2018 Term 2
Term 2 --- Week 1
Term 2 --- Week 1
8 : David : 2013 AMC Junior exam pages 1 and 2, also
Q.4 from ToT JA-39-aut (doors & keys problem).
9 : William : 2013 AMC Junior exam pages 1 and 2,
a proof of explicit formula for Fibonacci numbers.
10 : Peter : Mathematical Structures part 1:
Mathematicians make up stuff - shapes, numbers
Shapes: curved spaces (straight lines may not look straight,
eg direct flights SYD to LON)
Numbers: N --> {0} U N --> Z --> Q (specification, equivalences) --> R --> C
Groups: axioms, examples & non-examples inc M_2(Z), rotation groups
11 : Zoltan : CAT 2013 Senior paper and (pages 76 and 77 of) problems from KoMaL.
12 : Ralph : p-adic integers, rationals, valuations, limit of sequences,
3, 33, 333, 3333, ... approaches -1/3, 5-adically.
Term 2 --- Week 2
Term 2 --- Week 2
8 : Tamiru : 2013 AMC Junior exam, pages 3 and 4
9 : Chris : Pick's Theorem.
10 : Peter + : Mathematical Structures part 2:
David Axioms & examples for rings & fields. Examples including Z, Z/nZ, Z/pZ=GF(p),
When is Z/nZ a field? (ab = 0 ==> a & b don't have inverses),
Z[x], Z[x]/(f(x)), GF(p^k), GF(2)[x]/(x^2+x+1) = GF(4),
Q, Q[x], Q[x]/(f(x)), Q[x]/(x^2-2) = Q(sqrt(2)), Q[x]/(x^2+1) = Q(i),
R, R[x]/(x^2+1) = R(i) = C
Mentioned that these "crazy" fields we'd invented "just for fun" have suddenly
become essential for the backbone of modern internet life & business.
11+12 : Zoltan : CAT 2013 Senior paper and (pages 76 and 77 of) problems from KoMaL.
Playing the Horrible Game/Ice Breaker/Swiss Contest
Term 2 --- Week 3
Term 2 --- Week 3
8 : Elizabeth : 2013 AMC Junior exam, pages 5 and 6,
followed by Polite numbers.
9 : Zoltan + : Part A of the CAT 2013 Intermediate paper,
Pages 22 and 23 of problems from KoMaL.
David Played NIM. Explained method for recognising winning/losing positions.
Strategy for winning from the latter.
10 : Tryon : Infinity.
11 : Peter : Math Structures 1:
Mathematicians make up stuff - shapes, numbers
Shapes: curved spaces (straight lines may not look straight,
eg direct flights SYD to LON)
Numbers: N --> {0} U N --> Z --> Q (specification, equivalences) --> R --> C
Groups: axioms, examples & non-examples, M_2(Z),
rotation groups - symmetry group of square
Axioms & examples for rings & fields. Examples including Z, Z/nZ
12 : William + : Q1 and Q2 from the Number Theory and Statistics sheet.
Tamiru
Term 2 --- Week 4
Term 2 --- Week 4
8 : William + : Introduction to groups: constructing D3, D4, subgroups,
Heath finite integer groups under + and *, isomorphisms
-> geometric interpretation of addition in a finite group,
open/closed operators, identity and inverses,
Q? When are the integers under multiplication a group?
9 : Zoltan : Part A of the CAT 2013 Intermediate paper,
Pages 22 and 23 of problems from KoMaL.
10 : David + : Q1, 2 from Number Theory and Statistics.
Tamiru Rational zeroes Theorem for solving integral polynomial equations.
11 : Peter : Math Structures 2:
Axioms & examples for groups, rings & fields.
Examples including Z, Z/nZ, Z/pZ=GF(p)
(when is Z/nZ a field?) Proved, in a group, identity & inverses are unique
Ring: ab = 0 ==> a & b don't have inverses (proof),
Proved Z/nZ is field <==> n is prime
Z[x], Z[x]/(f(x)), GF(p^k), GF(2)[x]/(x^2+x+1) = GF(4), GF(8)
Q, Q[x], Q[x]/(f(x)), Q[x]/(x^2-2) = Q(sqrt(2)), Q[x]/(x^2+1) = Q(i),
R, R[x]/(x^2+1) = R(i) = C
Mentioned that these "crazy" fields (GF(p^k), algebraic number fields) we'd
invented "just for fun" have suddenly become essential for the backbone of
modern internet life & business.
Started on Z_p (mentioned Q_p)
12 : Tryon : Infinity.
Term 2 --- Week 5
Term 2 --- Week 5
8 : William : 2013 AMC Junior exam, pages 7 and 8.
9 : Tamiru : Rational zeroes Theorem for solving integral polynomial equations.
Q1 from Number Theory and Statistics.
10 : Zoltan : Part A of the CAT 2013 Intermediate paper,
Pages 22 and 23 of problems from KoMaL.
11 : Tryon : Infinity
12 : Peter : Math Structures 1:
Mathematicians make up stuff - shapes, numbers
Shapes: curved spaces (straight lines may not look straight,
eg direct flights SYD to LON)
Numbers: N --> {0} U N --> Z --> Q (specification, equivalences) --> R --> C
Groups: axioms, examples & non-examples, M_2(Z),
rotation groups - symmetry group of square
Axioms & examples for rings & fields. Examples including Z, Z/nZ
Term 2 --- Week 6
Term 2 --- Week 6
8 : William : Showed the basic theorems of groups --
Uniqueness of the identity and inverses from the axioms.
9 : Elizabeth : Polite numbers, followed by
Q2, Q4, Q6 from this Factors, multiples, primes problem sheet.
10 : Zoltan : Part A of the CAT 2013 Intermediate paper,
Pages 22 and 23 of problems from KoMaL.
11 : Ralph : p-adic integers, rationals, valuations, limit of sequences,
3, 33, 333, 3333, ... approaches -1/3, 5-adically.
anomaly, bound, catastrophe, dilemma, experiment --> space quantisation
N in Z_p, 0 in Z_p, -1 in Z_p --> Z in Z_p.
12 : Peter : Math Structures 2:
Axioms & examples for groups, rings & fields.
Examples including Z, Z/nZ, Z/pZ=GF(p)
(when is Z/nZ a field?) Proved, in a group, identity & inverses are unique
Ring: ab = 0 ==> a & b don't have inverses (proof),
proved Z/nZ is field <==> n is prime
Z[x], Z[x]/(f(x)), GF(p^k), GF(2)[x]/(x^2+x+1) = GF(4), GF(8)
Q, Q[x], Q[x]/(f(x)), Q[x]/(x^2-2) = Q(sqrt(2)), Q[x]/(x^2+1) = Q(i),
R, R[x]/(x^2+1) = R(i) = C
Mentioned that these "crazy" fields (GF(p^k), algebraic number fields) we'd
invented "just for fun" have suddenly become essential for the backbone of
modern internet life & business. Started on Z_p (mentioned Q_p)
Term 2 --- Week 7
Term 2 --- Week 7
8 : Zoltan : Part A of the CAT 2013 Intermediate paper.
9 : David : Allan's The Number on the Tomb problem sheet.
10 : William : Discussion of entropy of data and compression,
Construction of the Huffman tree for some of the English alphabet.
Talk about unique decoding, prefix-free-ness. (See English frequencies).
11 : Ralph : Completed the proof that Q is in Q_p. Def'ns of countable and dense subsets.
Recalled Z, Q are countable, Z is not dense in R, Q is dense in R.
Proved Z_p and Q_p are uncountable. Z is dense in Z_p and Q is dense in Q_p.
12 : Peter : RSA Cryptography, Integer factoring algorithms,
trial division, quadratic sieve, number field sieve.