 The fact that we can solve a quadratic equation by using a formula that sometimes requires us to find things that are impossible to find leads to a very important idea in mathematics, the notion of a complex number. In the 16th century, the Italian mathematician Geralamo Cardano posed the following problem. Find two numbers that add to 10 and multiply to 40. Now if we try to solve this problem, we'll let our numbers be x, and since they have to add to 10, the other one will be 10 minus x. Since we want them to multiply to 40, we want to make sure that x times 10 minus x equals 40. So now let's try and solve this equation. If in doubt, multiply out, and that gives us a quadratic equation. We'll put it in standard form and then apply the quadratic formula. And uh-oh, we're trying to take the square root of a negative number. And negative numbers don't have square roots. And at this point you're probably saying, eh, big deal. So Cardano poses a problem that doesn't have an answer. I can pose questions that don't have an answer. What's the airspeed of a laden swallow? But the reason that Cardano is important in this is he didn't stop here and say, oh, no solution. What he did instead was to point out that if you treated this square root of a negative number as a number, then this expression does give you two numbers that add to 10 and multiply to 40. Now he did point out that the manipulations were as subtle as they were useless. But it turns out they are quite useful. So let's consider the following complex question. While there's no reason to require every equation have a solution, it's nice when they do. And so for now we'll claim that square root negative 60 is a number, just not a real one. We'll say it's an imaginary number. And we can go a little bit further. If n is a negative number, if n is less than zero, then we say that square root of n is a pure imaginary number. This means that the square root of any negative number is a pure imaginary number. So there's a lot of pure imaginary numbers. But let's see if we can simplify this a little bit. So we might note that square root of a times b is the square root of a times the square root of b provided that a and b are both positive. Now could we extend this so that square root of negative n is square root of negative one times square root of n? And this would make our lives much easier because then the only pure imaginary we really have to worry about is square root of negative one. And the quick answer to this question is we can do this if we're careful. And that gives us the following result. Suppose n is greater than zero, then the square root of minus n is going to be the same thing as square root of negative one times square root of n. And because of this factorization property, all pure imaginaries can be rewritten using square root of negative one. And so for convenience, we'll define i, short for imaginary, to be equal to the square root of negative one. So for example, let's say we run into square root of negative twenty-five in a dark alley one night. And we say, eh, you're not so scary, square root of negative twenty-five by our theorem. That's square root negative one times square root of twenty-five. But wait, square root of twenty-five isn't so bad, that's just five. And this square root of negative one, we could call that i as our imaginary unit. And so square root of negative twenty-five is really i times five. Now multiplication, when we include imaginaries in complex numbers, is still commutative. So i times five is the same as five times i. And we usually write the i as the second factor. So instead of writing i times five, which is correct, we would more frequently write this as five i. Now it's vitally important to remember that the rule square root of minus n equals i times square root of n only works for n greater than zero. If we forget, we produce nonsense. So, for example, one is negative one times negative one times one. Eh, okay, I could buy that. I take the square root, so I get square root one equals square root negative one times negative one times one. Still good. I have a theorem that says if I have square root of negative n, I could rewrite that as square root negative one times n. Now sure, this theorem has some fine print, but we could ignore that. But if we do, we're doing something we're not actually allowed to do. Square root negative one is i, but because we've already taken a forbidden step, we really should stop and turn back. But if we keep going, that's i times i times square root of negative one. That's i squared times one. Really bad idea to keep going. Since i is the square root of negative one, remember that means that i is the number when squared gives us negative one. So we know that i squared is equal to negative one. And we end up by saying that one is equal to negative one. Um, wait a minute, that's not true. And the reason that this is a problem is that we based it on a forbidden operation. We cannot break up a product of negative numbers under the square root. What about numbers like our solution minus 10 plus square root negative 60 over 2? The first thing to remember is that because this is a fraction, I can split the fraction apart. This negative 10 over 2, I can just write as negative 5. And because this is square root of negative 60, I can split off that negative one times 60. And that becomes i square root of 60. And again, we typically write the i after the fact. And what this leads to is the following. You'll notice that this number consists of a real number plus a real number times i. And so let a and b be real numbers. Then the expression a plus bi is a complex number with real part a and imaginary part b times i. And the introduction of complex numbers allows us to solve quadratic equations in any situation. So for example, we want to solve the equation x squared plus 6x plus 13 equals 0. So the quadratic formula gives us and we get to this point negative 6 plus or minus square root of negative 16 over 2. So we can rewrite that as negative 6 plus or minus i square root of 16. Square root of 16 is equal to 4. So this equation solution is negative 6 plus or minus 4i over 2. We'll split our solutions negative 6 over 2 plus 4i over 2 or negative 6 over 2 minus 4i over 2. And we'll simplify negative 3 plus 2i or negative 3 minus 2i.