In this lesson you will investigate the multiplication of two complex numbers `v` and `w` using a combination of algebra and geometry. First, remember that you can represent any complex number `w` as a point `(x_w, y_w)` on the complex plane, where `x_w` and `y_w` are real numbers and `w = (x_w + i*y_w)`.
Although one more commonly sees expressions like `v = (a + b*i)` or `w = (c + d*i)`, in this lesson we choose to connect the real and imaginary components of a complex number to its horizontal and vertical coordinates on the complex plane. We will also make significant use of the modulus (distance from the origin) and the argument (angle from the positive real axis) of a complex number, which we will represent as `(r_w, theta_w)` to emphasize the connection to polar coordinates.
Because `w=(x_w + i*y_w)`, you can use the distributive property to write `v*w = v*(x_w + i*y_w) = v*x_w + v*i*y_w`.
In this expanded form, `x_v` and `y_v` are both real numbers. This enables you to use geometry on the complex plane to multiply complex numbers, based on three geometric techniques:
- geometrically multiplying a complex number by `i` (to construct `v*i`),
- geometrically multiplying a complex number by a real number (to construct `v*x_w` and `v*i*y_w`), and
- geometrically adding two complex numbers (to construct the sum `v*x_w + v*i*y_w`).
In the following activities, you will develop these three techniques and use them to find an elegant way to multiply two complex numbers.
- What happens when you dilate a complex number by a scale factor like `3`, `0.5`, or `-2`? Describe the result numerically.
- What happens when you rotate a complex number by an angle such as `pi`/`2`, `pi`, or `3 pi`/2? Describe the result numerically. How can you use your results to multiply a complex number by `i`?
- Start with two complex numbers and apply what you’ve learned to construct their product: `v·w = v*x_w + v*i*y_w`
- Starting with the construction of `v*w` in Part 4 based on algebra, find a much simpler way to multiply two complex numbers by using just two steps: the dilate-rotate method. Then investigate whether this method is commutative with respect to `v` and `w`? (That is, does `w*v = v*w` for complex numbers?) Second, is the composition itself commutative? (That is, does rotate-dilate give the same result as dilate-rotate?)
1 Dilate a Complex Number
In this activity you will explore the effect of dilation of complex numbers about the origin, and describe your results numerically.
You will start by dilating real numbers using several different scale factors, follow up by dilating imaginary numbers, and conclude by dilating complex numbers.
As a result of your investigation, you will describe the numeric effect of dilation on both the rectangular `(x + iy)` and the polar `(r, θ)` forms of expressing the complex number.
- Page 1: What happens when you dilate a number on the real axis by a scale factor like `3`, `0.5`, or `-2`? Describe the result numerically.
- Page 2: What happens when you dilate a number on the imaginary axis by a scale factor? Describe the result numerically.
- Page 3: What happens when you dilate an arbitrary complex number (anywhere on the complex plane) by a scale factor?
- Describe the numeric effect of dilation on both the rectangular `(x + iy)` and the polar `(r, θ)` forms of expressing the complex number being dilated.
2 Rotate a Complex Number
In this investigation you will explore the effect of rotation of complex numbers about the origin, and describe your results numerically.
On the first three pages, you will rotate real, imaginary, and complex numbers by `pi/2`, by `pi`, and by `3 pi/2` radians.
On the fourth page you'll rotate a complex number by an arbitrary angle.
On all four pages, you will use a point on the unit circle to determine the angle of rotation.
As a result of your investigation, you will describe the numeric effect of rotations on both the rectangular `(x + yi)` and the polar `(r, θ)` forms of expressing the complex number. You will also describe how to multiply a complex number by `i`.
- Page 1: Rotate a complex number. Describe how rotation changes the modulus and argument of the original number being rotated.
- Page 2: Rotate a complex number by `pi/2`, `pi`, and `3 pi/2` radians. Describe the numeric effect of each rotation on the rectangular `(x + yi)` form of the original number.
- Page 3: (Optional) Rotate a complex number v by an arbitrary angle. Describe the rectangular `(x + yi)` form of the rotated image in terms of `x_v`, `y_v`, and `phi`.
- Conclusion1: Based on your results, how can you use rotation to multiply a complex number by `i`?
- Conclusion2: (Optional) How does your result on the page 3 investigation relate to the rectangular coordinates of `phi`?
3 Multiply Two Complex Numbers
In this activity you will investigate the multiplication of two complex numbers `v` and `w` using a combination of algebra and geometry. First, remember that you can express `w` as `w=(x_w + i*y_w)`.
Using this formula for `w`, you can use the distributive property to write `v*w = v*(x_w + i*y_w) = v*x_w + v*i*y_w`.
In this expanded form, `x_w` and `y_w` are both real numbers. This enables you to use geometry on the complex plane to multiply the complex numbers `v·w`, based on three geometric techniques to:
- geometrically multiply a complex number by `i` (to construct `v*i`), and
- geometrically multiply a complex number by a real number (to construct `v*x_w` and `v*i*y_w`),
- geometrically add two complex numbers (to construct the sum `v*x_w + v*i*y_w`).
On pages 1–3, you'll review these three techniques.
On page 4 you'll use the techniques to multiply two complex numbers `v` and `w`, and you'll investigate the geometric relationships connecting the complex numbers `v`, `w`, and `v·w`.
4 Multiply by Transforming
In this activity you will explore some additional interesting questions about how complex multiplication works.
- Page 1: You will review the similar-triangle construction that helps to justify the dilate-rotate method of multiplying complex numbers.
- Page 2: You'll do the similar-triangle construction again, but this time without all the other construction lines.
- Page 3: Complex multiplication made simple! Just dilate-rotate one of the vectors using the polar coordinates of the other as `s` and `phi`.
- Page 4: Is complex multiplication commutative? Does `v*w = w*v`?
- Page 5: Is the dilate-rotate composition commutative? Does dilate-rotate give the same result as rotate-dilate?
- Page 6: Experiment on your own! What if the vectors are attached to the real axis? to the imaginary axis? What other investigations can you try?
5 The Fine Print
Requirements:
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Release Information
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This web site is based in part upon work supported by the National Science Foundation under NCSU IUSE award 1712280 (July 2017 through December 2020) and KCP Technologies DRK-12 award ID 0918733 (September 2009 through August 2013). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
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