New Foundations syllabus
========================
This syllabus was written down a couple of years ago by John Jones and
myself. The aim is to chase foundations, and its implicit ideological
program of "sets with structures" from Year 1, Term 1, and to replace
it with concrete computational algebra, stressing its links with other
subjects, and lending support to Analysis, Diffn Eqns, Math by Computer
and later courses. A main point is that there should be ABSOLUTELY NO
set theory or formal definitions of abstract algebra.
This course could be taught by 2 lectures a week backed up by
worksheets over 15 weeks.
1. Complex numbers
------------------
a. Calculating with complex numbers (e.g., conjugate, absolute value,
inverse)
b. De Moivre's formula exp(i\theta) = cos\theta + i sin\theta; deriving
trig formulas from de Moivre's formula
c. The fundamental theorem of algebra
d. The relation between the coefficients and roots of a (complex)
polynomial
e. Surds and roots of unity
2. Polynomials and power series
-------------------------------
a. Calculating with polynomials (long multiplication and long division)
b. Euclid's algorithm for polynomials
c. The remainder theorem (link with 1c)
d. Symmetric polynomials (link with 1d)
e. Calculating with power series (formal calculations of course, no
convergence)
f. Power series, exp and trig (link with 1b)
g. Calculations with Taylor series (again no convergence)
Optional topic:
h. Expansion of 1/(1-t)^n, binomial coefficients and the binomial
theorem
3. Vectors
----------
a. Calculating with vectors (i.e., in R^n)
b. Dot product, orthogonality, complementary subspace
c. Linearly independent vectors in R^n and choosing coordinates (or
axes)
Optional topics:
d. Vector product in R^3
e. Rotations, reflections and isometries of R^n
4. Matrices
-----------
a. Calculating with matrices, application to 2x2 and 3x3 linear
equations
b. Product and inverse matrices
c. Matrices as functions R^n -> R^n, application to linear equations
d. Eigenvalues and eigenvectors
e. Diagonalising matrices assuming existence of e-values (i.e., over C)
(link with 1c-d); an nxn matrix can be diagonalised if and only if it has
n linearly independents eigenvectors
f. Application to solving a system of first order linear ode's
Optional topics:
g. Orthogonal matrices
h. Diagonalising symmetric matrices by orthogonal matrices
i. Application to conic sections
5. Symmetries
-------------
a. Permutations and symmetric polynomials (link with 1d, 2d)
b. Rotations of cube and regular tetrahedron as matrices (link with
4a, 4b, 4d)
c. The symmetries of a regular polygon in the plane and the regular
polyhedra in 3-space (link with 1e)
6. Arithmetic
-------------
a. Integers and rational numbers, calculating mod n (notion of
working modulo equivalence, but no equivalence classes or cosets)
b. Irrational numbers
c. Algebraic numbers (link with 1d-e)
d. Transcendental numbers
e. Euclid's algorithm for integers, the af+bg = hcf property, (link
with 2b)
f. Primes and unique factorisation; there are an infinite number of
primes; how to generate a list of prime numbers, optional: tests for
prime numbers
g. Cyclotomic integers (very concrete i.e., consider the set of
complex numbers of the form ... . You can add them and mutliply them
and you get another complex number of the same form.
Optional topics:
h. Failure of unique factorisation for cyclotomic integers
i. The proof of Fermat's last theorem for the prime 3 using cyclotomic
integers.
7. Permutations
---------------
a. Calculating with permutations
b. The sign of a permutation, even and odd permutations,
determinant of a square matrix (link with 4a, 4d, 4e)
c. Discrimant of a polynomial (links with 1c-d, 2d, 3c)
d. Cycles and the cycle decomposition of a permutation (link with 5c)
-------------
From miles Mon Jun 11 14:57:58 2001
Subject: Foundations next year
To: "C.T. Sparrow"
Date: Mon, 11 Jun 2001 14:57:58 +0100 (BST)
CC: A.P.Simpson@warwick.ac.uk
Dear Colin,
as you'll find out at Warwick, I'm always fighting against various
aspects of our teaching system, most of which is garbage based on trying
to remember what we taught 25 years ago. I've never had any success in
these battles, but it doesn't stop me trying. The whole of our algebra and
geometry teaching is in a shambles, and John Jones and I have discussed at
various times what we should put in place to replace it. In particular, we
have this ridiculous course called Foundations in which we take people, many
of whom can't confidently add fractions, and teach them about equivalence
relations and Venn diagrams, so that they conclude that adding fractions and
getting the right answer is not part of modern math. I circulated this to
mastaff a couple of weeks ago, but it has recently been brought to my
attention that this course has been assigned to you (together with Adrian
Simpson from Sci Educ., who teaches the non-math students). I don't expect
you to change over to our syllabus this year, but you might bear some of the
following points in mind.
Best, Miles