Saturday, September 21, 2019

The History of Imaginary Numbers is a Soap Opera

I first read about this history in Journey Through Genius, by William Dunham, in the chapter titled Cardano and the solution of the cubic. It reads like a soap opera in which the math is done for glory, not for any possible connection to the real world. The people who came up with imaginary numbers thought they were fictitious elements in the process of solving a cubic, and never expected them to have any real meaning. Turns out they do. Imaginary numbers help scientists describe electrical current and probability distributions, among other things.

Years ago, on Living Math Forum, a mom wrote in to ask for help. Her son had asked: “The square root of 1 is 1, so what's the square root of -1 ?” That inspired me to write a math poem, Imaginary Numbers Do the Trick.

Recently I was having a lovely discussion with my editor, Maria Droujkova, and another author, about math storytelling. I realized this topic might possibly make for a good children's story. I'm working on it now. As I think about it, I'm not sure how to find the right age range. The math seems like it requires high school, but the story could interest younger kids, I think.

The History
Here's a short version:
  • Scipione del Ferro solves equations of the form  x3 + mx  = n (called depressed cubics). On his deathbed, in 1506, he passes his method on to his student, Antonio Fior.
  • Niccolo Tartaglia boasts that he can solve cubics of the form x3 + mx2  = n, so in 1535, Fior challenges him with 30 depressed cubics. (These challenges were a common feature of life as a mathematician in 1500's Italy, and provided a way for mathematicians to get more recognition and paying students.) Tartaglia's return problem list to Fior has a variety of problems. Tartaglia does not yet have a solution for the depressed cubic, and sweats it, working feverishly to try to figure it out. At the last moment, he succeeds, and solves all 30 problems. Fior does not do so well, and is humiliated.
  • Gerolamo Cardano comes to Tartaglia, asking him to disclose his method. He begs repeatedly, and Tartaglia, now Cardano's guest in Milan, finally concedes. Cardano takes an oath of secrecy. Tartaglia writes his solution in cipher, as a poem (!). 
  • Cardano takes on a brilliant student, Ludovico Ferrari, with whom he shares the secret. Together, they solve the general cubic, and then Ferrari goes on to solve the quartic. But all their work depends on reducing these to the depressed cubic, which Cardano has sworn not to tell about.
  • Cardano and Ferrari travel to Bologna, and are able to inspect the papers of ... Scipione del Ferro, where they find the solution. Cardano figures that relieves him of his oath and publishes, in his 1545 book, Ars Magna. He gives both del Ferro and Tartaglia credit, but Tartaglia is furious.
  • In the book, Cardano lays out the steps for solving the general cubic. But in doing so, he introduces a mystery. The depressed cubic x3 - 15x  = 4 clearly has solutions x = 4 and x = -2+-√3. And yet the formula found by Ferro, Tartaglia, Cardano, and Ferrari includes a √-121 for this equation. Cardano threw up his hands at the mystery. It was explored but not truly understood 30 years later by Rafael Bombelli. It took another almost two centuries for Euler to finally solve the mysteries of complex numbers.

    Here's a nice write-up I found online, but it suggests different facts than the version in Dunham's book. I will keep reading while I write, so I can hopefully get my facts right.

    My Request
    I'm looking for kids who would like to read my draft versions and tell me what parts they like. If you have kids who understand (at all) the notion of a square root and the idea of what a cubic equation is, would you ask them if they'd like to read my story? (I would, of course, mention them in my book if it gets published.) You, or your kids, can email me at mathanthologyeditor on gmail.

    Just a Beginning
    The Saga of the Imaginary Numbers
    “Mom, I’ve been thinking… If the square root of 1 is 1, what is the square root of -1?”

    “What a fun thing to think about, Althea! What have you figured out so far?”

    “I know that when I square 1 I get 1, and that’s why the square root of 1 is 1. But when I square negative 1, I get 1 too, so shouldn’t the square root of 1 be negative 1 too? But how can it be two things?”

    “Hmm, that’s a strange one, isn’t it? I think there are too many ones in this for me to keep track of things. Let’s switch to 3.

    "I’m going to try to say what you said, but with 3 and 9. 3 squared is 9, so the square root of 9 is 3. But negative 3 squared is still 9, So why isn’t the square root of 9 equal to negative 3 also? Is that basically the same question you asked?”

    “Yes. Except the square root of 9 can’t have two answers. Can it?”

    “Well, somebody a long time ago decided that there should be just one answer for the square root of a number. And so we say that there is the square root of 9 and also the negative square root of 9.”

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