonometric (Polar) Form on a Graph
  • The trigonometric form of “x + yi” is r(cos θ + i sin θ)
              -This can be derived from earlier equivalences. Because when we had x + yi, we found x = r cos θ and
               y = r sin θ, we can replace x and y with r cos θ and r sin θ, respectively:
                        x + yi = (r cos θ) + (r sin θ)i
              -By factoring out the "r" and multiplying by the "i," this turns into:
                        r(cos θ + i sin θ)
  • r = Modulus or Absolute Value
              r = (x2 + y2)1/2
              r = must be NON-negative
  • θ = Argument of the complex number
              -Any angle coterminal with θ is also an argument for the same complex number
              tan θ = y/x -> θ = arc tan (y/x)

Rectangular (Standard) Form

  • Rectangular form is "x + yi"

How to Change from Rectangular Form to Trigonometric Form

Change from Rectangular to Trigonometric Form on a Graph
  • If A = 2 + 2i
              -First, find "r." Remember, "r" - called the modulus - is the absolute value of the hypotenuse formed by
               sides "x" and "y"
                        r = √(22 + 22)
                           r = √(4+4)
                           r = √8
                           r = 2√2
              -Next, find θ. Remember, θ is called the argument, and is found through the following equation:
               tan θ = y/x because the tangent of the angle formed by "r" and the "x-axis" equals the opposite side
               divided by the adjacent side (i.e., the y-value divided by the x-value).
                        θ = arc tan (y/x)
                           θ = arc tan (2/2)
                           θ = arc tan (1)
                           θ = 45°
  • Then trigonometric form is found by plugging in "r" and "θ":
              A = r(cos θ + i sin θ)
              A = 2√2(cos 45° + i sin 45°)

How to Change from Trigonometric Form to Rectangular Form

Change from Trigonometric to Rectangular Form on a Graph
  • If B = 3√3 (cos 330° + i sin 330°)
              r = 3√3
              cos 330° = √(3/2)
              sin 330° = -1/2
  • Then 3√3 (√(3/2) + -1/2 i) -> 9/2 - i(3√3)/2

How to Express Complex Numbers in Proper Trigonometric Form

Proper Trigonometric Form on a Graph
  • Always remember a few essentials about proper trigonometric form:
              -The modulus (r) must always be non-negative
                        It is the absolute value of the diagonal from the point itself to the origin.
              -The parenthetical expression must be of the form: cos θ + i sin θ.
                        Make sure each term is written as a positive amount.
  • Example: z = 2(cos 30° - i sin 30°)
              -First, express z in rectangular form:
                        2(√(3/2) - 1/2 i) -> √3 - 1i
              -Thus, on a graph, this would consist of moving √3 units to the right, 1 unit down, resulting in a point in
               Quadrant IV.
                        r = √((√3)2 + 12) -> √4 -> √2
                        Using tan θ = y/x, we derive:
                             tan θ = 1/√3 -> arctan 1/√3 = -30° -> hence, θ = -30°
              -Finally, substitute:
                        z = 2[cos (-30°) + i sin (-30°)]

Multiplication & Division in Trigonometric Form

  • NOTE: While rectangular form makes addition/subtraction of complex numbers easier to conceive of, trigonometric form is the best method of conceiving of complex for multiplication/division purposes.
  • If you intend to multiply two complex numbers, z1 = r1 (cos θ1 + i sin θ1), and z2 = r2 (cos θ2 + i sin θ2), the product is derivable by following a few simple steps:
              -Multiply the moduli to find the product modulus: r1 times r2
              -Add the arguments to find the sum argument: cos (θ1 + θ2) + i sin (θ1 + θ2)
              -Multiply the product modulus by the sum argument: r1r2 [cos (θ1 + θ2) + i sin (θ1 + θ2)]
  • To divide two complex numbers:
              -Divide the moduli to get the quotient modulus: r1/r2
              -Subtract the arguments to get the difference argument: cos (θ1 – θ2) + i sin (θ1 – θ2)
              -Multiply the quotient modulus by the difference argument: r1/r2 [cos (θ1 – θ2) + i sin (θ1 – θ2)]
  • Example:
               z1 = √(3/2 + (1/2)i
               z2 = -2 – 2i
               Find z1 * z2:
                        (1) Express each in trigonometric form
                             z1 = 2(cos 30° + i sin 30°)
                             z2 = 2√2(cos 225° + i sin 225°)
                        (2) Multiply moduli:
                             2 * 2√2 = 4√2
                        (3) Add arguments:
                             cos(30° + 225°) + i (sin 30° + 225°)
                        (4) Triangular Form = 4√2 [cos(30° + 225°) + i (sin 30° + 225°)]
                             4√2[cos (255°) + i (sin 255°)]
               To find in Rectangular Form, evaluate the cos 255° and sin 255° and simplify:
                        4√2 [cos (255°) + i (sin 255°)]
                             With the sum and difference formulae:
                                 cos (a+b) = cos a cos b – sin a sin b
                                 sin (a+b) = sin a cos b + sin b cos a
                             With calculator:
                                 cos 255° = -.2588
                                 sin 255° = -.9659
                        -1.464 – 5.464i

DeMoiver's Theorem

  • By repeating the multiplication procedure outlined just above, one may derive DeMoivre’s Theorem, which allows us to compute powers and roots of complex numbers.
  • To illustrate, if we were to continue to multiply z = r (cos θ + i sin θ) by itself, we’d get:
               z2 = r2 (cos 2θ + i sin 2θ)
               z3 = r3 (cos 3θ + i sin 3θ)
               z4 = r4 (cos 4θ + i sin 4θ)
  • For negative exponents, it unfolds in the following pattern:
               z-1 = r-1 [(cos(-θ)) + i sin (-θ)]
               z-2 = r-2 [(cos(-2θ)) + i sin (-2θ)]
  • Formally stated as a rule, DeMoivre’s Theorem reads:
               zn = rn (cos nθ + i sin nθ)
  • EXAMPLE:
               (1 + √3i)5
                        -In trigonometric form:
                             2(cos 60° + i sin 60°)
                        -Apply DeMoivre's Theorem:
                             25 [cos 5(60°) + i sin 5(60°)]
                             32 (cos 300° + i sin 300°)
                             32 (1/2 + i(-√3)/2)
                             16 – (16√3)i

Roots of Complex Numbers

Calculating roots of complex numbers
  • Some basics about visualizing the roots of complex numbers:
               -The n roots of a complex number all lie on the circle formed within the complex plane with center at the origin and radius = (r)(1/n)
               -The n roots on said circle are all equally spaced, beginning at K = 0 and proceeding until k = n-1, progressing at arguments (i.e., intervals) differing by 360°/n
  • Formula:
               -Given any positive integer, n, then the nonzero complex number z (where z = r (cos θ + i sin θ)) has exactly n distinct nth roots, given by the following equation, in which k = 0, 1, 2,...., (n-1):
                        W (sub k) = (r)^n [cos (θ/n + k * 360°/n) + i sin (θ/n + k * 360°/n)]
  • Example: Find the 6th roots of 5 + 12i
               (1) Write 5 + 12i in trigonometric form:
                        r = √(52 + 122) = 13
                        θ = arctan (12/5) ~ 67.38°
                        Thus, 5 + 12i = 13(cos θ + i sin θ)
               (2) Since we're looking for sixth roots (n = 6), we replace n with 6 and simplify:
                        W (sub k) = 131/6 [cos(θ/6 + k * 360/6) + i sin (θ/6 + k * 360/6)]
                        W (sub k) = 1.533 [cos (67.38/6 + k * 60) + i sin 11.23 + 60k)]
                        W (sub k) = 1.533 [cos(11.23 + 60k) + i sin (11.23 + 60k)]
               (3) Plug in the various values of k up to (n-1) where n = 6 (because of the sixth roots).
                        K = 0 -> 1.504 + .299i
                        K = 1 -> .493 + 1.451i
                        K = 2 -> -1.01 + 1.153i
                        K = 3 -> -1.504 + -.299i
                        K = 4 -> -.493 + -1.451i
                        K = 5 -> 1.01 + -1.153i
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