480a.htm MATH 480-01 (43128): Topics in History of Mathematics JB-387, TuTh 6-7:50PM SYLLABUS Spring 2013 John Sarli JB-326 (909)537-5374 [email protected] TuTh 11AM-1PM, or by appointment Text: V.S. Varadarajan, Algebra in Ancient and Modern Times (AMS Mathematical World Vol. 12, 1998) ISBN 0-8218-0989-X Prerequisites: MATH 252, MATH 329, MATH 345, MATH 355 This is a mathematics course structured around historical developments that produced our current understanding of algebraic equations. Rather than being comprehensive, we will focus on a few ideas dating from antiquity that will help explain the interpretation of solutions to algebraic equations in terms of roots of polynomials. This modern interpretation required the systematic development of numbers from their early representation as geometric quantities to their formalization, more than two thousand years later, as structured algebraic systems. One of our objectives is to come to an understanding of why we teach algebra the way we do. The above text will be used as a guide. We will not cover all of it but you should read as much of it as possible. I will supplement its topics with notes that will appear on my website www.math.csusb.edu/faculty/sarli/ along with this syllabus. In order to maintain a seminar approach to this course (active participation through discussion) the grading will be based on just four components, 25% each: 1) First Written Project; 2) First Exam; 3) Second Written Project; 4) Final exam. As the course progresses I will make suggestions for suitable project topics, some of which will derive from exercises in the text. The exams, for which you may use your notes, will be require you to implement some of the basic mathematical techniques that we will develop in class. The date for the First Exam and the due date of the First Written Project will be announced in class and recorded on the website. Guidelines for project format will also be posted there. After assessing your performance on each of the four components, course grades will be assigned as follows: Some important dates: April 2, University closed April 12, Late add period ends April 19, Last day to drop w/o record 480a.htm[6/10/2013 4:14:11 PM] 480a.htm May 2, First Exam; First Project due May 31, University closed June 6, Last day of class (Second Project Due) Thursday, June 13, Final Exam Constructible Numbers Before the invention of the number line, real quantities were represented by geometric constructions, typically with straightedge and compass. More than an entire book of Euclid's Elements is devoted to geometric arithmetic consistent with his postulates for the development of plane geometry. For example, given a unit length one can construct a square whose side has this length. Then, by the Pythagorean Theorem, a diagonal of this figure has the property that the square of its length is . From our perspective we would say that is a constructible number. It is natural to ask just what numbers are constructible. This is made precise by stating axioms of constructibility consistent with the proof methods of Euclidean geometry. The purpose of these axioms is to determine which points in the plane are considered to be constructible. From these constructible points we can then define constructible numbers. For example, the distance between two constructible points is considered to be a constructible length (which we would associate with a non-negative real number). Any two distinct points may be chosen and designated constructible, and the distance between them taken as the unit length. The intersection of two constructible figures results in constructible points. The line or segment determined by two constructible points is a constructible figure. A circle with a constructible point as center and a constructible length as radius is a constructible figure. The following theorems are easily derived from these four axioms: The line parallel to a given constructible line and passing through a given constructible point not on the given line is a constructible figure. The perpendicular bisector of a constructible line segment is a constructible figure. The circle determined by three constructible points is a constructible figure. (If the three points are collinear then the resulting "circle" is clearly a constructible line.) These three theorems were all Euclid needed to do arithmetic with constructible numbers. By arithmetic we mean the operations of . Though Euclid did not have access to the coordinate plane, we can use that setting to describe the arithmetic of constructible numbers succinctly. In fact, we can take it further by interpreting the ordered pair as the complex number . This formalism was not fully developed before the time of Gauss, but we will see that the arithmetic of constructible complex numbers simplifies many of the historical discoveries we will study. To get started, let us take the points and as the two points in axiom . These correspond to the complex (real) numbers and . By axiom the line we call the real axis is constructible, and by axiom the circle we call the unit circle is also constructible. Then, by axiom the number is constructible (as is any integer), and by T the line we call the imaginary axis is constructible. Now 480a.htm[6/10/2013 4:14:11 PM] 480a.htm suppose that and are constructible real numbers, that is, they have been constructed on the real axis. Exercise. Show that is constructible, as are and . Now suppose and . The numbers and are clearly constructible, by axiom . By T the circle through , , and is constructible. From Euclid (III.35) we have where is the length of the segment from to the other intersection of the circle with the real axis. It follows that the product is a constructible number. Similarly, by constructing the circle through the constructible numbers , , and we see that is also a constructible number. We have shown: The collection of constructible real numbers is closed under the arithmetic operations. In other words, we can perform arithmetic within the set of constructible real numbers. A set of numbers with this property is called a field, to use modern terminology. Note also that the set of rational numbers is a field consisting entirely of constructible numbers. Exercise. Use T to show that the complex number is constructible if and only if the real numbers and are constructible. Deduce that the set of all constructible complex numbers is a field and develop formulas for the sum, difference, product and quotient of two constructible complex numbers. Geometers of Euclid's time did not have the concept of complex numbers but they discovered many constructions that had an impact on modern algebra. Here is one that allows us to find the reciprocal of any complex number although it was originally applied only to constructible points: Consider a constructible circle centered at with radius and let be a constructible point inside the circle, that is, . The line through and is constructible as is the line through perpendicular to line . Since the points and where this perpendicular intersects the circle are constructible, the segment between any two of these four points is constructible. Also, the tangents to the circle at and are constructible (why?), and these two tangents meet on the line , hence at a constructible point . In the early nineteenth century mathematicians began referring to as the inversion of in the given circle and interpreted this construction as the generalization of reflection in a line. Note that the right triangles , , and are similar to one another (and analogously with in place of ). This produces many relations among these constructible lengths, in particular Thus, the product of the distances from the center of the circle to two points related by inversion is the square of the radius of the circle. The symmetrical relationship between and suggests there is an analogous construction if the constructible point is given to be outside the circle, that is, (see suggested project 5, below). Perhaps because they did not have a theory of number systems as we understand it, the ancients were aware of some of the limitations of geometric arithmetic. In particular, they devoted a lot of mathematics to the attempt to resolve the following four questions regarding constructibility using only straightedge and compass: 480a.htm[6/10/2013 4:14:11 PM] 480a.htm Q Is it possible to trisect an arbitrary angle? Q Is it possible to construct any regular polygon? Q Is it possible to construct a square with area equal to that of a given circle? Q Is it possible to construct a cube with volume twice that of a given cube? Eventually it was proved that each of these questions has a negative answer, but the proofs required an understanding of roots of polynomials that was not available to the ancients. Questions of constructibility in general are answered by the following theorem, the result of work by Pierre Wantzel (1814-1848): If a complex number is constructible then it is a root of an irreducible polynomial with integer coefficients whose degree is a power of . So Q has a negative answer because, for example, constructing a cube with volume would require that be constructible. But is a factor of any polynomial that has as a root. For Q , consider the unit circle, whose area is . A square with side is not constructible because is transcendental, that is, not a root of any polynomial with integer coefficients. (This was proved by Ferdinand Lindemann (1852-1939) working from ideas that Charles Hermite (1822-1901) used to prove that the natural log base is transcendental.) As for Q , it can be shown that the above theorem has the following corollary: Let be the number of integers less than or equal to that are relatively prime to .