WEBVTT mathematics/math-analysis/selhorst-jones
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Hi--welcome back to Educator.com.
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Today, we are going to talk about complex numbers.
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At this point, we know a lot about factoring polynomials and finding their roots.
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Still, there are some polynomials we can't factor.
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There is no way to reduce them into smaller factors, so they are irreducible; and they simply have no roots.
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There is just no way to solve something like x² + 1 = 0.
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There are no roots to x² + 1, because for that to be true, we would need x² = -1.
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But when you square any number, it becomes a positive; if we square +2, then that becomes +4.
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But if we square -2, then that is going to become +4, as well; the negatives hit each other, and they cancel out.
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This pattern is going to happen for any negative number and any positive number, and 0's are just going to stay 0's.
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So, there is no way we can square a number and have it become negative.
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But what if that wasn't the whole story--what if there was some special number we hadn't seen, that, when squared, does not become positive?
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That is an interesting idea; if that is the case, we had better explore--let's go!
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We are looking for a way to solve x² = -1; in other words, we are looking for something that is the square root of -1.
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We are looking for something that, when you square it, gives out -1.
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So, here is a crazy idea: why don't we just make up a new number?
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We will try something really crazy, and we will just create a number out of whole cloth.
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We will imagine a special number that becomes -1 when squared; so we are making a new number.
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And since we are using our imagination to think of this new thing, we will call it an imaginary number, and we will denote it with the symbol i.
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We know that i = √-1, whatever that means...which means that, when you square i, you get -1.
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The square root of 4 is 2, because when you square 2, you get 4.
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So, when you square i (since it is the square root of -1), you get -1.
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These two ideas are how we are going to define this new thing.
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When you are writing it, I would also recommend writing it with a little curve on the bottom, just so you don't get it confused.
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Sometimes, if you are writing quickly, you might end up not even putting a dot,
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at which point it would be hard to see whether you meant to put down 1 or you meant to put down i.
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So, I recommend putting a tiny little curve at the bottom, and then a dot like that, when you are writing it by hand.
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That way, you will have some way of being able to clearly see that you are talking about the imaginary number, and not a normal number.
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With this new idea of i, we can solve the original equation: since i is equal to √-1, we can just take the square root of both sides.
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And remember: when you take the square root of both sides, you have to introduce a plus/minus.
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The square root of both sides of an equation...plus/minus shows up, no matter what.
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So, we get x = ±i; let's check it really quickly.
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If we have positive i, squared, then that is going to be equal to i², which is equal to -1, as we just had here.
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And our other possibility, if we have (-i)²...well, negative times negative becomes positive.
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So, we have positive i squared; but i² is, once again, -1; so it checks out--both of these are, indeed, solutions.
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So, we have found how to solve x² = -1.
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We have this new idea of taking a square root of -1, which means, when we square that thing,
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this thing that we have just created, the imaginary number, we will have -1.
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We can now use this idea to take the square root of any negative number.
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We are not just limited to taking the square root of -1; we can do it on anything.
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We just separate out that √-1; that will become an i; and then we take the root as normal.
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For example, √-25 would become...we pull out -1, so that is the same thing as 25 times -1.
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So, we separate this out; we break it into two square roots (a rule we are allowed to do): times √-1.
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So, √-1 becomes i; √25 is 5; so we have gotten 5i out of that.
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We look at the square root of -98; well, that is the square root of 98, times -1 inside.
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So, we can separate that out, and we will get √98 times √-1.
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√-1 will become i; but what is the square root of 98?
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I am not quite sure, so we need to break it up a bit more; look at how we can break that into its multiplicative factors.
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98 is the same thing as 49 times 2; the square root of 49 is 7; 7 times 7 is 49; so we have 7√2i.
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Finally, the square root of -60; we can see that as 60 times -1, so we separate out the -1 from √60, so this becomes i.
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What is √60? How can we break that into its factors?
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Well, we have a 6 times a 10; that is still not quite enough, though, so we break that up some more.
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We have 2 times 3 for 6, and 2 times 5 for 10, still times i; we see we have a 2 here and a 2 here,
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so we can pull them out, because they come as a pair.
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2 times √(3 times 5); we can't do that, so we might as well just turn it into one number, 2√15 times i.
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So, we can now take the square root of any negative number by having this idea of i, the imaginary number.
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With this idea of the imaginary number in place, we can create a new set of numbers, which we will call the complex numbers.
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And we denote it with ℂ; if you are writing that by hand, you make a normal C, and then you make a little vertical line like that.
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We still want to be able to talk about the real numbers, which, remember, we denote with this weird ℝ symbol.
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So, we will need them to appear along with i; so we need to have real numbers show up along with this imaginary number.
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And so, we just saw that we can have imaginary numbers that had this real component multiplied against them.
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We took the square root of -25, and we got 5i; so we are going to have to have some number times i;
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and we will also want to have a real part, a; so we will have the real part that will be a.
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a is the real part; and then we will have the imaginary part that will be bi; bi is the imaginary part.
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And a and b are both real numbers; they are just coming out of that real set, like usual.
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This gives us an entirely new form of number: as opposed to just being stuck with real numbers,
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we have a way of having a real number and a complex number, both interacting with each other.
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They will come together as a package; and this is the idea of the complex numbers.
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Two complex numbers are equal when the parts in one number are equal to the parts in the other.
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If we have a + bi, and we are told that that is equal to c + di, then that means the real parts have to be equal.
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So, we know that a and c are equal.
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Also, we know that the imaginary parts have to be equal; both parts have to be equal for this equality to hold.
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bi and di are the same thing, which means that b and d must be the same thing, since clearly i is going to be the same thing on both sides.
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Great; all right, so how do we do our basic arithmetic with these things?
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Addition and subtraction first: we add and subtract complex numbers pretty much like we are used to.
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a + bi plus c + di just means we are going to combine our real parts (they will become a + c);
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and our imaginary parts (bi and di) will come together, and we will get (b + d) all times i; bi + di becomes (b + d)i.
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The same thing over here; if we have a + bi minus c + di, then we have (a - c), and the bi's do the same thing with the di's.
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So, we are going to have b - d, because remember: there is that minus symbol there; so it is b - d there.
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a + bi minus c + di...we can also think of that as just distributing this negative sign, like we did before.
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So, we are now adding a -c and a -di; that is one way of looking at subtraction.
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So, real parts add and subtract together, and imaginary parts add and subtract together.
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Other than the fact that they stay separate, it pretty much works like normal.
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So, as long as they stay separate--they stay on their two sides--imaginaries can't interact directly with the reals;
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reals can't interact directly with the imaginaries when we are keeping it in addition and subtraction--it is pretty much just normal.
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Let's look at two examples: 5 + 2i plus 8 - 4i--our 5 and 8 will be able to interact together, because they are both real numbers.
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And let's color-code that, just so we can see exactly what is going on.
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So, with our colors before, with red representing reals again, we have 5 + 8, and then it will be +, and then our imaginaries interact as well; 2i - 4i.
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5 + 8...we get 13; and 2i - 4i (because this was a -4i here)...we get -2i, which we could also write as 13 - 2i.
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13 + -2i and 13 - 2i mean the same thing; great.
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If we did it with subtraction, (5 + 2i) - (8 - 4i), then we can distribute this negative sign; and we get -8,
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and then we will cancel out the minus there, and we get + 4i; so 5 - 8 + 2i + 4i.
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5 - 8 becomes -3; plus 6i; great.
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All right, multiplication is pretty much going to work very similarly, as well.
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Now, notice: we have two things in this sort of factor-looking form; so we do it as a FOIL expansion.
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We are going to do that same idea of distribution.
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We will multiply again in the same way that we distributed before.
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a will multiply on c, and a will multiply on di; so we get ac and adi.
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And then, bi will multiply on c, and bi will multiply on di; so we will get...bi on c gets us bci, and bi on di will get us bdi².
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Now, remember: i² = -1; so when we have this i² here, it cancels out, and it is like we have subtraction.
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So, we put together our real components now: bd and ac gets us ac - bd, because it was -bd.
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And we have our imaginary components, adi and bci, so we have (ad + bc)i.
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Remember, this i² equals -1, so it will just transform during the process of simplification.
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Now, you could memorize this formula right here, but I wouldn't recommend memorizing this formula right here,
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because you already know the FOIL expansion, and as long as you can remember i² = -1, that will keep it easier.
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That is the better way to do it.
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All right, let's see some examples: (1 + 2i)(5 - i): 1 times 5 becomes 5; 1 times -i becomes -i;
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2i times 5 becomes 10i; 2i times -i becomes -2...and we have two i's, so 2i².
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Once again, i² is equal to -1; that cancels out and becomes a plus.
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So, 5 - i + 10i + 2: 5 and 2 combine to give us 7; and -i + 10i combine to give us + 9i.
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Great; the next one is (6 + 10i)(5 + 3i): 6 times 5 is 30; 6 times 3i is 18i; 10i times 5 is 50i; and 10i times 3i is 30i².
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Once again, remember: i² is -1, so we have -30, at which point our -30 and positive 30 cancel each other out.
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And we are left with just 18i + 50i, so we get a total of 68i.
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Great; the final one is division; now, division is a little more tricky.
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Consider if we had (10 - 15i)/(1 + 2i): now, at first, we might think,
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"Oh, we have 10 and 1; we have -15 and 2i; so we will get 10/1 and -15/2,
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because the i's will cancel out"; but that would be wrong.
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We have to divide by the entire denominator, not just bits and pieces.
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For example, to see why this has to be the case, imagine if we had 5 + 5, over 3 + 2; that is really 10/5, which equals 2.
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But we could get confused and think that that was going to be 5/3 + 5/2; but division does not distribute like that.
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We are not allowed to do that; so we can't do the same thing here with our 1 + 2i.
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We can't break it up and distribute the pieces, because it is nonsense in real numbers; so it is definitely going to be nonsense in the complex.
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What if we break it up, and we put 1 + 2i onto the 10, and then we put 1 + 2i on the -15i separately?
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We get 10/(1 + 2i), and we get + -15/(1 + 2i).
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Well, that is true; we broke it up; we can do that with normal things.
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We could have it, if we wanted to, going back to 5 + 5 over 3 + 2--we could have that as 5/(3 + 2) + 5/(3 + 2).
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But that doesn't help us; we still have to divide, ultimately, by this 1 + 2i.
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We don't know how to divide by a complex number yet; that is our problem.
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Simply put, we have no idea how to divide by complex numbers; that is our problem.
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Addition and subtraction made natural sense; real numbers stuck together; imaginary numbers stuck together.
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FOIL was able to allow us to do multiplication--we just did normal distribution, and we remembered the rule that i² becomes -1.
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But division...we don't have a good understanding of what it means to divide by a complex number.
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That is tough; now, what we could do, if there was some clever way to get rid of having a complex number in the denominator--
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if we could somehow make it into an alternate form where we disappeared the complex number in the denominator-- we would be good.
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Hmm...to figure out this clever method that we want, first notice something you might have seen while we were working on quadratics.
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If you have (x - 2)(x + 2), you get x², and then + 2x, but also - 2x;
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since we have the -2 and the +2 here, they end up canceling each other out, and so we are left with just x² - 4.
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And there is no middle term with just x; there is no x that shows up.
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We are able to get rid of it, and have only the doubled and then no x whatsoever.
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We can expand this idea to complex numbers: we do a similar pattern, (1 - 2i)(1 + 2i); let's work that out.
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We get 1 - 2i + 2i - 4i²; so +2i and -2i cancel each other out, because we have the negative here and the plus here.
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And then, we have 4i², so that will cancel and become a plus; and we get 1 + 4 = 5.
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So, we have been able to figure out a way to multiply this thing and get just a real number.
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So, if we use this pattern, (a + bi)(a - bi), you automatically get a real number.
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When you multiply it out, it results in a number that has no imaginary part.
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You can get that i to disappear entirely, and get something that is completely real.
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This idea we call the **complex conjugate**; it comes up often enough, and it becomes important enough,
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that we give it a special name (complex conjugate); and we also give it a special symbol.
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We denote it with a bar over the number; so if we have a + bi as our complex number,
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we can talk about its conjugate with a bar over it; and that is a - bi.
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The conjugate of a + bi is a - bi; and what is the conjugate of a - bi?
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Well, we just flip it again, back to a plus, vice versa; a + bi is the conjugate of a - bi; the conjugate of a - bi is a + bi.
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They just end up flipping between each other, as long as we are doing conjugates.
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Notice that, whenever you multiply a complex number by its conjugate, it always results in a real number.
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So, by multiplying with the conjugate, we can get rid of imaginary things; we can get rid of it.
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Multiply by a conjugate; you always get a real number out of it.
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So, let's look at that: a + bi times a - bi: we get the a², and we will get - abi + abi;
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plus here, minus here; those cancel out; i² is a -1, so that causes that to become a plus.
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So, we will end up with a² + b², and no i; a and b were just real numbers, so we have something that is entirely real.
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We started with imaginary things, but by multiplying these together,
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choosing carefully what we had, we were able to knock out imaginaries entirely and get something that is just real.
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With this idea of complex conjugates in mind, we can now deal with division.
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We simply turn the denominator into a real number by multiplying top and bottom of the denominator's conjugate.
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We want to get the bottom to turn into just a real number, because we know how to divide by reals; you just put it on a fraction.
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So, with c + di, we need to multiply c - di.
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Of course, we can't just multiply the bottom because we feel like it;
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so we also have to multiply the top by c - di, as well, because something over itself is always 1
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(as long as you didn't start with 0/0; then the world explodes).
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But as long as it is something over something, and that something isn't 0, you get 1.
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So, c - di over c - di...we can do that; we just trust, intrinsically, that dividing something by itself is 1.
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That is the nature of division; that is the point of it.
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So, ac + bd + bc - ad times i over c² + d²; that is what it will end up simplifying to.
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And we could work this out, and we would see that this formula ends up working out.
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I don't want you to memorize this formula; I don't even really see a good point to working through it.
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The important thing to know is: just remember to multiply top and bottom.
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Remember to multiply top and bottom by the denominator's conjugate.
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This idea of being able to multiply by a conjugate--that is the really cool thing.
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You could memorize a formula, but it is not going to help you to memorize a formula, because it is hard to recall a formula like this.
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It is much easier to remember that I have division of a complex number.
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I multiply by the conjugate, because I want to get rid of real numbers.
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I have to multiply top and bottom, though, because of course, if you did otherwise, you would just be playing fantasy.
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You have to multiply by the same thing on the top and the bottom to keep it what you started with.
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All right, let's see an example: (10 - 59) over (1 + 2i).
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We want to multiply by the conjugate of (1 + 2i) (if we wanted to, we could express that with a bar all over the top of it), which would be (1 - 2i).
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We multiply by that, and we know we will have gotten to just a real number.
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1 - 2i multiplies top and bottom; and it does have to come in parentheses, because it is a whole thing multiplying some other whole thing.
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You don't just get multiplied bits and pieces.
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We work that out: 10 times 1 gets us 10; 10 times -2i gets us -20i; -15i times 1 gets us -15i; -15i times -2i gets us +30i².
00:18:27.600 --> 00:18:45.800
What is on the bottom? We have (1 + 2i)(1 - 2i); 1 times 1 gets us 1; + 2i, - 2i; those will cancel out; -2i + 2i, and then -4i².
00:18:45.800 --> 00:18:51.900
Remember, i² becomes -1; so we cancel out like that.
00:18:51.900 --> 00:18:55.500
And then, we also see that -2i + 2i cancel each other out.
00:18:55.500 --> 00:19:01.500
What does this become next? We combine things: 10 - 30, our real parts on the top, become -20.
00:19:01.500 --> 00:19:21.900
-20i - 15i becomes -35i; what is on the bottom? 1 + 4 is 5, so we can divide -20/5, minus 35i/5; so that gets us -4 - 7i.
00:19:21.900 --> 00:19:28.500
Great; all right, now that we understand the basics of how to work with complex numbers,
00:19:28.500 --> 00:19:31.800
we are now at a point where we can actually see how to factor irreducible quadratics.
00:19:31.800 --> 00:19:36.000
It is now possible for us to factor previously irreducible quadratics and find their roots.
00:19:36.000 --> 00:19:40.300
So, x² + 1, we see, is now factored into (x + i) and (x - i).
00:19:40.300 --> 00:19:49.500
Let's check this: we get that this would be equal to x² - ix + ix - i².
00:19:49.500 --> 00:19:53.600
Oops, ix; I didn't write that whole thing.
00:19:53.600 --> 00:20:00.900
Those cancel each other out; i² becomes + 1, so we get x² + 1.
00:20:00.900 --> 00:20:06.400
Sure enough, it checks out; and we have found a way to be able to factor this thing that, before, we could not factor.
00:20:06.400 --> 00:20:10.600
It used to be irreducible, but now we see that, through the complex numbers, it is not irreducible at all.
00:20:10.600 --> 00:20:16.600
It is totally factorable; we can revisit the quadratic formula and use it to find the roots of these supposedly irreducible quadratics.
00:20:16.600 --> 00:20:21.100
What used to be irreducible for us is no longer, so we can use the quadratic formula.
00:20:21.100 --> 00:20:25.900
Previously, we couldn't use it when b² - 4a was less than 0, because there was no square root of a negative number.
00:20:25.900 --> 00:20:32.100
But now we know that that just means an imaginary number; so if our discriminant, b² - 4a, shows it is less than 0,
00:20:32.100 --> 00:20:36.000
then that means, not that we have no answers, but that we just have imaginary answers.
00:20:36.000 --> 00:20:42.400
Cool; furthermore, because of the ± √(b² - 4ac) part in the quadratic formula,
00:20:42.400 --> 00:20:45.700
we see that complex conjugates must come in conjugate pairs.
00:20:45.700 --> 00:20:54.200
If b² - 4ac was less than 0, so this gives out stuff times i, then we have this ± thing;
00:20:54.200 --> 00:21:00.500
so it is going to be plus stuff(i), minus stuff(i); so we have one version that is a +i and one version that is a -i.
00:21:00.500 --> 00:21:03.800
That is what happens when we are doing a conjugate pair.
00:21:03.800 --> 00:21:14.200
We have a + bi; its conjugate is a - bi, so if we have stuff + stuff(i) and minus stuff(i), that is what we have right there.
00:21:14.200 --> 00:21:19.600
All right, so if we have a polynomial where we know that a + bi is a root
00:21:19.600 --> 00:21:24.800
(that is to say, when you plug it in you get 0), then we know that a - bi has to also be a root;
00:21:24.800 --> 00:21:28.500
these things come in conjugate pairs all the time.
00:21:28.500 --> 00:21:33.900
So, we talked a lot about the complex numbers; but we probably have this nagging question in the back of our head.
00:21:33.900 --> 00:21:39.600
Are they real? They are clearly not real numbers, because we are saying that they are not the real numbers,
00:21:39.600 --> 00:21:46.100
which are numbers like 5, 0, π, √2...we have been working with them all up until now.
00:21:46.100 --> 00:21:50.600
But are they real--are they legitimate--are they something that we really can use,
00:21:50.600 --> 00:21:52.700
and not be thinking that we shouldn't be using these?
00:21:52.700 --> 00:22:03.300
I mean, they have the word "imaginary" in their definition; do we really want to be trying to do science or math with something that is inherently imaginary?
00:22:03.300 --> 00:22:07.500
Let's think about this: what does it mean for a number to be legitimate?
00:22:07.500 --> 00:22:13.500
What is this idea of a number being a legitimate number that is valid for science, valid for math?
00:22:13.500 --> 00:22:17.400
Now, we probably all agree that 1, 2, 3...those are totally valid.
00:22:17.400 --> 00:22:21.600
You could pick up one rock; you could pick up two rocks; you could pick up three rocks.
00:22:21.600 --> 00:22:26.000
We could actually have these things in our hands and say, "Look, I have that many objects."
00:22:26.000 --> 00:22:32.200
And we might not be able to pick up 5 billion rocks, but we can get this idea that we could count that many rocks in front of us.
00:22:32.200 --> 00:22:37.200
So, that seems pretty valid; these nice whole numbers are perfectly reasonable.
00:22:37.200 --> 00:22:45.900
But what about 1/2 being a real number? 1/2 seems pretty valid, because we could take a pizza, and we could cut the pizza in half.
00:22:45.900 --> 00:22:52.300
We take a pizza; we cut it down the middle; and now, all of a sudden, we have two chunks of pizza.
00:22:52.300 --> 00:22:58.200
1 here; 1 here; we are left with two objects that come together to form a whole.
00:22:58.200 --> 00:23:04.200
But at the same time, we could say, "Well, this is one object, and this is one object; so it is 1 and 1."
00:23:04.200 --> 00:23:10.500
But we could also say it is 1/2 of what we originally started with, and it is 1/2 of what we originally started with.
00:23:10.500 --> 00:23:17.200
So, it is 1/2 and 1/2; so it is a little bit more questionable that this is valid,
00:23:17.200 --> 00:23:21.500
because can you actually pick up a half-object? No, it is an object in and of itself.
00:23:21.500 --> 00:23:25.500
But it is connected to other things, so it is not perfectly valid--not as valid as the rocks.
00:23:25.500 --> 00:23:29.900
We can grab rocks; we can hold rocks; but we can definitely believe in half-numbers.
00:23:29.900 --> 00:23:34.500
We can believe in rational numbers; we can believe in fractions; it seems reasonable.
00:23:34.500 --> 00:23:38.900
Well, OK, what about something even more slippery--what about the negative numbers?
00:23:38.900 --> 00:23:45.100
Negatives are pretty bad; or we could go even worse, and we could talk about irrational numbers.
00:23:45.100 --> 00:23:48.500
How can you possibly hold -1 rock? What does that mean?
00:23:48.500 --> 00:23:51.300
Can you hold √2 rock? Can you hold π rock?
00:23:51.300 --> 00:23:53.500
You can't hold these things in your hand; so are they valid?
00:23:53.500 --> 00:24:00.500
We can't cut a π slice of pizza; we can't cut a √2 slice of pizza; what does this mean now?
00:24:00.500 --> 00:24:04.800
Are they really valid? We certainly used them a lot before--we are used to using them.
00:24:04.800 --> 00:24:10.700
So, they seem reasonable in that way; but are they things that are real in the real world?
00:24:10.700 --> 00:24:16.000
Don't worry; they are not illegitimate; it doesn't mean that they are illegitimate because you can't hold them in your hand.
00:24:16.000 --> 00:24:18.400
We can use them to represent things in the real world.
00:24:18.400 --> 00:24:23.900
We can talk about an object falling with negative numbers; we can say it is going a negative height.
00:24:23.900 --> 00:24:28.300
As opposed to a positive height, where it goes up, it goes a negative height, where it goes down.
00:24:28.300 --> 00:24:34.300
Or maybe you have $100 in your bank account, but then you pull out 150; that leaves you with negative $50 in your bank account.
00:24:34.300 --> 00:24:37.600
So, you have an overdrawn bank account; we talk about that with negative numbers.
00:24:37.600 --> 00:24:41.100
So, that seems pretty reasonable; we could also talk about √2.
00:24:41.100 --> 00:24:44.100
√2 is able to connect the sides of a square.
00:24:44.100 --> 00:24:57.800
If we have a square where all of the sides are the same on our square, then the connection between one side and the diagonal is side times √2.
00:24:57.800 --> 00:25:00.100
So, we can figure that out from the Pythagorean theorem.
00:25:00.100 --> 00:25:01.900
That makes sense; there is some stuff going on.
00:25:01.900 --> 00:25:06.300
Or if we go ahead and we look at a circle, we will see π showing up.
00:25:06.300 --> 00:25:14.900
If we want to talk about the circumference of a circle (pardon my circle; it is not quite perfect), it will be π times 2 time the radius.
00:25:14.900 --> 00:25:20.400
So, there is π showing up; or, if we wanted to talk about the area, it will be π times the radius squared.
00:25:20.400 --> 00:25:22.500
So, there are relationships going on in circles.
00:25:22.500 --> 00:25:29.100
And circles are real-life things; we see circles in lots of places; we see spheres and other circular objects in lots of places in real life.
00:25:29.100 --> 00:25:33.800
So, it seems reasonable to count √2, π, -1...they are all valid numbers,
00:25:33.800 --> 00:25:39.300
not because we can hold it in our hand, but because we can use it for totally reasonable things.
00:25:39.300 --> 00:25:45.400
So, ultimately, these numbers are "real" (not to say real numbers, but "real" numbers, numbers that we believe in),
00:25:45.400 --> 00:25:49.900
because they have meaning--because they are useful for something.
00:25:49.900 --> 00:25:56.800
A number is valid, not because we can hold it in our hands, but because it is useful and/or interesting.
00:25:56.800 --> 00:26:02.000
That is what makes a number a valid number that we want to work with--because we can either use it in real things,
00:26:02.000 --> 00:26:05.500
or it is really interesting and fascinating--it is telling us cool stuff.
00:26:05.500 --> 00:26:11.600
After all, math is a language; and in language, we can talk about things that aren't just concrete.
00:26:11.600 --> 00:26:21.100
You can talk about things like "cat" and "tree"; but at the same time, you can also express abstract concepts--things like "justice" and "freedom."
00:26:21.100 --> 00:26:24.700
You can walk down the street, and you can point at a cat, and you can point at a tree.
00:26:24.700 --> 00:26:30.400
But you can't really hold a justice in your hand; and you can't say, "Oh, look, here is a freedom."
00:26:30.400 --> 00:26:33.900
They are not things that you can hold; they are not tangible, real things.
00:26:33.900 --> 00:26:38.000
They are abstract concepts that require us to think in this other way.
00:26:38.000 --> 00:26:43.400
And that is how the numbers work: 1, 2, 3...they are representing concrete things that we can really hold.
00:26:43.400 --> 00:26:49.400
But we can also talk about abstract ideas, like √2 or π, that are telling us relationships that are really useful.
00:26:49.400 --> 00:26:52.800
We might not be able to hold it in our hand; but it is still a really useful idea.
00:26:52.800 --> 00:26:59.500
So, it is just as valid; "cat" and "justice" are both valid things, because they are useful to us.
00:26:59.500 --> 00:27:04.200
They represent something worthwhile; they represent something interesting.
00:27:04.200 --> 00:27:08.300
It is the exact same way with the complex numbers; this is how it is with the complex numbers.
00:27:08.300 --> 00:27:13.800
You can't hold i rocks in your hands; you can't hold 52i in your bank account.
00:27:13.800 --> 00:27:21.300
But they still have validity; there is still meaning there; they are still valid; they still have meaning.
00:27:21.300 --> 00:27:25.700
In fact, they have direct connections to the real world; so that might be our other issue:
00:27:25.700 --> 00:27:32.900
"OK, I can believe in the fact that numbers get to be valid when they are interesting; but are they useful--can we use them in the real world?"
00:27:32.900 --> 00:27:37.100
Sure enough, you can: complex numbers show up a lot in electrical engineering.
00:27:37.100 --> 00:27:40.500
They show up in advanced physics; and they show up in other fields of science.
00:27:40.500 --> 00:27:43.100
They also show up in lots of advanced mathematics.
00:27:43.100 --> 00:27:47.600
If you are interested in mathematics--in the really, really high, interesting stuff--complex start to show up a lot.
00:27:47.600 --> 00:27:50.800
They are totally valid; you can prove real things; they are really meaningful.
00:27:50.800 --> 00:27:56.800
By using complex numbers, we can actually model real-world phenomena; and we can make accurate predictions.
00:27:56.800 --> 00:28:05.000
Complex numbers are proven to be useful; we can actually use a complex number and get truth out of it that we can then measure in the real world.
00:28:05.000 --> 00:28:10.000
You don't get a complex number of things; but you can have a complex number help you on your way
00:28:10.000 --> 00:28:14.400
to finding an accurate measurement, to finding something and predicting something that actually works.
00:28:14.400 --> 00:28:20.300
So, complex numbers are totally valid in terms of being useful in the real world, and also just as a thought construct.
00:28:20.300 --> 00:28:29.400
In many ways, the name "imaginary" is unfortunate: they are not imaginary in terms of "they don't count; they aren't really there."
00:28:29.400 --> 00:28:34.400
They are just imaginary because the name stuck; there is no reason that they are less valid than real numbers.
00:28:34.400 --> 00:28:37.600
They aren't less valid; they are just as valid as any other number.
00:28:37.600 --> 00:28:42.700
They are not real numbers, which is to say they are not ℝ; they are not those numbers that we talked about before.
00:28:42.700 --> 00:28:45.600
But the complex numbers can still represent reality.
00:28:45.600 --> 00:28:48.400
So, they are not real numbers, but they still show reality.
00:28:48.400 --> 00:28:54.600
They are imaginary, but only in name; they are actually things that can be used to show real life.
00:28:54.600 --> 00:28:57.600
They tell us all sorts of useful things, and they are pretty cool.
00:28:57.600 --> 00:29:06.300
Complex numbers are legitimate and valid; they are not real numbers, but they are "real" in the sense that they are a part of the real world.
00:29:06.300 --> 00:29:11.500
All that said, nonetheless, complex numbers are not going to be something that we will see a lot.
00:29:11.500 --> 00:29:14.800
They are totally legitimate; they are valid; but we won't see much of them.
00:29:14.800 --> 00:29:18.600
Complex numbers tend to be connected to advanced math, for the most part.
00:29:18.600 --> 00:29:23.100
And so, it is really going to be more advanced math than we want to study right now.
00:29:23.100 --> 00:29:28.400
So, if you keep going in math, or you keep going and see some really high-level science at some point in a few years,
00:29:28.400 --> 00:29:31.900
you will probably end up seeing complex numbers be used for real things.
00:29:31.900 --> 00:29:37.800
But right now, we are just sort of saying, "Oh, look--complex numbers! That is cool," and we are moving on to something else.
00:29:37.800 --> 00:29:43.100
So, most math courses--especially courses at this level--will limit themselves to just the real numbers,
00:29:43.100 --> 00:29:50.700
because if they go too far, it will get too complex (get the joke?).
00:29:50.700 --> 00:29:55.700
Unless a question specifically asks about complex numbers, or they were directly mentioned in the lesson
00:29:55.700 --> 00:29:58.400
(such as this one), just stick to the real numbers.
00:29:58.400 --> 00:30:02.800
You really want to just stick to the real numbers, unless you are working specifically with the complex numbers,
00:30:02.800 --> 00:30:05.200
or you have been told to work specifically with the complex numbers.
00:30:05.200 --> 00:30:08.400
We will briefly play with complex numbers in a couple of lessons in this course.
00:30:08.400 --> 00:30:13.800
But they are something best explored later on in a more advanced mathematics course, or an advanced science course.
00:30:13.800 --> 00:30:19.100
Thus, in general, limit yourself to using just the real numbers, ℝ, for now.
00:30:19.100 --> 00:30:22.800
And really, that is going to be pretty easy, because it is what you are used to doing.
00:30:22.800 --> 00:30:27.400
You are used to just working with the real numbers; so it is not going to be hard to just go back to working with the real numbers,
00:30:27.400 --> 00:30:30.600
because it is what you have been doing for years and years and years.
00:30:30.600 --> 00:30:32.100
All right, we are ready for some examples.
00:30:32.100 --> 00:30:38.000
Simplify (25 - 45i)/(-3 + 4i); remember, we need to multiply by the conjugate.
00:30:38.000 --> 00:30:44.200
The conjugate to -3 + 4i, which we could denote with a bar over all the top of it, is equal to -3...
00:30:44.200 --> 00:30:48.100
and then we flip the sign on the imaginary part, so it will be - 4i.
00:30:48.100 --> 00:30:55.700
So, we want to multiply this by (-3 - 4i)/(-3 - 4i).
00:30:55.700 --> 00:31:04.300
Now, notice: you have to put parentheses around all of this, because the whole thing is multiplying--not just bits and pieces, but the whole thing.
00:31:04.300 --> 00:31:18.400
So, we work this out; 25 times -3 becomes -75; 25 times -4i becomes -100i; -45i times -3 will become positive 45...
00:31:18.400 --> 00:31:35.900
that is 3 times 5 off of 150, or + 135i; -45 times -4 becomes positive, so we will get 4 times 5 off of 200, so 180i.
00:31:35.900 --> 00:31:50.900
Divided by...-3 times -3 gets us positive 9; -3 times 4i gets us + 12i; +4i times -3 gets us -12i; 4i times -4i gets us -4i².
00:31:50.900 --> 00:31:56.200
So, we see that we have -12i + 12i, and also when we have i², it becomes positive.
00:31:56.200 --> 00:32:03.500
Oops, I accidentally made a typo here: -45 times -4i will become + 180i².
00:32:03.500 --> 00:32:08.200
So, cancel out that i²; we get -180; now, let's combine things.
00:32:08.200 --> 00:32:18.200
-75 - 180; that will get us -255; -100i + 35i will get us +35i; what is on the bottom?
00:32:18.200 --> 00:32:29.200
4²...we missed that; sorry--one more mistake; 4 times 4 gets us -4²i²,
00:32:29.200 --> 00:32:37.200
so 4² gets us 16; 9 + 16 is in our division, so divide by 25.
00:32:37.200 --> 00:32:44.500
-255 + 35i; divide by 25; we notice that we can pull out a 5 from all of these; this is 5 times 51.
00:32:44.500 --> 00:32:52.400
This is 5 times 7; this is 5 times 5; so we go through and cancel one of the 5's on all of them.
00:32:52.400 --> 00:33:07.500
And we are left with -51 + 7i, all over 5, which, if we wanted to, we could alternately represent as -51/5 + 7/5 i,
00:33:07.500 --> 00:33:10.400
keeping our imaginary part and our real part completely separate.
00:33:10.400 --> 00:33:14.400
Both of these are totally legitimate answers; we would know what we were talking about in either case.
00:33:14.400 --> 00:33:22.100
All right, the second example: Given that x = -2 + i is a root to the below polynomial, find the other root and verify both.
00:33:22.100 --> 00:33:26.900
Remember: if x = -2 + i is one of our roots, the conjugate is also the case.
00:33:26.900 --> 00:33:34.800
So, x bar, the conjugate of x being -2 + i, is going to be...what is the conjugate of that?...-2 - i.
00:33:34.800 --> 00:33:40.700
So, we know what the other root is; the other root is -2 - i, and our first root is -2 + i.
00:33:40.700 --> 00:33:46.800
We are guaranteed that a complex conjugate must be the other root, from what we talked about earlier.
00:33:46.800 --> 00:33:49.400
So now we are told to verify both of them.
00:33:49.400 --> 00:33:51.400
There are two different ways we can verify this.
00:33:51.400 --> 00:33:57.900
First, we could verify this through factors; we could show that, if we were to use these as factors...
00:33:57.900 --> 00:34:05.800
because remember, knowing a root tells you a factor; remember, if we know that there is a root at k,
00:34:05.800 --> 00:34:11.200
then we know that there is a factor, (x - k); so if we know that there is a root at (-2 + i),
00:34:11.200 --> 00:34:18.500
then we know that there is a factor of (x - -2 + i), following that same pattern of x - k.
00:34:18.500 --> 00:34:21.400
It is just that k, in this case, is two things.
00:34:21.400 --> 00:34:27.200
That is times (x - (-2 - i)) for our other factor.
00:34:27.200 --> 00:34:31.500
So, if we can multiply these two factors together, and we can get x² + 4x + 5,
00:34:31.500 --> 00:34:34.800
then we will have verified that those must be the roots, because they are the factors,
00:34:34.800 --> 00:34:38.500
and there is this deep connection between roots and factors; you can go either way.
00:34:38.500 --> 00:34:45.900
So, let's work this out: simplify the insides first: x minus a negative will become + 2 - i;
00:34:45.900 --> 00:34:52.800
times x minus a minus will become + 2 + i; we can start working this out.
00:34:52.800 --> 00:35:00.100
x times x becomes x²; x times 2 becomes + 2x; x times i will become + ix.
00:35:00.100 --> 00:35:09.200
2 times x will become + 2x; 2 times 2 will become + 4; 2 times i will become + 2i; -i times x will become -ix;
00:35:09.200 --> 00:35:16.300
-i times 2 will become -2i; -i times +i will become -i².
00:35:16.300 --> 00:35:20.100
-i² becomes +1, because the i² cancels out.
00:35:20.100 --> 00:35:22.200
And now, let's work through and see this.
00:35:22.200 --> 00:35:25.200
So, let's simplify this: x²: how many other x²'s do we have?
00:35:25.200 --> 00:35:30.300
That is the only one, so we get x² + 2x; how many other x's do we have?
00:35:30.300 --> 00:35:37.400
We have x there, 2x there, and no other x's; so we put those all together, and we get + 4x.
00:35:37.400 --> 00:35:42.400
ix's--how many ix's do we have? We have that ix and that ix, so ix - ix.
00:35:42.400 --> 00:35:47.500
They cancel each other out, and they completely nullify each other; so we don't have to put them down at all.
00:35:47.500 --> 00:35:51.800
How many constants do we have? 4 there; don't forget the 1 that came out of our i².
00:35:51.800 --> 00:35:56.200
So, we have 4 + 1, because it flipped the sign; that is + 5.
00:35:56.200 --> 00:36:03.000
And then 2i - 2i; once again, they nullify each other, so we get x² + 4x + 5; it checks out; great.
00:36:03.000 --> 00:36:04.300
We found the answer.
00:36:04.300 --> 00:36:10.800
The alternate way that we could do this is: we could do this by verifying that they are, indeed, roots.
00:36:10.800 --> 00:36:22.500
So, we could do this another way by showing that they are roots; let's start by showing that x = -2 + i is a root.
00:36:22.500 --> 00:36:32.300
We plug that in; x², (-2 + i)², plus 4(-2 + i), plus 5.
00:36:32.300 --> 00:36:44.000
(-2 + i)² becomes: -2 times -2 becomes positive 4; -2 on i, plus i on -2, becomes -4i; i on i becomes + i².
00:36:44.000 --> 00:36:53.300
And then, continue on: plus 4 on -2 becomes + -8; 4 on i becomes +4i; and pull down the 5.
00:36:53.300 --> 00:37:03.100
i² becomes -1; notice that we have + 4i - 4i, so they eliminate each other here and here.
00:37:03.100 --> 00:37:11.700
4 - 1 becomes 3; -8 + 5 becomes -3; and we get 0...sure enough, that is a root, because it produces 0.
00:37:11.700 --> 00:37:24.300
The other one: let's plug in x = -2 - i; we plug that one in: (-2 - i)² + 4(-2 - i) + 5.
00:37:24.300 --> 00:37:33.400
-2 times -2 is positive 4; -2 on -i and -i on -2 get us + 4i; -i on -i gets us + i².
00:37:33.400 --> 00:37:44.600
Plus -8, minus 4i, plus 5...so we see that we have a positive 4i here and a negative 4i here; they eliminate each other.
00:37:44.600 --> 00:37:56.400
We have this i²; it becomes -1; so 4 and -1 gets us 3; -8 and 5 gets us -3, which, once again, equals 0; so they are both roots.
00:37:56.400 --> 00:38:00.200
There are two different ways to do it: we can show that these are the factors that would be given by those roots,
00:38:00.200 --> 00:38:05.700
and when you multiply those factors, you get back exactly to where you started; that checks out.
00:38:05.700 --> 00:38:11.400
Or alternately, we can do it by roots and show that when you plug that in, you get the zeroes; so that checks out.
00:38:11.400 --> 00:38:16.900
Great; the third example: Factor x² - 8x + 19.
00:38:16.900 --> 00:38:21.900
Well, we know that this is probably going to involve complex numbers; it is probably a little bit hard to figure it out in terms of complex numbers.
00:38:21.900 --> 00:38:25.100
But can we find the roots? Sure enough, we can find the roots.
00:38:25.100 --> 00:38:29.500
Let's find roots, and then we will use the roots to give us factors.
00:38:29.500 --> 00:38:33.200
Remember: once you know roots, you know factors; so we find the roots first.
00:38:33.200 --> 00:38:36.800
We can just use the quadratic formula, because now we can use it on anything.
00:38:36.800 --> 00:38:39.600
We don't have to worry about if it is a complex or not.
00:38:39.600 --> 00:38:43.900
The discriminant won't hold us back, because now we can just get imaginary answers, as well.
00:38:43.900 --> 00:38:54.700
We have x =...the roots occur at [-b ± √(b² - 4ac)]/2a.
00:38:54.700 --> 00:39:01.800
And hopefully, you were able to say that out loud before I said it, to yourself, because really, you want to have that one memorized.
00:39:01.800 --> 00:39:04.200
I said it the last time we talked about the quadratic formula.
00:39:04.200 --> 00:39:09.300
The quadratic formula comes up enough in math and science that it is ultimately something you really want to have memorized.
00:39:09.300 --> 00:39:12.800
All right, so what is our b? Our b is -8.
00:39:12.800 --> 00:39:27.500
So, we plug that in: [-(-8) ± √((-8)² - 4 (what is our a? our a is a 1) (1) (times...what is our c? c is 19)(19)...
00:39:27.500 --> 00:39:33.000
let's move that square root over all of the way; 2 times...a is 1 again, so 2 times 1.
00:39:33.000 --> 00:39:47.200
That equals -(-8) (gets us positive 8), plus or minus the square root of...64; what is 4 times 19? that is 76, so minus 76; all over 2.
00:39:47.200 --> 00:39:55.200
We divide out the 2, so we will get 8/2; that gets us 4; plus or minus the square root of 64 - 76; that will still be over 2.
00:39:55.200 --> 00:39:56.900
Let's put it over that, just so we don't forget that.
00:39:56.900 --> 00:40:12.100
64 - 76 gets us -12; so we have 4 ± √-12...so we can pull that out as an i, so we will get √12 i, over 12,
00:40:12.100 --> 00:40:25.300
equals 4 ± √12...what is √12? √12 we can see as √4(3), which equals 2√3,
00:40:25.300 --> 00:40:36.000
so plus or minus 2√3 i, over 2; look, we have 2 and 2; those cancel out, and we are left with all of our roots.
00:40:36.000 --> 00:40:42.200
They are when x is equal to 4, plus or minus the square root of 3, times i.
00:40:42.200 --> 00:40:45.900
Those are our roots; however, those aren't our factors.
00:40:45.900 --> 00:40:50.100
We want to find what the factors are; so let's get that in another color.
00:40:50.100 --> 00:41:03.400
If we know that our roots are 4 ± √3i, remember: if you know k is a root, then that tells you x - k is a factor.
00:41:03.400 --> 00:41:17.600
So, in this case, our roots are x = 4 + √3i, and x = 4 - √3i, which is good, because they came as a conjugate pairing there.
00:41:17.600 --> 00:41:20.300
So, those are both of our possibilities; those are both of our factors.
00:41:20.300 --> 00:41:34.300
x - k: our factors will be x minus this one right here, so minus (4 + √3i)...not that whole thing...
00:41:34.300 --> 00:41:46.600
I put that parenthesis on the wrong place; i...the parentheses close there; times (x - this thing here, (4 - √3i).
00:41:46.600 --> 00:41:50.400
So now, let's simplify it, so we can get the factors in a nice, slightly-simpler form to look at.
00:41:50.400 --> 00:42:01.800
x - 4 - √3i and x - 4 + √3i; we have factored it by being able to do that.
00:42:01.800 --> 00:42:03.900
And if we wanted to, we could also expand this and check this.
00:42:03.900 --> 00:42:08.300
And we would be able to show that that is, indeed, exactly what it is; great.
00:42:08.300 --> 00:42:15.100
The final example: What is i³, i⁴, i⁵, i⁶, i⁷, i⁸, etc.?
00:42:15.100 --> 00:42:18.500
What pattern appears as we go through these powers of i?
00:42:18.500 --> 00:42:21.000
Let's take a look at how we work through it.
00:42:21.000 --> 00:42:25.300
If we have i¹, just plain i, we have i.
00:42:25.300 --> 00:42:26.700
That is just what it is; it is just i.
00:42:26.700 --> 00:42:30.900
What about when we have i²? Well, by definition, that was -1.
00:42:30.900 --> 00:42:32.900
So, let's see the way it keeps going as we take this up.
00:42:32.900 --> 00:42:39.400
i³...we multiply the -1 by one more i, so we would get -1 times i, or just -i.
00:42:39.400 --> 00:42:47.600
i⁴ would be equal to...i times i gets us -i²; -i²(i²) cancels, and we get positive 1.
00:42:47.600 --> 00:42:57.900
-i² cancels, and we get positive 1; so we are left at 1, just a plain + 1.
00:42:57.900 --> 00:43:04.100
What if we keep going? i⁵ is equal to...well, we multiply by 1, so it is just i, once again.
00:43:04.100 --> 00:43:10.400
i⁶ would be equal to i², multiplying by one more i, which we know is -1.
00:43:10.400 --> 00:43:17.600
i⁷ is equal to i³, which is equal to...we already figured this out; that was -i.
00:43:17.600 --> 00:43:22.100
i⁸...well, that is going to be equal to i⁴, because we just multiply the one above.
00:43:22.100 --> 00:43:26.800
We already figured out what i⁴ is; that is going to be positive 1.
00:43:26.800 --> 00:43:28.300
Let me make that plus sign a little clearer.
00:43:28.300 --> 00:43:31.400
i⁹...if we just kept going, we would have i⁵;
00:43:31.400 --> 00:43:37.400
we already figured out what i⁵ was--that was i¹, which is just i; and so on, and so on, and so on.
00:43:37.400 --> 00:43:47.400
So, the pattern repeats every 4.
00:43:47.400 --> 00:43:51.800
What we need to do is: we basically need to divide by 4 and see what we have.
00:43:51.800 --> 00:44:03.100
What we can do is divide the exponent of i by 4; then, what do we do next?
00:44:03.100 --> 00:44:08.000
Let's do a quick check: if we did i⁹, 4 goes into 9 how many times?
00:44:08.000 --> 00:44:13.600
It goes in twice; so we would have 8; 9 - 8 is 1, so we would get a remainder of 1.
00:44:13.600 --> 00:44:34.300
So then, you look at the remainder, and that tells you that it is equal to i to whatever-you-just-figured-out-your-remainder-is.
00:44:34.300 --> 00:44:40.700
So, for example, if we wanted to figure out what i^80 is (which is divisible by 4),
00:44:40.700 --> 00:44:47.500
we can see that is just i to the 4 times 4 times 4 times 4 times 4; if we figure that out for i^80,
00:44:47.500 --> 00:44:51.800
then we can figure out that what that is equivalent to...by 4...how many times does that go into 80?
00:44:51.800 --> 00:45:00.100
4 goes into 8 twice, so that gets us 8 - 0; bring down the 0; 0; we get 20, and our remainder is 0.
00:45:00.100 --> 00:45:08.200
So, that would be the equivalent of i⁰, which is just the same thing as i⁴, which is +1.
00:45:08.200 --> 00:45:12.000
So, that is how you want to do it if you are given a really, really, really large i.
00:45:12.000 --> 00:45:16.300
It is just a question of if you divided it by 4--what would be left over? What would be the remainder?
00:45:16.300 --> 00:45:21.200
And if you end up having a remainder of 0, then it fit perfectly, so it ends up coming out just as 1.
00:45:21.200 --> 00:45:23.400
All right, great; we will see you at Educator.com later.
00:45:23.400 --> 00:45:28.300
And we will finally see how complex numbers tell us something about polynomials, more than just quadratics.
00:45:28.300 --> 00:45:31.400
We will see how they are deeply connected to everything that we have been talking about.
00:45:31.400 --> 00:45:34.700
It will be so deep that it is called the fundamental theorem of algebra.
00:45:34.700 --> 00:45:36.000
All right, see you later--goodbye!