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Carbon copy complex number

- Mathematics began as a way to quantify our world, to measure land, predict the motions of planets, and keep track of commerce. Then came a problem considered impossible. The secret to solving it was to separate math from the real world, to split algebra from geometry and to invent new numbers so fanciful they are called imaginary. Ironically, 400 years later, these very numbers turn up in the heart of our best physical theory of the universe. Only by abandoning math's connection to reality could we discover reality's true nature. (light music) In 1494, Luca Pacioli who is Leonardo da Vinci's math teacher publishes "Summa de Arithmetica," a comprehensive summary of all mathematics known in Renaissance Italy at the time. In it, there's a section on the cubic, any equation which today we would write as ax cubed plus bx squared plus cx plus d equals zero. People have been trying to find a general solution to the cubic for at least 4,000 years, but each ancient civilization that encountered it, the Babylonians, Greeks, Chinese, Indians, Egyptians, and Persians, they all came up empty-handed. Pacioli's conclusion is that a solution to the cubic equation is impossible. Now, this should be at least a little surprising, since without the X cubed term, the equation is simply a quadratic. And many ancient civilizations had solved quadratics thousands of years earlier. Today, anyone who's passed eighth grade knows the general solution. It's minus b plus minus root b squared minus four ac all over two a. But most people just plug and chug into this formula completely oblivious to the geometry that ancient mathematicians used to derive it. You know, back in those days, mathematics wasn't written down in equations. It was written with words and pictures. Take, for example, the equation x squared plus 26x equals 27. Ancient mathematicians would think of the x squared term like a literal square with sides of length x. And then 26x, well, that would be a rectangle with one side of length 26 and the other side of length x, and these two areas together add to 27. So how do we figure out what x is? Well, we can take this 26x rectangle and cut it in half. So now I have two 13x rectangles and I can position them so the new shape I create is almost a square, It's just missing this section down here. But I know the dimensions of this section. It's just 13 by 13. So I can complete the square by adding in a 13 by 13 square. Now, since I've added 13-squared or 169 to the left-hand side of the equation, I also have to add 169 to the right-hand side of the equation to maintain the equality. So now I have this larger square with sides of length X plus 13, and it is equal to 196. Now the square root of 196 is 14. So I know that the sides of this square have...