The Imaginary Part: Not By Logic But By Intuition

I decided to write today a little more directly about mathematics.  A book I’m working on led me to review a story I like very much, the incubation and birthing of the complex number.  The story has been told many times (Dantzig’s Number and Nahin’s An Imaginary Tale to name just two).  But most of my students have not been taught anything about how this other kind of number came to be, or they’ve been told the wrong story.

In the book Number, Tobias Dantzig takes note of an essay written in the late sixteenth century by Franciscus Viete in which he proposes the use of vowels to represent unknown quantities in an algebraic expression, and consonants to represent the given magnitudes (Descarte later revised the idea so that x,y,z are used for the unknowns and a,b,c for the givens).  Merely writing down these numberless expressions, allowing them to be, Dantzig suggests, freed the algebraist of their prejudice for only natural number solutions to problems.  Without any forsight, the path to the refinement of the number concept was opened.

But more revealing of the kind of blind searching we never associate with mathematics is the story of the complex number and the imaginary unit.  Before the logical foundation of the real number was even close (a 19th century development), a path to the complex number was opened.  A very short version of the story is this: The 16th century mathematician, Scipione del Ferro found a formula for the solution to a depressed cubic equation (one with no x squared term).  He could use the formula (which involved a number of roots) to find the one real solution to the cubic.  A short time later, Girolamo Cardano (known as Cardan) found a way to extend this idea to the solution of all cubics.  Unlike del Ferro, when Cardan worked with the square roots in the formulas, he allowed himself to work with negative numbers inside the root.  He is quoted in Rudin as saying, “Putting aside the mental tortures involved,”  everything works out.  And later, “So progresses arithmetic subtlety the end of which, as is said, is as refined as it is useless.”   The mathematician Rafael Bombelli (also 16th century) took note of the role complex conjugates were playing in finding roots, and proceeded to develop rules for operations on these yet to be understood thoughts.   One could say he found the complex number domain, a territory that would remain uncharted for some time (a bit more than 200 years).   In 1673 John Wallis began a geometric interpretation of the numbers but it was not until 1797 that Caspar Wessel sees clearly what they are.

Dantzig says the following about the development of mathematical ideas:

Distant outposts were acquired before the intermediate territory had been explored, often even before the explorers were aware that there was an intermediate territory.  It was the function of intuition to create new forms; it was the acknowledged right of logic to accept or reject these forms, in whose birth it had no part. But the decisions of the judge were slow in coming, and in the meantime the children had to live, so while waiting for logic to sanctify their existence, they throve and multiplied.