The Cult of Pythagoras: Math and Myths

7. Euler’s Imaginary Mistakes

101 Like zero, negative numbers have sometimes been used to derive apparent contradictions. For example, consider the following: These steps seem to prove the impossible equation, that 1 is equal to its opposite . We expect that something in the sequence of operations must be a mistake. What is it? I will give an original solution to this apparent paradox, and to do so, I’ll first explain the forgotten arguments of a famous mathematician , Leonhard Euler. The paradox above involves operations with square roots of negative numbers, the so-called imaginary numbers. While nowadays mathematicians value these numbers as being as legitimate as any others, for centuries they argued about them. These numbers seem to represent impossible opera- −1 1 = 1 −1 −1 1 = 冪莥 冪莥 1 −1 7 EULER'S IMAGINARY MISTAKES √ −1 √ 1 = √ 1 √ −1 √ −1 √ −1 = √ 1 √ 1 −1 = 1 102 E U L E R ’ S I M AG I N A RY M I S TA K E S tions that often led to paradoxes or contradictions, to perplexing effects, like magic. Hence, many mathematicians refused to use these numbers, and others ridiculed them with words such as “imaginary,” “false,” “unreal,” “absurd,” “nonexistent,” “sophistic,” “unintelligible,” “merely auxiliary quantities,” “impossible numbers,” “quantities that exist merely in the imagination,” “figments ,” “beings of reason,” “unexecutable operations,” “nonsense,” “jargon,” “incorrect forms,” “mistaken forms,” “mere algebraic forms,” “expressions not susceptible of any immediate application,” “hieroglyphs,” “monstrous,” “chimeras,” “fictitious beings that cannot exist nor be understood,” and so forth.1 They even used expressions referring to “evil,” “witches,” and “tortures .” It was not normal for mathematicians to use such nasty expressions, but one reason they did so was that they were so very sure that the notions they were criticizing were so very wrong. Still, since it was possible to extract roots of positive numbers, equations often arose having instead negative numbers in radicals. So mathematicians struggled to make sense of these expressions. Sometimes they disagreed about whether a particular operation was possible; other times they disagreed about the results of some operations. There is something pleasant in anecdotes about great mathematicians who made silly mistakes. Case in point: historians and mathematicians alike sometimes claim that Leonhard Euler, of all people, was confused about how to multiply imaginary numbers. His unlikely slips were published in his famous Complete Introduction to Algebra of 1770. As the story goes, Euler thought that the product rule √a × √b = √(ab) (1) is valid regardless of whether a and b are positive or negative. Mathematicians say that if the radical signs mean the “principal square root operation” (so that √4 = 2), then Euler was wrong because his rule seems to say √  −4 √  −9 = √     −4        ×       −9 = √  36 = 6 whereas mathematicians now say √  −4 √  −9 = √ −1 √ −1 √ 4 √ 9 = (−1)(2)(3) = −6 Again, that’s if we use “principal square roots.” If instead we interpret the signs to mean the “unrestricted root operation” (so that √4 = ±2), then mathematicians say that Euler was still wrong, because √  −4 √  −9 = √4√9(i2 )= ±6(−1) = 6± E U L E R ’ S I M AG I N A RY M I S TA K E S 103 which is not equal to Euler’s √(–4 × –9) = √(36) = ±6. Hence, for over two hundred years, writers have said that for negative numbers a and b the correct rule is √a × √b = −√(ab). (2) One way or another, mathematicians say that Euler was wrong, that he was just confused or mistaken.2 But actually, it was the mathematicians who were confused by Euler’s words. When Euler composed his Algebra, controversies still abounded regarding the rules of how to operate with negative and imaginary numbers. Such numbers were still often demeaned as “impossible.”3 In 1758, Francis Maseres had published his Dissertation on the Use of the Negative Sign in Algebra, part of his bid for the Lucasian Chair of Mathematics at Trinity College, the job that Newton once held. Maseres rejected the use of isolated negative numbers and also of imaginaries. In 1765, François Daviet de Foncenex denounced as useless the representation of imaginary numbers as constituting a line perpendicular to a line of negatives and positives.4 And Euler himself was at the center of a dispute on the question of the logarithms of negative numbers, in opposition to Jean d’Alembert, Johann Bernoulli, and others.5 By 1770, the symbol i was not yet widely used...