In this blog series, Leo Corry, author of A Brief History of Numbers, helps us understand how the history of mathematics can be harnessed to develop modern-day applications today. In this second post, looking specifically at the impact of imaginary numbers on our comprehension of the physical world, Leo explores how theorems change over time.
Few elementary mathematical ideas arouse the kind of curiosity and astonishment among the uninitiated as does the idea of the ‘imaginary numbers’, an idea embodied in the somewhat mysterious number i. This symbol is used to denote the idea of, namely, a number that when multiplied by itself yields -1. How come? We were consistently told in the early years of secondary school that two numbers of the same sign, positive or negative, when multiplied by each other always yield a positive number. Now we are told that this is not always the case, and here we have this ‘number’ for which the rule does not hold? What kind of number is this if it breaks such a fundamental law about the multiplication of numbers? Is the mystery and the apparent contradiction solved just by calling i an ‘imaginary number’?
The truth is that when one is introduced to imaginary numbers, it’s not the first time that a fundamental idea previously taught, is turned upside down. Think, in the first place, about the negative numbers. When learning in primary school the job of subtracting numbers, a typically curious child may ask the teacher how to subtract, say, six from four. A typically cautious teacher would answer, “well, … ehem, … that can’t be done.” Indeed, it makes a lot of pedagogical sense to let the pupil acquire good mechanical skills in performing the operations without having to worry about such nuances, and there is no immediate need to bring up confusing issues such as the idea of negative numbers. All of this can be clarified later on, simply by telling the child that, “well … yes … actually, we can subtract 6 from 4 with the help of a new idea, the idea of negative numbers.” Most children tend to pass this experience without a lasting negative impact, though perhaps for some this is the beginning of the kind of post-traumatic symptoms so commonly associated in our society with the study of mathematics.
At any rate, the difficulty that typically arises immediately after becoming aware of the existence of negative numbers relates to the rule ‘minus times minus yields plus’. It really takes time until one becomes used to this strange rule, and, let’s be frank, many intelligent people never really come to believe it, and much less to understand its justification. And then comes the news that the number i breaks that rule.
The story of imaginary numbers is interesting not only because it touches upon the most central topic of mathematics, numbers and their properties, but also because there is a dramatic parallel between, on the one hand, the path that the individual student crosses before reaching a clear understanding of the topic and, on the other hand, the historical path that the world of mathematics at large had to cross for the same purpose. It may sound strange at first, but in general, central developments in the history of mathematics happen as the more complex ideas give way to simpler ones. This is opposite to the way in which we are typically taught, namely from the simple to the complex.
Roots of negative numbers started to surface repeatedly when the great mathematicians of the Renaissance worked out solutions for equations involving cubes and fourth powers of the unknown. The most prominent of these was Girolamo Cardano (1501-1576). The techniques he developed led to correct answers to the problems that he investigated, but the intermediary stages often involved calculations with roots of negative numbers such as:
For Cardano equations such as x2 + 1 = 0, or even the simpler one x + 3 = 0 have no solutions. Just like we were initially told in school. But his techniques were making roots of negative numbers, as in the example above, ever more conspicuous and unavoidable. He continued to look at them as ‘sophistic’, ‘subtle’ and ‘useless.’ Nevertheless his mathematical curiosity did not let him just ignore them. He searched ways to apply to these numbers the same formal mathematical procedures he had considered to be legitimate for integers and fractions, while at the same time “putting aside the mental tortures involved.”
The conceptual status of imaginary numbers was successfully clarified only slowly in the centuries following. They became central to our conception of mathematics at large by the mid-nineteenth century. No less interesting is the fact that this apparently artificially concocted idea also became fundamental for physics. Many of the central pillars of modern physics, such as electrodynamics, cannot even be conceived without imaginary numbers.
Featured image credit: Formula mathematics psychics by markusspiske. CC0 public domain via Pixabay.