Complex numbers introduction
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Voiceover:Most of your mathematical lives you've been studying real numbers. Real numbers include things like zero, and one, and zero point three repeating, and pi, and e, and I could keep listing real numbers. These are the numbers that you're kind of familiar with. Then we explored something interesting. We explored the notion of what if there was a number that if I squared it I would get negative one. We defined that thing that if we squared it we got negative one, we defined that thing as i. So we defined a whole new class of numbers which you could really view as multiples of the imaginary unit. So imaginary numbers would be i and negative i, and pi times i, and e times i. This might raise another interesting question. What if I combined imaginary and real numbers? What if I had numbers that were essentially sums or differences of real or imaginary numbers? For example, let's say that I had the number. Let's say I call it z, and z tends to be the most used variable when we're talking about what I'm about to talk about, complex numbers. Let's say that z is equal to, is equal to the real number five plus the imaginary number three times i. So this thing right over here we have a real number plus an imaginary number. You might be tempted to add these two things, but you can't. They won't make any sense. These are kind of going in different, we'll think about it visually in a second, but you can't simplify this anymore. You can't add this real number to this imaginary number. A number like this, let me make it clear, that's real and this is imaginary, imaginary. A number like this we call a complex number, a complex number. It has a real part and an imaginary part. Sometimes you'll see notation like this, or someone will say what's the real part? What's the real part of our complex number, z? Well, that would be the five right over there. Then they might say, "Well, what's the imaginary part? "What's the imaginary part of our complex number, z? And then typically the way that this function is defined they really want to know what multiple of i is this imaginary part right over here. In this case it is going to be, it is going to be three. We can visualize this. We can visualize this in two dimensions. Instead of having the traditional two-dimensional Cartesian plane with real numbers on the horizontal and the vertical axis, what we do to plot complex numbers is we on the vertical axis we plot the imaginary part, so that's the imaginary part. On the horizontal axis we plot the real part. We plot the real part just like that. We plot the real part. For example, z right over here which is five plus three i, the real part is five so we would go one, two, three, four, five. That's five right over there. The imaginary part is three. One, two, three, and so on the complex plane, on the complex plane we would visualize that number right over here. This right over here is how we would visualize z on the complex plane. It's five, positive five in the real direction, positive three in the imaginary direction. We could plot other complex numbers. Let's say we have the complex number a which is equal to let's say it's negative two plus i. Where would I plot that? Well, the real part is negative two, negative two, and the imaginary part is going to be you could imagine this as plus one i so we go one up. It's going to be right over there. That right over there is our complex number. Our complex number a would be at that point of the complex, complex, let me write that, that point of the complex plane. Let me just do one more. Let's say you had a complex number b which is going to be, let's say it is, let's say it's four minus three i. Where would we plot that? Well, one, two, three, four, and then let's see minus one, two, three. Our negative three gets us right over there. That right over there would be the complex number b.