Polar Coordinates and Complex Numbers
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
Imaginary numbers arise frequently in mathematics, but in order to do much with them we need to know more about the complex plane and the rectangular form of complex numbers. The complex plane is a medium used to plot complex numbers in rectangular form, if we think as the real and imaginary parts of the number as a coordinate pair within the complex plane.
Recall that the real numbers can be visualized as points on a line. And we call this the number line. You put 0 in the middle. You decide where 1 is and then you've defined the length 1 and you can put 2, 3, 4 and so on, any of the integers on the number line. And then of course fractions fit in between and irrational numbers fit on the line too. Every real number has a place a unique place on the number line.
Now recall complex numbers are numbers that can be expressed in the form a+bi where a and b, these coefficients are real numbers and i is the special number the square root of -1. How do we visualize complex numbers? We graph them on the plane. And we, the way we do it is we plot a+bi as ab. I want to show you some examples. For example plot 3+2i. a is 3 b is 2 so I would just plot 3 2. So here's 3, 1, 2. That would be the point 3+2i. And sometimes it's nice to draw a line connecting it to the origin, and of course the origin in the complex plain is still zero.
Let's plot another point -2. -2 is a complex number even though it's also a real number. It's -2+0i. So the imaginary part of this number is 0. Let's plot this. -2+0i is -2 0 right here. I'll just write -2 and I can also connect it to the origin.
So it's important to note that numbers that are right on the horizontal axis are real numbers and this is therefore called the real axis.
Now let's take a look at another example. 4i. This can be written in the form 0+4i. So the real part is 0. When we plot this, we plot it as 0 4. 0 4 would be 1, 2, 3, 4 here. So this is 4i. And so just as the horizontal axis is the real axis, the vertical axis is the imaginary axis. So think about the horizontal axis. This this is the number line basically, exactly the same as what we had before, only now we've got numbers above it and numbers below it. All of which are imaginary. None of these numbers are real.
Okay. One more example 1-3i. That would be plotted as 1 -3. So over 1 down 3. So this is 1-3i.
That's it. Thats how we plot complex numbers. We call this the complex plain again the horizontal axis is the real axis. The vertical axis is the imaginary axis.