Offered: Spring 1998 Instructor: Kevin J. Mitchell Time: Class: M-W-F 1:20 to 2:30 in Napier 201 Text: Complex Variables with Applications by Brown & Churchill
Even more fundamental for European mathematicians was the problem of negative numbers. It was not until the seventeenth centurythat negative numbers became known in Europe through Arab texts, though the Hindus had used them as early as the seventh century. Why were negative numbers problematic? Consider this argument given by Antoine Arnauld (1612--1694). We wish to claim that -1:1 = 1:-1. Of course -1 is less than 1. How can a smaller number be related to a larger one in the same way that a larger number is related to a smaller one?
Without quite understanding negative or irrational numbers, mathematicians began to struggle with what we now call complex or imaginary numbers. Again the motivation was a search for solutions. The quadratic equation x^2 - 2 = 0 at least has irrational solutions, but the equally simple equation x^2 + 1 = 0 has no real solutions, rational or irrational. This fact follows directly from the axiom system for real numbers which guarantees that for any real number x, we have x^2 is non-negative. But then x^2 + 1 > 0. That is, no real number x satisfies x^2 + 1 = 0.
As early as the sixteenth century mathematicians began to explore the properties of nonreal solutions to quadratic equations. The symbol i was used to denote the "imaginary'' solutions of x^2 + 1 = 0. Acting as if the usual algebraic laws were valid here, mathematicians such as Jerome Cardan (1501--1576) began to use square roots of negative numbers to solve all sorts of "impossible'' problems. In his Ars Magna of 1545 Cardan poses the following problem. Find two numbers whose sum is 10 and whose product is 40. Of course, this requires us to find solutions to the quadratic equation x(10 - x) = 40, or equivalently, x^2 - 10x + 40 = 0. Using elementary calculus, the quadratic x^2 - 10x + 40 has an absolute minimum of 15 at x = 5, as long as x is constrained to be a real number. Therefore, no real solutions to this problem exist. Cardan was not deterred, however, and he obtained the nonreal "solutions'' that we today would denote by 5 + sqrt(-15) and 5 - sqrt(-15), that is, 5 + sqrt(15)i and 5 - sqrt(15)i. You can verify that Cardan's answers are correct, acting on faith that the usual laws of algebra apply to such quantities.
Cardan and others at this time operated without any careful or formal theory of complex numbers by blindly applying the ordinary rules of algebra as needed. But some mathematicians completely rejected the notion of such "imaginary'' solutions to equations. Felix Klein wrote:
Below, I have outlined the readings for the term. I hope that we can stick this schedule, as the material on residues is a fitting place for the course to end. I will assign practice problems out of the text. I have an Answer Manual for the text and you are welcome to use it. The main component of your grade for the course will be a series of 12 to 15 graded homework assignments. While you may discuss this work with classmates, I expect that the work you hand in will be your own. The graded homework assignments will contain both routine and more challenging problems which extend the material we are discussing in class. I don't expect that you will be able to "solve" every problem. There will be two out-of-class self-scheduled exams. You will have 90 minutes to work on these exams at a time during the day when it is convenient for you. There will also be a final exam.
Your course grade will be calculated as follows: Homework 45%,
2 Tests 15% each, Final Exam 15%, and Class Participation 10%.
Because of the nature of this course, its assignments, and its assessment,
your attendance and participation is both crucial and required.
If you must miss a class for some reason beyond your control, talk to me
about it in advance.
M & F 11:00 to 12:00 T & T 11:45 to 12:45I am often in my office much of the day; drop in to get hints or help with course assignments or just to chat.