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Complex number forms review

Review the different ways in which we can represent complex numbers: rectangular, polar, and exponential forms.

What are the different complex number forms?

Rectangulara+bi
Polarr(cos(θ)+isin(θ))
Exponentialreiθ

Rectangular form

a+bi
The rectangular form of a complex number is a sum of two terms: the number's real part and the number's imaginary part multiplied by i.
As such, it is really useful for adding and subtracting complex numbers.
We can also plot a complex number given in rectangular form in the complex plane. The real and imaginary parts determine the real and imaginary coordinates of the number.
Want to learn more about complex number rectangular form? Check out this video about the complex plane and this video about adding and subtracting complex numbers.

Polar form

r(cos(θ)+isin(θ))
Polar form emphasizes the graphical attributes of complex numbers: absolute value (the distance of the number from the origin in the complex plane) and angle (the angle that the number forms with the positive Real axis). These are also called modulus and argument.
Note that if we expand the parentheses in the polar representation, we get the number's rectangular form:
r(cos(θ)+isin(θ))=rcos(θ)a+rsin(θ)bi
This form is really useful for multiplying and dividing complex numbers, because of their special behavior: the product of two numbers with absolute values r1 and r2 and angles θ1 and θ2 will have an absolute value r1r2 and angle θ1+θ2.
Want to learn more about complex number polar form? Check out this video.

Exponential form

reiθ
Exponential form uses the same attributes as polar form, absolute value and angle. It only displays them in a different way that is more compact. For example, the multiplicative property can now be written as follows:
(r1eiθ1)(r2eiθ2)=r1r2ei(θ1+θ2)
This form stems from Euler's expansion of the exponential function ez to any complex number z. The reasoning behind it is quite advanced, but its meaning is simple: for any real number x, we define eix to be cos(x)+isin(x).
Using this definition, we obtain the equivalence of exponential and polar forms:
reiθ=r(cos(θ)+isin(θ))

Want to join the conversation?

  • leaf green style avatar for user Abby Hilsman
    How can you calculate the arctan of e.g. 2(sqrt3)/6 without a calculator?
    (7 votes)
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    • cacteye blue style avatar for user Jerry Nilsson
      𝜃 = arctan(2√3∕6) ⇒
      tan 𝜃 = sin 𝜃∕cos 𝜃 = 2√3∕6 = √3∕3 = 1∕√3

      So, we have the ratio
      sin 𝜃 : cos 𝜃 = 1 : √3, which means that if sin 𝜃 = 𝑎, then cos 𝜃 = 𝑎√3

      From the Pythagorean identity we have
      sin²𝜃 + cos²𝜃 = 1 ⇒ 𝑎² + 3𝑎² = 1 ⇒ 𝑎 = ±1∕2

      So,
      sin 𝜃 = ±1∕2, cos 𝜃 = ±√3∕2, which means that the angle we're looking for is either in the first quadrant (sin 𝜃, cos 𝜃 > 0) or the third quadrant (sin 𝜃, cos 𝜃 < 0).

      Either way we're dealing with a right triangle whose legs are of lengths
      1∕2 and √3∕2 and whose hypotenuse is 1.
      By reflecting this triangle over its longer leg, we get an equilateral triangle (in this case all three sides are equal to 1), and thereby we know that the original triangle must be a 30-60-90 triangle.

      Now we can use soh-cah-toa to find out that
      sin 30° = 1∕2 and cos 30° = √3∕2

      So, in the first quadrant 𝜃 = 30°, or in radians 𝜃 = 𝜋∕6,
      and in the third quadrant 𝜃 = 𝜋∕6 − 𝜋 = −5𝜋∕6
      (26 votes)
  • leaf green style avatar for user alex.bertke18
    how do you go from 5(cos90+isin90) polar form to rectagular form?
    (0 votes)
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  • aqualine sapling style avatar for user Brie Fossling
    I've been trying to understand how to find ALL polar coordinates for a week and it's getting a little bit frustrating. Could someone explain to me, given polar coordinates of a point, how to find ALL the coordinates? The thing that doesn't make me understand is this 2nπ thing or adding the 2nπ. I just don't get the method....
    (4 votes)
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    • aqualine ultimate style avatar for user Kadmilos
      If you have a solution at π/2, then you have solutions every 2π.
      π/2 + 2π is a solution, π/2 + 2π + 2π is a solution. Because rotating by 2π is a complete rotation around the circle.

      So if you have
      3Θ = π/4 +2πn
      Θ=(1/3)(π/4 + 2πn)
      Θ= π/12 + (2/3)πn

      Then it is a matter of picking the right n to get the solution in the required range
      (7 votes)
  • blobby green style avatar for user smotyer
    I think you should add one more step between being at let's say: 8cos(120) and then all of a sudden saying well that's clearly -4sqrt3 or something. It's not explained in the video either. On my calculator it just tells me -6.928 and I don't know how to get the 'exact' answer. Can you add an extra step or just allow for an answer like -6.928 to also be considered correct? Thank you so much for this free resource. I don't want to seem ungrateful I just don't get how to do that part!
    (5 votes)
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  • piceratops seed style avatar for user Marlene Bond
    How is A complex number question such as z=10 and theta = 210 degrees answered with a square root? it says finding a is: a = 10cos210degrees

    and the answer is:
    a=-5sqrt3

    Well, why must the answer be -5sqrt3 and not -8.660254038 rounded to -8.660

    We feel we are missing something somewhere to not know this.

    Help, please.
    (3 votes)
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  • male robot donald style avatar for user Akshat
    What is the use of the exponential form i didn't understand it ?
    (2 votes)
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  • male robot hal style avatar for user Joseph Arcila
    Why the product of two numbers with angles theta1 and theta2 will have an angle theta1+theta2 ?
    and Why the argument of z^n = n*theta ?
    Thanks!
    (4 votes)
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  • aqualine ultimate style avatar for user Deniz Boloni-Turgut
    For the exercise on polar and rectangular forms of complex numbers, can we use a calculator or is there a way to do the calculations by hand?
    (2 votes)
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  • blobby green style avatar for user Karan Mehta
    Can we take out polar form of pure imaginary numbers
    (2 votes)
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  • aqualine ultimate style avatar for user Elliott Bork
    do you use theta in degrees or radians?
    (2 votes)
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