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Visualizing complex number multiplication

Learn how complex number multiplication behaves when you look at its graphical effect on the complex plane.

What complex multiplication looks like

By now we know how to multiply two complex numbers, both in rectangular and polar form. In particular, the polar form tells us that we multiply magnitudes and add angles:
=r(cos(α)+isin(α))s(cos(β)+isin(β))=rs[cos(α+β)+isin(α+β)]\begin{aligned} &\phantom{=}r(\cos(\alpha) + i\sin(\alpha)) \cdot s(\cos(\beta) + i\sin(\beta))\\\\ & =rs[\cos(\alpha + \beta) + i\sin(\alpha + \beta)] \end{aligned}
One great strength of thinking about complex multiplication in terms of the polar representation of numbers is that it lends itself to visualizing what's going on.
What happens if we multiply every point on the complex plane by some complex number z? If z has polar form r, left parenthesis, cosine, left parenthesis, theta, right parenthesis, plus, i, sine, left parenthesis, theta, right parenthesis, right parenthesis, the rule outlined above tells us that every point on the plane will be scaled by a factor r, and rotated by an angle of theta.

Examples

For z, equals, square root of, 3, end square root, plus, i, equals, 2, left parenthesis, cosine, left parenthesis, 30, degrees, right parenthesis, plus, i, sine, left parenthesis, 30, degrees, right parenthesis, right parenthesis, multiplying z would scale everything by a factor of 2 while rotating by 30, degrees, like this:
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For z, equals, start fraction, 1, divided by, 3, end fraction, minus, start fraction, i, divided by, 3, end fraction, the absolute value of z is
square root of, left parenthesis, start fraction, 1, divided by, 3, end fraction, right parenthesis, squared, plus, left parenthesis, start fraction, 1, divided by, 3, end fraction, right parenthesis, squared, end square root, equals, start fraction, square root of, 2, end square root, divided by, 3, end fraction
and its angle is minus, 45, degrees, so multiplying by z would scale everything by a factor of start fraction, square root of, 2, end square root, divided by, 3, end fraction, approximately equals, 0, point, 471, which will mean shrinking, while rotating minus, 45, degrees about the origin, which is a clockwise rotation.
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For z, equals, minus, 2, which has absolute value 2 and angle 180, degrees, multiplication rotates by a half turn about the origin while stretching by a factor of 2.
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Another way to think about these transformations, and complex multiplication in general, is to put a mark down on the number 1, and a mark down on the number z, and to notice that multiplying by z drags the point for 1 to the point where z started off, since z, dot, 1, equals, z. Of course, it must do this in a way which fixes the origin, since z, dot, 0, equals, 0.
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Khan Academy video wrapper
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Isn't it interesting how facts as simple as z, dot, 1, equals, z and z, dot, 0, equals, 0 can be so helpful in visualizing complex multiplication!

A visual understanding of complex conjugates

Let's look at what happens when we multiply the plane by some complex number z, then multiply the result by its conjugate z, with, \bar, on top:
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If the angle of z is theta, the angle of the complex conjugate z, with, \bar, on top is minus, theta, so the successive multiplications have no total rotation. We can see this by the fact that the spot that started on 1 ultimately lands on the positive real number line.
What about the magnitude? Both numbers have the same absolute value, vertical bar, z, vertical bar, equals, vertical bar, z, with, \bar, on top, vertical bar, so the total effect of multiplying by z then z, with, \bar, on top is to stretch everything by a factor of vertical bar, z, vertical bar, dot, vertical bar, z, with, \bar, on top, vertical bar, equals, vertical bar, z, vertical bar, squared.
Of course, this fact is simple enough to see with the formulas, since left parenthesis, a, plus, b, i, right parenthesis, left parenthesis, a, minus, b, i, right parenthesis, equals, a, squared, plus, b, squared, equals, vertical bar, a, plus, b, i, vertical bar, squared, but it can be enlightening to see it in action!

What complex division looks like

What happens if we divide every number on the complex plane by z? If z has angle theta and absolute value r, then division does the opposite of multiplication: It rotates everything by minus, theta and scales by a factor of start fraction, 1, divided by, r, end fraction (which means shrinking by a factor of r).

Example 1: Division by square root of, 3, end square root, plus, i

The angle of square root of, 3, end square root, plus, i is 30, degrees, and its absolute value is 2, so everything rotates by minus, 30, degrees, which is clockwise, and scales by a factor of start fraction, 1, divided by, 2, end fraction (which means shrinking by a factor of 2).
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Example 2: Division by start fraction, 1, divided by, 3, end fraction, minus, start fraction, i, divided by, 3, end fraction

The angle of start fraction, 1, divided by, 3, end fraction, minus, start fraction, i, divided by, 3, end fraction is minus, 45, degrees, and its absolute value is
square root of, left parenthesis, start fraction, 1, divided by, 3, end fraction, right parenthesis, squared, plus, left parenthesis, start fraction, 1, divided by, 3, end fraction, right parenthesis, squared, end square root, equals, start fraction, square root of, 2, end square root, divided by, 3, end fraction
So now everything rotates by plus, 45, degrees, and is scaled by a factor of start fraction, 3, divided by, square root of, 2, end square root, end fraction, approximately equals, 2, point, 121.
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You may have noticed that these divisions can also be seen as taking the dot that sits on top of z and placing it over 1.

Relating the visualization of complex division with the formula

To compute start fraction, z, divided by, w, end fraction, where let's say z, equals, a, plus, b, i and w, equals, c, plus, d, i, we learned to multiply both numerator and denominator by the complex conjugate of w, start overline, w, end overline, equals, c, minus, d, i.
start fraction, z, divided by, w, end fraction, equals, start fraction, a, plus, b, i, divided by, c, plus, d, i, end fraction, equals, start fraction, a, plus, b, i, divided by, c, plus, d, i, end fraction, dot, start fraction, c, minus, d, i, divided by, c, minus, d, i, end fraction, equals, start fraction, left parenthesis, a, plus, b, i, right parenthesis, left parenthesis, c, minus, d, i, right parenthesis, divided by, c, squared, plus, d, squared, end fraction, equals, start fraction, z, dot, start overline, w, end overline, divided by, vertical bar, w, vertical bar, squared, end fraction
In other words, dividing by w is the same as multiplying by start fraction, start overline, w, end overline, divided by, vertical bar, w, vertical bar, squared, end fraction. Is there a visual way to understand this?
Suppose w has angle theta and absolute value r, then to divide by w, we must rotate by minus, theta and scale by start fraction, 1, divided by, r, end fraction. Since start overline, w, end overline, the conjugate, has the opposite angle from w, multiplying by start overline, w, end overline will rotate by minus, theta, like we want. However, multiplying by start overline, w, end overline scales everything by a factor of r, when we need to go the other way, so we divide by r, squared, equals, vertical bar, w, vertical bar, squared to correct.
For instance, this is what directly dividing by 1, plus, 2, i looks like:
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And here is what it looks like to first multiply by its conjugate, 1, minus, 2, i, then to divide by the square of its magnitude vertical bar, 1, plus, 2, i, vertical bar, squared, equals, 5.
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The end result of both is the same.

Want to join the conversation?

  • male robot johnny style avatar for user maier.juerg
    As these short videos look nice I am at a complete loss what I should be able to learn or understand from looking at it?
    (62 votes)
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  • leaf green style avatar for user George
    The document opens with: "By now we know how to multiply two complex numbers, both in rectangular and polar form. In particular, the polar form tells us that we multiply magnitudes and add angles". Where in the sequence was this taught? I totally missed this and do not understand.
    (31 votes)
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    • duskpin ultimate style avatar for user stalris
      I'm on the same boat as you. I don't think he directly mentioned it in any video.

      He did show how to divide two complex nubers in polar form by first converting them to exponential form. You can extend that to multiplication.

      Polar form -> Exponential Form:
      r(cos(α) + sin(α)i) = r * e^(iα)
      s(cos(β) + sin(β)i) = s * e^(iβ)

      (r * e^(iα)) * (s * e^(iβ)) =
      r * s * e^(αi + βi) =
      r * s * e^( (α + β) * i )

      From there you can convert the exponential form back into polar form where the modulus(?) = r * s and the argument = (α + β) giving us:

      r * s (cos(α + β) + sin(α + β) * i)
      (6 votes)
  • blobby green style avatar for user Murtaza Badshah
    I'm even more confused after reading this section...
    (26 votes)
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  • piceratops ultimate style avatar for user Michel Vermonden
    By now we know how to multiply two complex numbers, both in rectangular and polar form. In particular, the polar form tells us that we multiply magnitudes and add angles:

    Which videos about multiplication is he referring to?
    (13 votes)
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  • spunky sam green style avatar for user Taha Anouar
    r(cos(α)+isin(α))⋅s(cos(β)+isin(β))
    ​=rs[cos(α+β)+isin(α+β)]
    how they calculated it ??
    (8 votes)
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    • piceratops tree style avatar for user VVCephei
      If you work out the multiplication of cos(a) + i*sin(a) and cos(b) + i*sin(b), you'll get:
      (cos(a)cos(b) - sin(a)sin(b)) + i*(sin(a)cos(b) + sin(b)cos(a))
      The expression in the first set of parentheses is a formula for cos(a + b), and the expression in the second set of parentheses is sin(a + b).

      Division requires an extra step, but you will get cos(a - b) + i*sin(a - b)
      (7 votes)
  • blobby green style avatar for user abadyelhertany2000
    this article is so confusing
    (9 votes)
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  • blobby green style avatar for user DC83
    Man I am having some difficulty keeping up with all the terminology etc and thinking in a mathematical sense. I can do all the exercises no problem, but I would really like to be able to think in a logical/mathematical sense and work through practice session using that rather than seeing patterns/repetition. Is there a book you guys would recommend to be able to think more objectively/logically?
    (5 votes)
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  • male robot hal style avatar for user Mike
    What does total rotation refer to in the sentence "successive multiplications have no total rotation"? It means that instead of going back to the starting point (the two angles do not "cancel each other out"), the point moves along the x-axis?
    (2 votes)
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    • female robot grace style avatar for user tyersome
      Net rotation might be a better phrasing.
      My understanding is that the two rotations are by the same degree but have opposite signs and so cancel out. The scaling factors however are the same and so are multiplied together. The net result is no rotation and scaling by the magnitude squared.
      (7 votes)
  • blobby green style avatar for user abnave.m
    I didnt get the case magnitude of multiplication by z and then z'.

    It says the total effect of multiplying by z and then z' is to stretch everything by a factor of |z|.|z'|=|z|^2.
    Then lt says, that above fact can be seen easily from formula:
    (a+bi)(a-bi)=a^2+b^2=|a+bi|^2
    Shouldn't it be:
    |(a+bi)(a-bi)|=|a^2+b^2|=|a+bi|^2
    Because, I guess, following is not true
    a^2+b^2=|a+bi|^2
    But, |(a+bi)(a-bi)|=|a+bi|^2 can be proved correct as follows:
    |(a+bi)(a-bi)|=|a^2+b^2| = (a^2+b^2)^2 (no imaginary part) ...[I]
    |a+bi|^2 = (a^2+b^2)^2 ...[II]
    From [I] and [II],
    |(a+bi)(a-bi)|=|a+bi|^2
    (2 votes)
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  • leaf yellow style avatar for user Anders Bogga
    What's the purpose of rotating and scaling the grid as well as the dot? Is it only to visualise the change or is it for another reason?
    (1 vote)
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