The Riemann Hypothesis
For s = a + bi with a > 1, the Riemann zeta function is given by the formula
(s) =
1
.
ns
n =1
This function can be extended in a natural way to be dened for all s except s = 1.
The extended function is equal to zero at all negat
Modular Arithmetic
Formalized and systematized by Gauss in Disquisitiones arithmeticae (1801).
m and n are congruent modulo p (m n (mod p) if p is a divisor of the dierence of m
and n.
Clock arithmetic.
How is it similar to or dierent from ordinary ar
Diophantine Equations
These are polynomial equations for which postive integer solutions are sought.
Diophantuss Arithmetica (250) consisted of solutions to one or two hundred Diophantine
equations, but neither complete solution sets nor systematic solu
Pythagorean Triples
Is more or less Proposition 29 in Book X of Elements (300 BC).
A Pythagorean triple is a triple of numbers (a, b, c) such that a2 + b 2 = c 2 .
A Pythagorean triple is primitive If a, b , and c have no nontrivial common factors.
Th
Denitions
Background: Natural numbers means numbers.
m|n, m divides n, m is a factor of n, m is a divisor of n, n is a multiple of m: There is a
number q such that qm = n.
m is prime if it has exactly two divisors. (Thus, 1 is not prime.)
m is composi
Number Theory
General
It is at this point that we encounter a change: people working on mathematics of no obvious
value to common people doing their daily work.
Number theory is used in cryptography and cosmology. Strong students with knowledge of
numbe
Computation
General
I mean mainly addition, subtraction, multiplication, and division. (Not algebraic problems.)
Abacus
Slide Rule
Mechanical devices post slide rule belong in History of Computer Science class.
Egyptians
Example of Multiplication thr
Real Numbers
Real analysis studies more pathological functions than complex analysis, so more secure
footing was necessary.
Notion of (positive) real number arose early as a ratio of lengths.
Existence of transcendental (non-algebraic) real numbers:
1
Real Numbers
Real analysis studies more pathological functions than complex analysis, so more secure
footing was necessary.
Notion of (positive) real number arose early as a ratio of lengths.
Existence of transcendental (non-algebraic) real numbers:
1
Zero
Recall rst used as a medial place-holder (but not alone) by Babylonians in 300 BC.
Mayans also had it about 300 BC.
Ptolemy used a genuine zero in 130 AD in a hybrid Greek/Babylonian system.
Brahmagupta in 628 AD used rst decimal zero, and this g
Other Numbers
Positive Rationals
Why are they natural objects to consider?
Sharing.
The problem of measurement.
Around 1000 BC, the Egyptians used unit fractions (and 2/3).
Unit fractions were written with an ellipse over the denominator.
Writing pr
Mesoamerican
Olmec = 1200 BC
Maya = 300 BC
Numeral system
Unknown date of origin
Vigesimal = base 20
But 18 20, 18 202 , etc.
Reminiscent of ? Chinese counting-rods? Mesopotamian?
Stacked vertically
Had a zero
Roman
753 BC
Threw out the Etrusca
Indus Valley
Modern day Pakistan
Aryans settled in about 1500 BC (supplanting previous civilization in place for a millennium).
Sanskrit
Brahmi numerals for 1 through 9 date to about 300 BC
Evolved into ours through time.
Adopted by Arab/Islamic Mat
Mesopotamia
Babylonian?
Land between the rivers: Tigris and Euphrates
Present-day Iraq
Oldest extant writings: ca. 3000 BC (cuneiform=wedge)
Oldest purely mathematical texts date to Old Babylonian or Hammurabic period (ca. 1900
BC)
Claims that syste
Numbers and Counting
General
Counting is arguably the most fundamental/primitive mathematical action.
Modern science of combinatorics. Counting arrangements.
E.g., how many ways can 67 be written as the unordered sum of positive integers?
More general
Intuitionists rejected the Law of the Excluded Middle, which says that statements are true
or false. Thus, while they accepted A ( A) as a logical identity, they didnt accept
( A) A as a logical identity.
Hilbert: Taking the law of the excluded middle
It implies the Banach-Tarski Paradox (presented by Banach and Tarski in 1924): A ball in
3-dimensional space can be decomposed into nitely many pieces (5, actually) in such a way
that those 5 pieces can be reassembled through rigid motions (translation a
Logical Issues in the 20th Century
The Paradoxes of Naive Set Theory
Most boil down to sophisticated versions of the question: Is the sentence This sentence is
false. true or false?
Recently, careful students of Cantor have made a strong case that Canto
Reaction
Kronecker, Cantors advisor, was extremely conservative on these issues, at times denying
the existence of irrational numbers. Cantor felt that Kronecker hindered his work on sets.
Poincar (1908): Later generations will regard set theory as a di
Along with other mathematicians of the late 1800s he was interested in the question of
representing functions with trigonometric series of the form
f (x) = a0 +
[an cos nx + bn sin nx].
n =1
In particular, he was interested in the question: Of what sets
Set Theory in the 19th Century
Its Genesis
Terminology
Bolzano gave us the word set in his 1851 publication of Paradoxes of the Innite.
Notation
Peano introduced the symbols for intersection and for union in 1888.
Peano introduced the symbol for memb
The Weierstrass Approximation Theorem: If f is continuous on [a, b], then there is a
sequence of polynomials that converges uniformly to f on [a, b].
The Weierstrass M -test: If |fk (x)| Mk on [a, b] for k = 1, 2, 3, . . ., and
verges, then
k=1
fk (x) c
Volterra constructed a function that was dierentiable everywhere but whose derivative
was not Riemann-integrable.
Textbooks
LHpital (1696)
Analyse des inniment petits pour lintelligence des lignes courbes (Analysis of the Innitely Small for the Underst
Calculus, Part 3
Whats a Function?
Euler (1748): [A] Function of a variable quantity is an analytical expression composed in
whatever way of that variable and of numbers and constant quantities. This roughly corresponds today to what we call an analytic
Characterizations of Riemann-integrability
Riemann
The norm of a partition is the width of its largest subinterval.
A function f is integrable if and only if, for any > 0 no matter how small, we can
nd a norm so that, for all partitions of [a, b] havi
It hasnt caught on.
Denite Integration
Denition
Leibniz (late 1600s): It was a sum of innitesimals.
Fourier (1822): Gave us our current notation of denite integral and dened it to be an
area.
Cauchy (1820s)
He began with a function f continuous on t
Calculus, Part 2
Dealing with the Innitely Small
Newton: Vanishing quantities. Errors, no matter how small, are not be considered in mathematics. (Even innitesimals cant just be dropped from the nal answer.)
Leibniz: Innitesimal quantities
Berkeley
Th
Calculus, Part 1
Predecessors of Newton/Leibniz
Area by Method of Exhaustion in antiquity
Eudoxus
Archimedes
India
Pedersen: Many methods were developed to solve calculus problems; common to most of
them was their ad hoc character. It is possible to