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Early Modern Europe and Disreputable Numbers



From yesterday's ancients we skip forward quickly (due to my ignorance and subjectivity, as well as the demands of my broader points) to the early modern era in Europe, specifically to a publication by a fellow named Rafael Bombelli, (pictured above) of the city-state of Bologna, in AD 1572.


Negative numbers were still new in Europe then, but the more advanced thinkers, including Bombelli, had a handle on that. By way of review: multiplying two negative numbers yields a positive. Multiplying a negative and a positive yields a negative, multiplying two positives yields a positive.

One consequence of these rules is that there are no square roots of negative numbers. No rational square roots and no irrational square roots either. No square roots of any level of rationality for negative numbers. Why? Because if you had a square root for a negative number then, by definition, you would have a situation in which the multiplication of that number by itself would yield that negative number. Yet negatives multiplied to each other yield positives, as do positives multiplied by each other, so in neither case can we get to a negative. 

Bombelli knew of another mathematician of his day, Cardano, who had evoked the square roots of negatives in order to solve what was known as the cubic equation (never mind what that was or is). Yet they were just an intermediary step in the solution, and Cardano was thoroughly unhappy by the fact that he had to use them even in an intermediary capacity in his work. Cardano said that he resorted to such overly subtle monstrosities only by way of “dismissing mental torture,” and once the equation was solved with their aid, he didn’t want to use them any more. Once he had gotten to the roof, he kicked the ladder away.

This bothered Bombelli. It is, in fact, because he was bothered by Cardano’s approach that he, Bombelli, rather than Cardano is the hero of this step in our story.

Bombelli said in 1572 that we should embrace the ladder. He gave the square roots of negatives a name of their own (not “imaginary” -- we’ll get to that – he called them the “plus of minus.”)  Bombelli said, in effect: let’s embrace the paradoxical quality of these things, these pluses of minus!

Nobody took him out to sea to do away with him, but the reaction of the mathematicians of the world was remarkably unenthusiastic.

Just as the greatest mathematicians of ancient Greece had been aghast at the idea of irrational numbers, those of early modern Europe were shocked by the notion of these odd plus-of-minus numbers, which the carefully devised rules governing the multiplication of negatives had said could not exist.

Among those who second Cardano’s feeling of distaste was the great philosopher Rene Descartes. Early in the 17th century, Descartes wrote that some equations have positive roots, some negative roots, and some “imaginary” roots. He meant the word “imaginary” as a derogation. But it has stuck. Like many put-upon sets of people, the imaginary numbers had come to embrace their denigration in order to win over the denigrators.

Say it loud! Say it proud! We’re imaginary!

Comments

  1. "... by a fellow named Rafael Bombelli, (pictured above)..."

    Isn't it rather Viete?

    ReplyDelete
  2. Oops, you're right. The illustration is of the French mathematician Francois Viete. How I made the mistake I don't know.

    ReplyDelete

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