Even Famous Mathemeticians Struggle with Imaginary Numbers

[Note: This post is only for students in Algebra II and above, who have seen Imaginary Numbers.]

[Also Note: If you are interested in reading further on this topic, it's put together here in a well-written article from the American Mathematical Monthly.]

You know the Radical Product Rule? It's the one that says:
√[ab]=√[a]*√[b]You probably saw that rule in class and it made some sense. Then you went home, did an assignment on Multiplying Radicals, and didn't think about it again. Here's the crazy thing about it: that equation is the result of dozens of Mathematicians collaborating over the course of decades.

When Leonhard Euler (pronounced like Oiler, as in a person who oils things for a living) wrote his highly influential Algebra textbook in 1770, he naturally included the Radical Product Rule -- no doubt you saw it, as well, in your own Algebra I textbook written by a Mathematician considerably less famous. However, it was unclear in his text exactly what were the rules for the Radical Product Rule.


Act Like You Know: e^iπ + 1 = 0
Euler seemed to imply that the Radical Product Rule worked for both positive and negative a and b. That would mean that

√[-a]*√[-b]

√[(-a)*(-b)]

√[a*b*(-1)*(-1)]

√[a*b*(⁺1)]

√[ab].
However we know -- by using Imaginary Numbers -- that

√[-a]*√[-b]

i*√[a]*i*√[b]

i²*√[a]*√[b]

-√[ab]
In fact, Euler was one of the major early explorers of i, although it appeared in his textbook that he had trouble simply multiplying it. (It has recently been demonstrated that Euler did in fact understand the Radical Product Property.) Unfortunately, the way he presents it in the book can be confusing (he mostly used words, rather than equations) and it actually sparked a debate among Mathematicians on two continents, in particular: Etienne Bézout and Sylvestre François Lacroix (French, early-/mid-1800s), and Jeremiah Day (American, early-1800s, President of Yale College).

More specifically, they were debating whether or not Euler had a mistaken understanding of the Radical Product Rule, since he apparently implied that
√[-a]*√[-b] = √[ab].Bézout and Lacroix argued against Euler, whereas Day came to his defense. In the process of the debate, however, some of the issues that Euler himself had to deal with came to light.

Consider the fact that in 1758 -- just over a decade before Euler wrote his book -- a rival Mathematician published a well respected essay in which he denies many important uses of Negative Numbers that we take for granted. Euler's rival, named Francis Maseres, and many others like him believed that Negative Numbers were absurd: I cannot hold negative two pencils in my hand! In 1796, another respected Mathematician, William Frend, referred to Negative Numbers as "ridiculous." With even Negatives open to debate at the time, Euler took baby steps in describing Imaginary Numbers in his textbook.

During the ensuing, decades-long debate on Negative Radicals, many of the properties of Radicals and Imaginary Numbers were developed that we take for granted today. For instance, the idea that
√[a²] = ±athat is, an equation might have multiple simultaneous answers, was developed beyond Euler's statements in his textbook. Matrices of those simultaneous solutions were important in the discussions as well. Not to mention that even using the √ to represent a Radical became standard: before then, Mathematicians had been using radicals with and without the line over the numbers inside -- which led to plenty of confusion -- and parentheses and exponents with radicals were out of hand. The debate two hundred years ago among scholars in different countries required the use of a single notation, which resulted in the one we use today.

There were decades of debate over just one equation -- and all the notation that comes with it -- that Algebra students often take for granted in high school. These are rich ideas. Too often I hear students complain about Imaginary Numbers not making sense, but there was a time when that was said of Negative Numbers as well. Imaginary Numbers not only come in handy to solve the problem of a negative inside the radical -- nor do they only have great significance in the sciences (for instance in Quantum Physics, my passion) -- but they helped Mathematicians rigorously understand a concept as elementary as our Radical Product Rule.