Complex numbers introduction
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Now that we know a little bit about the imaginary unit i, let's see if we can simplify more involved expressions, like this one right over here. 2 plus 3i plus 7i squared plus 5i to the third power plus 9i to the fourth power. And I encourage you to pause the video right now and try to simplify this on your own. So as you can see here, we have various powers of i. You could view this as i to the first power. We have i squared here. And we already know that i squared is defined to be negative 1. Then we have i to the third power. I to the third power would just be i times this, or negative i. And we already reviewed this when we first introduced the imaginary unit, i, but I'll do it again. i to the fourth power is just going to be i times this, which is the same thing as negative 1 times i. That's i to the third power times i again. i times i is negative 1. So that's negative 1 times negative 1, which is equal to 1 again. So we can rewrite this whole thing as 2 plus 3i. 7i squared is going to be the same thing, so i squared is negative 1. So this is the same thing as 7 times negative 1. So that's just going to be minus 7. And then we have 5i to the third power. i to the third power is negative i. So this could be rewritten as negative i. So this term right over here we could write as minus 5i, or negative 5i, depending on how you want to think about it. And then finally, i to the fourth power is just 1. So this is just equal to 1. So this whole term just simplifies to 9. So how could we simplify this more? Well we have several terms that are not imaginary, that they are real numbers. For example, we have this 2 is a real number. Negative 7 is a real number. And 9 is a real number. So we could just add those up. So 2 plus negative 7 would be negative 5. Negative 5 plus 9 would be 4. So the real numbers add up to 4. And now we have these imaginary numbers. So 3 times i minus 5 times i. So if you have 3 of something and then I were to subtract 5 of that same something from it, now you're going to have negative 2 of that something. Or another way of thinking about it is the coefficients. 3 minus 5 is negative 2. So three i's minus five i's, that's going to give you negative 2i. Now you might say, well, can we simplify this any further? Well no, you really can't. This right over here is a real number. 4 is a number that we've been dealing with throughout our mathematical careers. And negative 2i, that's an imaginary number. And so what we really consider this is this 4 minus 2i, we can now consider this entire expression to really be a number. So this is a number that has a real part and an imaginary part. And numbers like this we call complex numbers. It is a complex number. Why is it complex? Well, it has a real part and an imaginary part. And you might say, well, gee, can't any real number be considered a complex number? For example, if I have the real number 3, can't I just write the real number 3 as 3 plus 0i? And you would be correct. Any real number is a complex number. You could view this right over here as a complex number. And actually, the real numbers are a subset of the complex numbers. Likewise, imaginary numbers are a subset of the complex numbers. For example, you could rewrite i as a real part-- 0 is a real number-- 0 plus i. So the imaginaries are a subset of complex numbers. Real numbers are a subset of complex numbers. And then complex numbers also have all of the sums and differences, or all of the numbers that have both real and imaginary parts.