a reading list, resources for students and teachers
These are not specifically about Complex Numbers, but rather mathematics as a creative process and about the process of invention and the introduction of new ideas to mathematics. They should help motivate the pure mathematics elements in the syllabus, and as such help motivate this topic in particular since it is in many ways something of the keystone in the number, trigonometry and geometry strands studied over the past years. These are stories of mathematical thinking in various guises. They may also prove useful as a background to other topics, especially pursuing the ideas in Bagni (2010) about learning from the historical process in order to develop the strategy for teaching a topic. The first mathematicians to deal with an issue are in some ways like our students, they did not have the benefit of hindsight.
A selection of books, narratives, some very new, some very old, in no particular order. Potential inspiration for teachers or their students, each certainly an inspiration for me. Such lists are inevitably personal.
Gleick, Chaos
James Gleick, (1987) Chaos, Making a New Science Macdonald & Co Ltd, London
The story of the discovery of a whole new field of mathematics, starting from a meterologist noticing a very strange behavior in a basic simulation of the weather in a 1960s computer. Deterministic, yet not predictable except by running the system until the moment in question.
Gleick has written a number of accounts of the process of science, here he tells the story of a journey in mathematics that was just as hard to believe as the 16th century mathemeticians found complex numbers hard to believe. This also illustrates the role of computers and programming in modern theoretical mathematics.
Penrose, The Road to Reality
Roger Penrose (2004). The Road to Reality, A Complete Guide to the Laws. Johnathan Cape, Randon House Group Ltd, London
Penrose is one of the top physicists of the 20th century, and he is telling the story as he saw it. His view is quite radical, and to him complex numbers are deeply part of the structure of space and time, in ways not yet fully understood.
It inspired at least one fellow pure maths major to take up the field, he was given it in high school. But he is smart, as a book it does require a very good mathematical sense even though he tries to minimise the formal parts. He has the rare pleasure of referring to equations and results named after himself. He writes very well indeed, but never wastes a word. It is not for the faint of heart.
He talks about how much the beauty and elegance of a concept guides this exploration today, how with much of the cutting edge of physics the mathematics is truly wild, and yet how often amazingly accurate models have come from this.
Very highly recommended for that very rare student that is seeking something more, and is already way over your head.
Lockhart, Arithmetic
Paul Lochkart (2017). Arithmetic. Havard University Press, Cambridge Mass, London, UK
In contrast to Penrose (2004) this book is an easy read, I bought it in the afternoon and had just finished it, taking notes, in the nearby cafe at closing time (a late-closer fortunately). A couple of years ago I had spent many many days in the same cafe with Penrose, at some points averaging about 2 pages an hour, and he filled over 1000 of those fascinating pages in telling his story.
Lockhart is a maths teacher rather than a theoretical physicist and his book is a like spending an evening in the pub with someone truly passionate about numbers. He is very careful about the underlying ideas in the pure-maths sense, but he also just as carefully avoids the vocabulary and symbols that would be used in, say, an undergraduate unit examining number theory.
This book is about the magic of numbers, he talks the way a teacher could talk to a stage 4 or stage 5 student, though he is addressing teachers rather than students. He writes a chapter where Penrose would use a paragraph, but this leaves the book full of very nicely contextualised source material for your classes. He develops numbers from counting through to combinations and permutations. Lockhart is the author of Locharts Lament (pdf)
Everett, Numbers
Caleb Everett, (2017); Numbers, the Making of Us: Counting and the Course of Human Cultures. Harvard University Press. harvard press
It is a very readable new book, a fascinating look at numbers, language and learning.
He is considering the origin of numbers, what is innately recognised in our brains and what instead depends on language, on symbols defined, before it can be developed as a concept. Essentially single, pair and triple seem to be recognised as abstract properties by the brain. He argues this is a very ancient ability, shared by most vertebrates at least, while anything beyond three is either a cultural construct or approximate.
His parents were missionaries in Brazil, he spent quite some time then (and since) with a society that does not have numbers at all. Of some thousands of languages only a very very few have no numbers, they do not count at all. He writes of what that observation shows of human numbers, language and such. In a society where numbers are very important we teach them very young. Maybe about 2yo kids are taught the words for numbers, but these are just words, they do not correspond to cardinality. Then they are taught succession, that one more than six is seven, etc, the numbers are given a sequence. Finally they are taught the procedure for counting, that if each object is named by each number in succession then the last label given is the size of the set. At this point they can identify abstract groups as having a size that is given a number, a label applicable to any such set, a handle we then use to develop, ultimately, the whole edifice of numerical mathematics.
Like i this was invented sometime(s), taught to others and spread because it was (very) useful to some kinds of societies. It is interesting to consider that geometry can exist happily without number, the people in the book live in communities of only a dozen or so people, there are about 700 of them altogether, they live in rainforest which is a very extreme barrier to travel (and are notoriously unwelcoming of strangers and outside ideas in any case) so they have had very little outside influence, and not much use for numbers. They make fine tools, weapons, boats and houses so they do possess geometrical thinking, just no numerical thinking. They have never taken on the invention that almost every other society on earth uses.
It is a little hard to believe, but a number of careful studies over a few decades has confirmed it.
Schrödinger, What is Life?
Erwin Schrödinger (1944). What is Life? republished as book, with Mind and Matter (1958), an autobiograhical sketch (1960) and an introduction by Roger Penrose, Cambridge University Press (1992).
Maths and science, with maths as a tool for theoretical exploration, rather than as a tool for detailed descriptions and number crunching. This was the paper that showed clearly that a gene must be a giant molecule and inspired Crick to search for its shape. One of the key physicists and mathematicians in modern science considers the numbers and makes some predictions that, a decade after this publication, are confirmed.
It is an important piece of history, but also shows mathematical thinking at its best. It is very accessible mathematically.
Galileo, Dialogue
Galilei Galileo (1632). Dialogue Concerning the Two Chief World Systems. Translation: Stillman Drake (1953), Regents of the University of California, Modern Library (2001)
This is the state of debate about physics and mathematics shortly after the tentative introduction of complex numbers (they would not play a part in physics for quite some time), and before the efforts of the next decades resolved a lot of the questions posed here. It is a compelling socratic dialogue and readily accessible to a high school reader. It shows mathematitions on the verge of inventing the mechanics and calculus that is taught in high school. What did they understand? What was close? Where were they still a long way off?
This book is much more readable the Copernicus (1543), who in many ways started this particular journey at exactly the time that complex numbers were first used.
Weeks, The Shape of Space
Jeffrey Weeks, (2002). The Shape of Space (Second Edition). CRC Press, Taylor & Francis Group, Boca Ranto, FL.
Dealing with abstractions, dimensions, topology, proofs in a way that is accessible, but new to, extension 2 students. It would help extend the spacial thinking of a student, in a rather different way to the complex numbers topic. But if you read Penrose, really they are very related topics.