Lesson: The Complex Plane

Question 2

1 point


There are some ways in which the complex numbers seem like R2. When we add them, it is done in a componentwise fashion, as we add the real parts together and the imaginary parts together, with neither calculation affecting the others. And if we multiply a complex number x + yi by a real number s, then we get sx + syi, which is reminiscent of scalar multiplication.

And so it is that when we want to graph complex numbers, we graph them in terms of their two components, the real part and the imaginary part, as though they were elements of R2. We label the x1-axis the real axis, and the x2-axis the imaginary axis, and we equate a complex number x + yi with the point (x, y).

Let’s look at an example. Here’s how we would plot the points 1 + i, (1 + i)-conjugate, which is 1 – i, -2 – 5i, (-2 – 5i)-conjugate, which is -2 + 5i, -3, and i.

Now, since complex addition is done componentwise, we still have a parallelogram rule for addition. And multiplication by a real number also works the same as in R2. But multiplication by a complex number does not have a counterpart in R2. Starting in the next lecture, we will use this visualization of the complex numbers to develop a new way of thinking about complex numbers that will provide us with a geometrical interpretation of complex multiplication, and has many other uses as well.

For now, let’s look at an example of graphing complex addition. So first, we can plot the number 2 + i and the number 1 + 3i. Well, then we see that the sum of (2 + i) and (1 + 3i) is 3 + 4i, which follows the same parallelogram rule for addition as we would see in R2. And we can also look at the scalar multiple of 3(2 + i), which becomes 6 + 3i, which you can see is simply 3 (2 + i)’s stuck end-to-end. However, if we looked at the complex product (2 + i)(1 + 3i), we would get -1 + 7i, and it is not immediately obvious how this point is related to (2 + i) and (1 + 3i).

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