We discussed CE/BCE notation for dating. (No year zero.) We discussed a coarse timeline of development for Homo sapiens. We discussed some of the the earliest mathematical objects/writings, namely:

The Ishango bone:    
Plimpton 322:    
The Rhind papyrus:    
The Moscow papyrus:    

We discussed an ancient algorithm for how to solve the simultaneous system $x+y=s, xy=p$ for $x$ and $y$, and explained why this was equivalent to solving a general quadratic equation.

We gave three proofs of the Pythagorean Theorem (the Bride's Chair proof from Euclid's Elements, one proof without words, and James Garfield's proof).

Here are 118 proofs of the Pythagorean Theorem. The proofs we saw in class are proofs 1, 3, and 5. The 13 remarks preceding the first proof are interesting.

Today, we discussed Section 1.3 of the book. Namely, we discussed Pythagorean triples and how to use the Chord and Tangent Method to develop a parametrization of all Pythagorean triples.

The session ended with a Challenge Problem: find all Pythagorean triples that form an arithmetic progression. The solution is here.

Irrational numbers.

Commensurability: We defined segments $a$ and $b$ to be commensurable if there is a segment $c$ and positive whole numbers $m$ and $n$ such that $|a| = m*|c|$, $|b| = n*|c|$. We mentioned that $a$ and $b$ are commensurable iff the ratio of their lengths is a rational number.

We mentioned Hipassus, whose is credited with discovering the existence of irrational numbers. We proved $\sqrt{2}$ is irrational by reductio ad absurdum using an arithmetical argument. We then examined a false proof by ChatGPT for the statement that $\sqrt{18}$ is irrational. Finally, we discussed a geometric proof that $\sqrt{2}-1$ is irrational, and used this to deduce that $\sqrt{2}$ is also irrational.

We reviewed the discussion on Irrational numbers from Jan 16. We defined the Golden Ratio and gave a geometric argument that it is irrational.

We began discussing the Euclidean algorithm in the following way. We started with the structure $\mathbb{N}=\langle \{0,1,2,\ldots\}; 0, S(x), +, \cdot\rangle$ and defined the additive order on $\mathbb{N}$:

$m\leq n$ iff $(\exists k)(m+k=n)$.

Then we defined the multiplicative order on $\mathbb{N}$:

$m\leq n$ iff $(\exists k)(m\cdot k=n)$.

We drew the Hasse diagrams for these orders and showed that these posets are lattices. This means that, in each poset, any two elements $x$ and $y$ have a least upper bound $x\vee y$ ( = the join of $x$ and $y$) and a a greatest lower bound $x\wedge y$ (the meet of $x$ and $y$).

In the additive order, $m\wedge n = \min(m,n)$ and $m\vee n = \max(m,n)$.

In the multiplicative order, $m\wedge n = \textrm{lcm}(m,n)$ and $m\vee n = \gcd(m,n)$.

The Euclidean algorithm explains how to compute the multiplicative meet and join using the additive meet and join. Quiz 1.

We discussed the facts that (i) the additive order on $\mathbb{N}$ is a well order. (ii) the additive order on $\mathbb{N}^+$ is a linear extension of the multiplicative order on $\mathbb{N}^+$. (iii) There is no infinite, strictly descending chain in $\mathbb{N}$ under either the additive order or the multiplicative order. These facts were shown to ensure that algorithms terminate. (Any algorithm on $\mathbb{N}$ that successively returns strictly smaller numbers must terminate, else an infinite, strictly descending chain of natural numbers will be produced.)

Next, we introduced the Euclidean Algorithm, and used it to show that $\gcd(21,8) = 1$. The following handout was circulated.

We worked through this handout

Quiz 2.

1. Bézout's identity. (Did it!)
2. Irreducibility implies primality in $\mathbb{Z}$. (Did it!)
3. The Fundamental Theorem of Algebra. (Did not get this far!)

We discussed the following topics.

1. The Fundamental Theorem of Algebra. (Every positive integer can be factored into primes in a unique way, up to the order of the primes.)
2. (Section 2.1) Euclid's Elements are presented in the form of “theorems”, which are accompanied by “proofs” from the “axioms”.
3. (Section 2.2) We explained why there can be at most 5 regular polyhedra (= polyhedra where any two sides are congruent regular polygons and the same number of sides meet at each vertex). We then discussed the fact that all 5 possibilities can be realized (= the Platonic Solids = tetrahedron, cube, octahedron, dodecahedron, icosahedron). We pointed out that every Platonic Solid has a dual solid. We discussed Luca Pacioli's construction of the icosahedron. We worked on this handout where we recorded some numerical data about the Platonic Solids.

We began a discussion of Euclid's Elements, including

1. a brief description of the contents,
2. a discussion of the five axioms of plane geometry,

Quiz 3.

We discussed straightedge and compass constructions, including:

1. Allowable tools.
2. Meaning of “construction”
3. Meaning of Constructible
   • points,
   • lines,
   • segments,
   • rays,
   • angles,
   • circles,
   • ETC.
4. Construction of coordinate axes.
5. Orthogonal projections, construction of lines orthogonal to a give line at a give point, assigning coordinates to points.
6. We made some references to Ordered Fields, but we will say more about them on Friday, Feb 6.

We showed that the problem of determining the constructible points in the plane can be reduced to the problem of determining the constructible real numbers. We explained why the constructible numbers form the least Euclidean field. We discussed why $\pi$ and $\sqrt[3]{2}$ are not constructible numbers.

Quiz 4.

1. a regular $n$-gon is constructible with straightedge and compass.
   
2. $\cos(2\pi/n)$ is a constructible number.
3. $\cos(2\pi/n)$ has minimal polynomial whose degree is a power of $2$.
4. $\phi(n)$ is a power of $2$.
5. $n=2^r\cdot p_1\cdots p_s$ for some $r\geq 0$ and distinct Fermat primes $p_1,\ldots,p_s$. We defined the Euler phi-function, $\phi(n)$, and Fermat primes. We listed the known Fermat primes, $3, 5, 17, 257, 65537$.

Quiz 5.

Some interesting side quests: we explained how continued fractions can be used to solve Bézout's Identity and also can be used to give optimal rational approximations to irrational numbers.

Community Quiz!

We defined congruences on $\mathbb{Z}$, and stated that every congruence on $\mathbb{Z}$ is equal to “congruence modulo $m$” for some nonnegative $m\in \mathbb{Z}$. We then discussed the Chinese Remainder Theorem following these notes.

We discussed the Chinese Remainder Theorem further, and continued working on these notes.

We began discussing the problem of solving polynomial equations. We spent most of the time discussing the Cardano formula following these slides.

We continued discussing the solution to the cubic formula following these slides.

Quiz 7.

1. the possible signs of an $n$-th root ($\sqrt[n]{x}$).
2. the quartic formula.

1. The fact that quintics can be solved by extraction of 2nd, 3rd, and 5th roots and Bring radicals.
2. The description of the Galois group of a polynomial $f(x)$.
3. The fact that $|\textrm{Gal}_{\mathbb{Q}}(f)|\leq n!$ when $n=\textrm{deg}(f)$.
4. We discussed the polynomial $f(x)=x^4+1$, and explained why some permutations of its roots cannot preserve all rational relations satisfied by the roots. (In fact, $|\textrm{Gal}_{\mathbb{Q}}(x^4+1)| = 4\ll 4!$, so most permutations of the roots of $x^4+1$ will not preserve all rational relations satisfied by the roots.)

No Quiz!

We completed our discussion of higher order polynomial equations in one variable. This included:

1. The definition of the Galois group of a polynomial over $\mathbb{Q}$.
2. The statement of the Galois Correspondence.
3. A description of $\textrm{Gal}_{\mathbb{Q}}(x^4+1)$, its intermediate subgroups, and the corresponding intermediate subfields between $\mathbb{Q}$ and $\mathbb{Q}[\alpha]$ for a primitive $8$th root $\alpha$ of $1$.
4. A brief description of a criterion for solvability by radicals for higher order polynomial equations in one variable.

We began discussing new developments in geometry between 1630-1900. We focused mostly on Projective geometry. We gave the axioms for projective planes and examined some examples, e.g. the Fano Plane, and $\mathbb{F}\mathbb{P}^2$ for a field $\mathbb{F}$.

We stated Desargues Theorem, and explained why it suggests the need for points at infinity and lines at infinity. We reviewed the construction of $\mathbb{F}\mathbb{P}^2$ from a field $\mathbb{F}$. We briefly indicated how points in $\mathbb{F}\mathbb{P}^2$ may be assigned homogeneous coordinates. (This is something explained more thoroughly in this handout.) We described a 9-element nearfield $\mathbb{NF}_9$ with the property that Desargues Theorem fails in $\mathbb{NF}_9\mathbb{P}^2$.

Quiz 8.

We spent time working on 1(a)-(c) of this handout.

Quiz 9.

We ended class by stating the Cayley-Bacharach “8 implies 9” Theorem.

We compared Euclidean Plane Geometry, Projective Plane Geometry, and Hyperbolic Plane Geometry. We explained how to determine the points, lines, incidence relation, betweenness relation, congruence of segments relation, and congruence of angles relation in the disk model of Hyperbolic Plane Geometry.

We explained why the Parallel Postulate does not hold in the disk model. We ended the lecture by reflecting on the implication $$ E_1\wedge E_2\wedge E_3\wedge E_4\Rightarrow E_5 $$ where $E_i$ is the $i$th axiom of Euclidean geometry. We asked: what does it mean for a statement to be provable from a set of hypotheses? How do we know if a proof system is sound? complete? decidable? What is the role of models/countermodels?

Quiz 10. (Last one!)

We said a few words about the life and work of David Hilbert. We then began discussing the influence of Hilbert's Problems. We started discussing the first problem in detail: the Continuum Hypothesis (CH). We described some of the ZFC axioms. We discussed countable versus uncountable sets. We explained the Cantor diagonalization argument, which shows that $[0,1]$ is uncountable. We defined ordinal and cardinal numbers.

We started our meeting by reviewing the axioms of ZFC.

Then, we briefly explained the work of Gödel and Cohen, which together establish the undecidability of CH.

We concluded by stating the possibilities for $|\mathbb{R}|$.

Final Exam review sheet

Final Exam review sheet

Final Exam review sheet