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:
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 ? If has polar form , the rule outlined above tells us that every point on the plane will be scaled by a factor , and rotated by an angle of .
For , multiplying would scale everything by a factor of while rotating by , like this:
For , the absolute value of is
and its angle is , so multiplying by would scale everything by a factor of , which will mean shrinking, while rotating about the origin, which is a clockwise rotation.
For , which has absolute value and angle , multiplication rotates by a half turn about the origin while stretching by a factor of .
Another way to think about these transformations, and complex multiplication in general, is to put a mark down on the number , and a mark down on the number , and to notice that multiplying by drags the point for to the point where started off, since . Of course, it must do this in a way which fixes the origin, since .
Isn't it interesting how facts as simple as and 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 , then multiply the result by its conjugate :
If the angle of is , the angle of the complex conjugate is , so the successive multiplications have no total rotation. We can see this by the fact that the spot that started on ultimately lands on the positive real number line.
What about the magnitude? Both numbers have the same absolute value, , so the total effect of multiplying by then is to stretch everything by a factor of .
Of course, this fact is simple enough to see with the formulas, since , 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 ? If has angle and absolute value , then division does the opposite of multiplication: It rotates everything by and scales by a factor of (which means shrinking by a factor of ).
Example 1: Division by
The angle of is , and its absolute value is , so everything rotates by , which is clockwise, and scales by a factor of (which means shrinking by a factor of ).
Example 2: Division by
The angle of is , and its absolute value is
So now everything rotates by , and is scaled by a factor of .
You may have noticed that these divisions can also be seen as taking the dot that sits on top of and placing it over .
Relating the visualization of complex division with the formula
To compute , where let's say and , we learned to multiply both numerator and denominator by the complex conjugate of , .
In other words, dividing by is the same as multiplying by . Is there a visual way to understand this?
Suppose has angle and absolute value , then to divide by , we must rotate by and scale by . Since , the conjugate, has the opposite angle from , multiplying by will rotate by , like we want. However, multiplying by scales everything by a factor of , when we need to go the other way, so we divide by to correct.
For instance, this is what directly dividing by looks like:
And here is what it looks like to first multiply by its conjugate, , then to divide by the square of its magnitude .
The end result of both is the same.
Want to join the conversation?
- 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)
- They are very helpful. It helps some to work through them multiple times. Even to draw the graphs on paper following the videos. Multiplication transforms Z = 1 +0i to a new complex grid with a new grid, W= 1 + 0i rotated from Z by a + bi.
These videos answer a question I asked in an earlier video on complex numbers.
My question and attempted answer are linked here, in the Dividing Complex Numbers video
- 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)
- 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)
- I'm even more confused after reading this section...(26 votes)
- 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)
how they calculated it ??(8 votes)
- 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)
- this article is so confusing(9 votes)
- 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?(6 votes)
- 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)
- 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)
- 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:
Shouldn't it be:
Because, I guess, following is not true
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],
- I think you are overthinking this!
The value of a real number squared (or in this case a sum of squares) is never going to be negative so
|a² + b²| = a² + b².(6 votes)
- 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)
- It helps to understand how the graph changes, if you do multiple points it is much more fun.(5 votes)