How Imaginary Numbers Were Invented

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 maths connection to reality could we discover reality's true nature 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 is a section on the cubic any equation which today we would write as ax cubed plus bx squared plus c x 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 pachioli'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 past eighth grade knows the general solution it's minus b plus minus root b squared minus 4 ac all over 2a 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 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 ...