Get free access to over 100,000 answers and learning tools with WyzAnt!



Search 82,181 tutors
FIND TUTORS

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
Sign up for free to access more trigonometry resources like . WyzAnt Resources features blogs, videos, lessons, and more about trigonometry and over 250 other subjects. Stop struggling and start learning today with thousands of free resources!
Don't wait until it's too late! Get help from our trigonometry tutors.
Leylee P.
Contact Now

Message Leylee P.

Send Leylee P. a message explaining your needs and you will receive a response by email.

Please enter the tutor's email address.
Please enter the student's email address.
Please describe how you heard about us.
I have read and agree to the terms of use. *
James W.
Contact Now

Message James W.

Send James W. a message explaining your needs and you will receive a response by email.

Please enter the tutor's email address.
Please enter the student's email address.
Please describe how you heard about us.
I have read and agree to the terms of use. *
Arth P.
Contact Now

Message Arth P.

Send Arth P. a message explaining your needs and you will receive a response by email.

Please enter the tutor's email address.
Please enter the student's email address.
Please describe how you heard about us.
I have read and agree to the terms of use. *