 In 1545, the Italian mathematician Geralama Cardano posed the problem, finding two numbers that add to 10 and multiply it to 40. Cardano found the answer 5 plus the square root of negative 15 and 5 minus the square root of negative 15. But what is the square root of negative 15? It can't be a positive number, since the square of a positive number is positive. And it can't be a negative number, since the square of a negative number is still positive. So, what could it be? A normal person would say that the problem has no solution. But Cardano wasn't a normal person. He was a mathematician. Putting aside the mental tortures involved, these values actually solve the equation. And today we refer to them as complex numbers. If a and b are both positive, then the square root of a, b can be rewritten as the square root of a times the square root of b. If we're careful, we can extend this concept to complex numbers. If n is greater than 0, then the square root of negative n is the square root of negative 1 times the square root of n. This means it will be useful to define i to be the square root of negative 1. So, we can simplify expressions like square root of negative 25. We can write negative 25 as negative 1 times 25. And, since 25 is greater than 0, then we can rewrite square root of negative 1 is i. And square root of 25 is 5. And it's traditional to put i as the second factor, although we don't always do that. Now Cardano's problem actually gives rise to the quadratic equation x squared minus 10x plus 40 equals 0, so let's solve it. And remember, factoring to solve a quadratic is almost always a waste of time. Use the quadratic formula. So, using the quadratic formula, we get the solutions. And let's do a little simplification. First of all, anytime we have a fraction whose numerator is a sum or difference, we can split it apart. This square root of negative 60 we can rewrite. We can rewrite square root of negative 1 as i. And we can simplify the radical square root 60 as... Where here we've written the i first. Now, a number like 5i is called a pure imaginary. Meanwhile, a number like 5 plus i root 15 is a complex number with real part 5 and imaginary part square root of 15. And note that the imaginary part is the real number multiplied by i and does not include the i. For example, let's identify the real and imaginary parts of 3 minus 4i. So remember in the complex number a plus bi, the real part is a, and the imaginary part is b. So the real part is 3. Meanwhile, remember a minus b is the same as a plus negative b. And so we can write 3 minus 4i as 3 plus negative 4i. So what we might call the imaginary component of the number is negative 4i, and so the imaginary part is negative 4. Or consider a number like 5 plus 12i. Let's write a number with the same real part and opposite imaginary part. The real part of the number is 5 and the imaginary part is 12. The opposite of the imaginary part is negative 12. So a number with the same real part and opposite imaginary part would be 5 minus 12i.