The Logarithm of −1: A Mini-Primary Source Project for Complex Variables Students

Author(s):

Dominic Klyve (Central Washington University)

By the mid-eighteenth century, logarithms were well understood by European mathematicians. Every positive number had a logarithm, and finding these values had proven useful, both for the pedestrian needs of calculation and to the theoretical development of mathematics. Perhaps it was inevitable that these mathematicians began to ask whether negative numbers also had logarithms—and of all negative numbers, it certainly made sense to start with log(-1). The mini-Primary Source Project (PSP) The Logarithm of -1 provides students with an introduction to the extension of the logarithm function to this new domain, and to the fundamental issues of mathematical “truth” that this extension raises.

Deciding whether negative numbers had logarithms at all—and then what the values of these logarithms were—proved a significant challenge to eighteenth-century mathematicians. Among the scholars involved in that debate were Jean le Rond d’Alembert (1717–1783) and Leonhard Euler (1707–1783). It would be difficult to find two more disparate mathematicians. Euler was a devout Christian, a modest and humble man, and one who was perhaps more comfortable with mathematics and the sciences than he was in the company of others. d’Alembert, on the other hand, was the epitome of the humanist French philosophe. He was brilliant, to be sure—among other things, he was one of the leaders of the movement of publish the whole of the world’s knowledge in a grand Encyclopédie (perhaps the best example of a pre-internet Wikipedia), and he worked successfully in fields ranging from history to philosophy.

A page from Henry Briggs’s 1617 Logarithmorum Chilias Prima showing the base-10 (common) logarithm of the integers 0 to 67 to fourteen decimal places.
Wikimedia Commons, public domain.

In addition to their vast personal differences, the two men disagreed on a number of mathematical and scientific questions, including the values of the logarithms of negative numbers. One of the most fascinating issues arising from their disagreement is a much larger question: who gets to decide what the value of log(-1) is? Most modern mathematicians would likely agree that some facts from our world are immutable—it’s difficult to imagine, for example, a world in which \(\sqrt{2}\) is rational. Other “facts” generally endorsed by our community of discourse, and taught with authority to our students, are certainly the conscious choice of human definition. For instance, whether the number 1 should be included in the list of prime numbers is clearly a matter of human definition—and experts have come to different verdicts about this question. The early twentieth-century computational number theorist D. H. Lehmer (1905–1991), for example, considered 1 to be prime [Lehmer 1914]. Nearly two thousand years earlier, Nicomachus (60–120) excluded both 1 and 2 from his prime number list.

To which of these categories does the value of log(-1) fall? Does the edifice of existing mathematics force an answer, or are we free to, as it were, use our judgment? This is a delicate kind of question, and it’s one that we don’t often present to our undergraduate students. In contrast, this PSP allows students to glimpse the contingent side of mathematics by following the debate between d’Alembert and Euler as it unfolded, via the original language and thought processes of their time.

It’s worth summarizing the central arguments of the two protagonists here, since both arguments are short and clean. Using the notation of the time, we use “log” to denote the natural logarithm. d’Alembert believed that \(\log(-1)=0\), and he had a few unrelated arguments for why this was so. The most elegant simply applies the existing laws of logarithms and follows them to their natural conclusion. In a letter to Euler (quoted in [Bradley 2007, p. 266]), d'Alembert wrote

All difficulties reduce, it seems to me, to knowing the value of \(\log(-1)\). Now why may we not prove it by the following reasoning? -1 = 1/ -1, so log(-1) = log 1 - log(-1). Thus 2log(-1) = log 1 = 0. Thus log(-1) = 0.

The reader is encouraged to consider this argument, and to see whether they find a flaw in it. (This author confesses he had difficulty finding any such flaws when it was first presented to him.)

An easy way to understand Euler’s ideas about this same question (although not the way that he originally expressed them) is to start with that most famous equation \[e^{\pi i} = -1.\] If we accept this, we need only take the natural logarithm of both sides to find that \(\log(-1) = \pi \cdot i\), a value most certainly different than the one d’Alembert proposed.

But Euler’s ideas went even further.

If we accept (as Euler did) that the equation \[e^{(2n+1)\pi i} = -1\] holds for any integer \(n\), then it would appear that log(-1), paradoxically, takes on the infinitely many different values given by

\[\log(-1) = (2n+1)\pi \cdot i. \]

Graph of complex multi-valued lograithmic function
A modern rendering of the logarithm function on the complex numbers. Designed by user Leonid 2, Wikimedia Commons – CC BY-SA 3.0.

We are returned to our larger question—who is the ultimate arbiter of mathematical truth? In this case, no amount of logic could move either party. In the end, the disagreement between d’Alembert and Euler about this question led to acrimony between the two mathematicians, spilling out into debates about everything from academic precedence to astronomy, and forever tearing asunder a longstanding friendship. While history ultimately declared Leonhard Euler to be the winner, the question of the value of log(-1), and the debate it engendered, still carries lessons for us today.

The complete project The Logarithm of -1 (pdf) is ready for student use and the LaTeX source code is available from the author by request. Instructor notes are provided to explain the purpose of the project and guide the instructor through implementation of the project.

This project is the twentieth in A Series of Mini-projects from TRIUMPHS: TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources appearing in Convergence, for use in courses ranging from first year calculus to analysis, number theory to topology, and more. Links to other mini-PSPs in this series appear below. The full TRIUMPHS collection also offers one other mini-PSP based on Euler's work along with three more extensive “full-length” PSPs for use in teaching complex analysis.

Acknowledgments

The development of the student project The Logarithm of -1 has been partially supported by the TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) project with funding from the National Science Foundation’s Improving Undergraduate STEM Education Program under Grants No. 1523494,1523561, 1523747, 1523753, 1523898, 1524065, and 1524098. Any opinions, findings, and conclusions or recommendations expressed in this project are those of the author and do not necessarily reflect the views of the National Science Foundation.

References

Bradley, R. E. 2007. Euler, D’Alembert and the Logarithm Function. In Leonhard Euler: Life, Work and Legacy, volume 5 of Studies in the History and Philosophy of Mathematics, edited by R. E. Bradley and E. Sandifer, 255–277. Amsterdam: Elsevier.

Lehmer, D. H. 1914. List of Prime Numbers from 1 to 10,006,721. Washington, DC: Carnegie Institution of Washington Publication No. 165.

Dominic Klyve (Central Washington University), "The Logarithm of −1: A Mini-Primary Source Project for Complex Variables Students," Convergence (June 2021)

Convergence

Tags:

History of Mathematics