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https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:complex/x2ec2f6f830c9fb89:complex-num/v/complex-number-intro

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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.

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Intro to complex numbers (video) | Khan Academy

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Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

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I have the complex number 3 minus 4i. I've plotted it on the complex plane. We see that the real part is 3, so we've gone 3 along the horizontal axis, or the real axis. And the imaginary part is negative 4, so we've gone down 4 along the vertical axis. So this right here is the point 3 minus 4i. Now what I want to think about is what the absolute value of 3 minus 4i is. And just as a reminder, absolute value literally means-- whether we're talking about a complex number or a real number, it literally just means distance away from 0. So the absolute value of 3 minus 4i is going to be the distance between 0, between the origin on the complex plane, and that point, and the point 3 minus 4i. So this distance right over here is going to be the absolute value of 3 minus 4i. So how can we think about that? Well, we could literally just set up a right triangle and then use the Pythagorean theorem. So let's think about it. If we wanted to set up a right triangle, the height here, the distance between 0 and negative 4, well, that distance is going to be 4. And then the base of this triangle, the distance between 0 and 3, is just going to be 3. And this is definitely a right angle. This is a horizontal line. This is a vertical line. We can now use the Pythagorean theorem to figure out the absolute value of 3 minus 4i. The distance between this point and 0-- it's the hypotenuse of this right triangle. So we just use the Pythagorean theorem. This side squared, 3 squared, plus this side squared, plus 4 squared, is going to be equal to the absolute value of 3 minus 4i squared, the absolute value squared. So 3 squared plus 4 squared, that's 9 plus 16, which is 25. So you get 25 is equal to the absolute value of 3 minus 4i squared. And we know if you take the absolute value of something, this is just a distance. It's going to be positive. So we want to take the positive square root, the principal square root, of both sides of this. And so we're going to be left with-- well, the principal square root, the positive square root of 25, is 5, is equal to the absolute value of 3 minus 4i. So another way of saying it, this thing right over here is going to be equal to 5. This distance right over here is equal to 5. Now, without having to draw it, one way you could just think about this is I'll have my real part. I have my imaginary part. I could literally take each of those parts, square them, take the sum, and take the square root. So another way of taking it, if you didn't want to visualize all this-- but this is really what we're doing. You could say, well, this is just going to be equal to take the real part squared. Take the imaginary part squared-- so let me write this. Add them together, and then take the square root, or the principal root. The principal root's just the positive square root. So that's going to be the square root of 9 plus 16, which, once again, is equal to 5.

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Absolute value of complex numbers (video) | Khan Academy

# data[].attributes.attachment_link.description
Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.