Graduate Mathematics From Scratch


Intro

This points to all the other math in other curriculums as a centralized 'redo it all from scratch' curriculum for anyone who wants to bring themselves up to date with the last hundred years of mathematics. Some of the choices were from this mathematical maturity talk and a talk on what is the purpose of education and who actually benefits. The rest of the choices were from the time I spent in undergrad where everything was a poorly defined subset of a much richer math universe and whenever someone asked about how this all worked we were told 'sometime later you'll learn this in graduate school' well that day is today.

Everything here is either free or can be obtained via library genesis or from a local library they can move books around branches to you.

Recommended prereqs

If you learn some basic programming somewhere this will be easier since things like higher-order functions (a derivative is a function that returns a function) and domain-specific languages will help to understand increasingly abstract math notation. There is such a course here.

How to do math

The simplest way to learn is to doubt everything you read and manipulate it yourself plugging in corner cases seeing if it's true. You can also build your own math models in something like scheme/lisp and play with them there, Sussman wrote 2 books on how to do this with differential equations

Solutions are nice in the beginning but you should always treat math like there is no solutions and you're figuring it out yourself from scratch. If you don't figure it out immediately it doesn't matter the entire act of proving something has you digging through definitions so you will have learned something. A proof is the most efficient possible argument why something is true.

Typical curriculums

  • High school

If you go to one of the few good high schools left you will likely use David Bressoud's AP calculus text which used to be one of those Finney textbook publishing mills that pumped out a new version every year for schools to buy until Bressoud got himself on to the AP development committee and became a co-author. He wrote Context for Calculus as a supplement for it explaining where all the topics come from and how they fit together similar to his Radical Approach to Real Analysis books which teaches modern analysis as it was invented.

AP or 'advanced placement' depending on which university/college you go to may be a waste of time. Where I live most universities have split up their introductory freshmen calculus sequence which used to be Calc I single and Calc II multivariable into junk like 'Mathematical Discovery 100' and 'Integration and Approximation' so even though you placed out of the basic course you still have to redo everything anyway.

A long time ago before they removed real geometry from the high school curriculum you would have to get up in front of the class and demonstrate a proof which was derived from previous proofs. The act of doing this a few hundred times over multiple grades according to Robert Harper at CMU in his OPLSS lectures is you end up learning how mathematical logic works. That's all gone now so you'll have to teach yourself.

  • Undergrad

Almost everyone is forced into the same calculus sequence because the university cannot trust your high school background and since you'll likely need calculus anyway for almost every field you go into they picked single-variable calculus as the dumping ground. There is usually some kind of mandatory 'introduction to proofs' style course now that is really just a discrete math course. A bachelors degree is now just high school 2.0 if you combine undergrad real analysis with the typical calculus sequence you learn 4 semesters of the Riemann integral only to be told on your first day of grad instruction they replaced that integral over 100 years ago and you basically wasted your time.

  • Masters (MD, JD, MScEng)

A Masters is usually a vocational school. A lawyer goes to law school, a doctor goes to medical school, an engineer spends an additional year or two in graduate school and typically has to produce a thesis in order to be licensed by some engineering association. Most schools don't require you to defend a thesis you simply submit it for grading. A thesis is an argumentative treatment of a general subject often just a presentation of learned information it's not the same as PhD dissertation though everyone interchanges thesis/dissertation now.

A non-thesis on-line Masters is essentially credential buying. The courses you take that I have seen are easier than the undergrad versions almost survey tier courses at some schools. You can buy a masters in computer science, math, almost anything for around $20-35k and do it all remotely. It's strange you can do this for a MSc but impossible to find any kind of real associates degree like why can't I take the Georgia Tech masters but only receive associates for it if I don't have an undergrad? Why must undergrad be mandatory? More evidence that undergrad is the new high school you can't avoid.

  • PhD

PhD means doctor of philosophy as traditionally there was no such titles as mathematician or scientist you were simply somebody engaged in the natural philosophy and doctor was a title bestowed to a teacher. You go directly into PhD from undergrad usually unless you live in UK or some place where you need a Cambridge style 9 month masters. The goal of graduate school/phd track is to produce original research overseen by an advisor and you have to defend your work to a group of experts.

I've seen two different styles of grad school. The first is the traditional style where you have to take 1-2 years of basic grad instruction to bring yourself up to date with modern mathematics and whatever other courses your advisor signs off on as you sometimes need permission to take courses you can't just enroll in whatever you want. Second year you write the PhD general exam a public exam in front of a committee and anyone else who wants to sit in. Terence Tao almost failed his general which he explained was due to him playing video games and not taking the work seriously. If you survive the general exam you start writing your dissertation proposal or you drop out and collect your masters degree.

The second style is you have to pass the same general exam but you are expected to start publishing papers immediately not waiting around until after your exams or proposal. This is what compsci graduates typically have to do. If you just sit in classes for 2 years they'll be wondering hey what's up with you, publish or perish.

Eastern math education

Some western private/charter schools now teach 'Singapore Math' which has nothing to do with actual Singaporean math and is just a trademarked name. If you look up what the gov of Singapore actually teaches in primary mathematics it's Polya's problem solving heuristics combined with inquiry-based learning meaning the students have to figure out everything themselves by investigating the problems and asking questions as they go just like you would in real life if you had to teach yourself. The direct instruction is to show the connections between different topics. Everything is assessed with tests but they aren't standardized nationally so schools can come up with their own. School starts at age 3 in nursery school and by 'Primary 6' or age 12 they have already completed up to Grade 10 in North America. There isn't a grade 12 as most students are enrolled in an associates degree to prepare for university or vocational school in Singapore.

If you have a kid and live in North America use AoPS Online (can get screenshots of the books from libgen) and go through the books with them or pay $300/mth for the Russian School of Math (meaning soviet era math) if there is one in your area they can physically attend. Old school soviet math always starts with the hardest problems at the beginning of the lesson and you as a group figure it out through detective work. You should also search for 'math circles' in your area.

Real life math

In my experience a professional environment where you work with math you don't get a problem to solve and go off spending hours or days by yourself. It's all collaborative. The biggest problem then becomes how to model real numbers in a computer and you spend all your time figuring that out as the actual math model is quite easy to build but getting it all optimized such as linear algebra is not. Even though libraries exist for all these things they are never good enough you always pay some logarithmic penalty for anything not in level 1 cache then it just gets worse from there until your models are doing massively complex work on vectors and matrices taking forever. That's when you start slipping in non-deterministic (probability) tricks many of which we'll learn here.

Curriculum

This seems like a lot but it's probably 40% of what a grad student would do in 2 years.

Arithmetic

Here we study the properties of numbers

  • Herb Gross Classical Arithmetic archived here with videos on YouTube
  • Proving all the laws of naturals, integers. rationals and real numbers from Tao's Analysis I
  • Undergraduate Number Theory
  • Undergraduate Complex Analysis
    • This will come up again in other courses

Algebra

Algebra as we know it today was invented 1200 years ago as a 'system of balancing' to figure out complex Islamic inheritance law and being able to add scalars to both sides of an equation to remove negative numbers. Before that Diophantus wrote several treatises on arithmetic which was really proto-algebra and again invented to solve Athens legal procedures about dividing of estates between family members. The most interesting aspect of Diophantus' work is that it's quite modern how it teaches you how to do math by solving problems giving examples of potential mistakes and dead ends. If you're interested read Brill's Companion to the Reception of Ancient Rhetoric chapter 26 The Rhetoric of Math on library genesis.

Options:

Calculus & Functional Analysis and Probability

Non-convex optimization requires functional analysis and probability models/definitions are much easier with measure theory.

Options:

  • Herb Gross Calculus Revisited lectures on MIT OCW covers the entire Thomas' Calculus book
  • Putnam Seminar to see some calculus proof strategies.
  • Axler's free book and these graduate lectures on measure theory, functional analysis and probability.

We can do Axler's functional analysis/measure theory book at the same time as Herb Gross' lectures because why not.

Geometry

The kind of Geometry that was used for thousands of years to teach mathematical reasoning.

Options:

  • Some of Paul Lockhart's book Measurement
  • Geometry I from the Middle East Technical University
    • Lectures to help us with Geometry from the Mathematical Circles Library
  • Graduate Algebraic Geometry (there's 2 more playlist)
    • This is for my own interest in statistics but it's an incredibly mind blowing art

If there's any time I'll go through Aristotle's Categories too.

What is Geometry

geometry.jpg
Figure 1: Geometry of mathematical finance uni.lu

Geometry is way of doing math. For a few thousand years you learned mathematical reasoning by going through Euclid's Elements and demonstrating the proofs by showing how earlier proofs apply to the one you are demonstrating. Isaac Newton used it to prove limits back when everyone was using infintesimals.

Isaac Newton and Geometry

I did some research on how Isaac Newton learned math, if you're curious expand details.

For geometry-themed entertainment try Isaac Newton on Mathematical Certainty and Method by Niccolo Guicciardini.

Isaac Newton entered Trinity College in 1661 for a law degree as a sub-sizar meaning subsidized tuition in exchange for being a domestic servant. Cambridge then was just a diploma mill there were seldom any lectures, the oral exams at the end of your degree basically optional and fellows (graduate students) took jobs as tutors to supplement their income. Undergrad degrees consisted of one of these tutors giving you a list of books to read. According to his Trinity College notebook which Cambridge still has, this curriculum was a large reading list of classics such as Du Val's four volume collected works of Aristotle like Organon and Metaphysics, Porphyry's Isagoge, Vossius's Rhetoric, Diogenes Laertius, Epicurus, Plato, and many Roman poets like Ovid. This was because of the recovery of Aristotle in the middle ages which shaped 'philosophy of natural sciences' courses for the next few centuries. There didn't exist titles like mathematician, practitioners then considered themselves philosophers. Newton soon stopped taking notes of his official curriculum, and must have read Walter Charleton’s Physiologia because he wrote down 37 headings on different pages that were questions to be answered investigating natural science topics, 18 of which came from Charleton's book and some from Aristotle. These were 'Of Atoms, Of a Vacuum, Of Vision' and many others. He gave each topic different sized gaps indicating how much he thought he needed to write about each. Everytime he would learn something about them he'd write notes under the heading like a comet position he once tracked.

According to Newton's notebook, and the writings of his close friend De Moivre, here is how Newton learned mathematics. Wandering the town fair in 1663 he comes across a book on astrology, and out of curiosity buys a copy. There's a figure he couldn't understand in the book because he didn't know trigonometry. He buys a book on trigonometry but couldn't understand the demonstrations because of a lack of geometry. At Cambridge he goes to the math department and everyone there is deeply immersed in the work of La Geometrie by Descartes. They tell him to buy Barrow's Euclid Elementorum an easier version of Euclid's Elements. He skims it and after finding what he needs for the astrology/trig book abandons the book as 'too trivial' (later has to go back and relearn classic geometry).

He returns to the math department and is given Oughtred's Clavis mathematicae or the key to mathematics writing he understands it except for the solutions of quadratic and cubic equations however his takeaway in his notes is that algebra can be used for exploration, which he starts doing by writing out hundreds of examples.

geometrie.jpg
Figure 2: Cartesian Geometry by Van Schooten rarebooks

Again he returns to the math department to ask for a new book, and they are all still immersed in Cartesian geometry so Newton, despite being told it was a very difficult book, borrows a 1659 two volume Latin translation of Descartes La Geometrie by van Schooten called Geometria, a Renato des Cartes with appendices and commentary by his students. This book was considered the state of the art of 17th century analytical geometry, it would be like reading a graduate text today by a leading expert that had an appendix full of lecture notes and PhD student dissertations.

This is the algorithm Newton used to read van Schooten's two volume book. He read a few pages or so, couldn't understand the text, and went back to the beginning. Went a little further then stopped again going back to the beginning. He repeats this loop by himself until he finally 'makes himself whole of Descartes'. Looking at the online copy of this 1659 version the Descartes geometry is about 104 pages and 450 or so pages of additional research/commentary. It is broken up into 3 parts and assuming he didn't have to reread each part everytime he looped, only the relevant part (~34 pages), it probably took him 3 months to finish van Schooten which is what his notebook shows that after a few months he was already doing research in analytical geometry trying to generalize Descartes. His notebook indicates he learned from the grad student commentaries how to transform a hard problem into a different simpler one.

Newton's notes which you can read online show him trying to generalize any math he read which often would lead him into algebraic corners where he would get stuck as there didn't exist at the time methods to solve/simplify. He seemed to assume everything he read was a special case of something more general such as the insight that integration is the reverse operation of differentiation so there must only be a theory of derivatives somewhere waiting to be discovered.

After Descartes he returned the borrowed copy and bought a different copy of Descartes for himself, and van Schooten's Exercitationes mathematicae libri quinque or Five Books of Mathematical Exercises to help him fill in the blanks of his algebra misunderstandings as he had a lot of mistakes in his notebooks regarding negative roots. Newton assumed the cubic parabola was the same shape in all quadrants but soon after corrected these mistakes. Descartes geometry wasn't like today's Cartesian plane it originally consisted of just one positive quadrant but van Schooten and his students had expanded it.

Descartes' La Geometrie is filled with little comments of encouragement as he was afraid nobody would read his work if it was too long so everywhere there is these reassurences like 'don't worry reader the following isn't too hard' or insisting the reader figure out everything to not deny them of the satisfaction of figuring out the geometry themselves. The different copy Newton buys has all these comments left in tact and Newton misunderstands one of these comments by Descartes about the equation of curves stating it is easy to find everything you want to know about a curve from it's equation and the reader need not be bothered by a lengthy demonstration when in fact this is an unsolved problem. Newton is annoyed he can't figure out this supposedly simple exercise in Descartes' book so breaks down the equation of a curve into many headings in his notebook and tries to generalize which led to him creating his own advanced analysis well beyond any other mathematician at the time.

In the winter of 1664 he reads Arithmetica Infinitorum by Wallis which is the arithmetic of infintesimals. Newton recognizes many of the sums are similar to what he read in Oughtred's book, known today as Pascal's triangle and in typical fashion seeks to generalize and invents the binomial theorem. Newton then reads Viete's Opera Mathematica which was another textbook by van Schooten compiling all the work of Viete such as Diophantine equations. In less than a full year Newton managed to bring himself up to date with the entire achievement of mid 17th century mathematics by himself and begins self-directed research writing out 22 headlines of 'problems' in his notebook and classifying them into groups regarding integration, analytic geometry and mechanics.

The plague shuts down the school from the summer of 1665 to spring of 1667 and Newton returns home, makes himself an office by building bookshelves for his now large library and spends all his time doing research with his new analytical tools building them into calculus. Describing his activities during the second plague year: "I am ashamed to tell to how many places I carried these computations, having no other business at the time, for then I took really too much delight in these inventions". There is notes he kept of calculating a logarithm to it's 52nd decimal point. This is basically the end of story for Newton's analysis research, sometime during the pandemic years he is satisfied with his calculus and abandons research in math to pursue his other questions in his notebook. There is a lot of notes about how tedious it was to do calculations before he came up with his analysis so we can assume he became satisfied that he had all he needed.

Newton believed the ancients had already figured everything out, and this information was lost over time after disasters had destroyed the information. He based this from his own experience living during the plague, the great fire of London and political upheaval. He believed their knowledge was encoded in myths written in Roman and Greek literature. He believed every myth was real but that their lives were embellished through story telling. A common alchemist practice for example was to interpret Ovid's Metamorphoses where the god of the forge/metalwork Vulcan catches his wife Venus and Mars locked in an embrace so traps them in a fine metallic net. The alchemists of the Royal Society that Newton belonged to frequently used the names of planets for metals so obtaining an alchemists manuscript by George Starkey he recreated this myth and ended up with an alloy with a strided net like surface. Newton also decoded Cadmus and the founding of Thebes from Ovid into practical lab instructions.

When he returns to school there is some kind of scholarship crisis where he has to perform a test to prove what he knows, and the Lucasian chair Isaac Barrow at the time was a self-taught expert in Euclidean geometry so tests Newton and finds he knows nothing about geometry. However he's still impressed with his analysis he's discovered so like Tao when he failed his generals is given a pass anyway with the condition that he learns Euclid, which Newton does.

20 years later he is offered to write the Principia, and wanting to prove his infintesimal calculus he tries uncovering the ancient analysis used in classical geometry. Pappus' book 7 contains a commentary about a lost group of tools and propositions that Euclid, Apollonius, Eratosthenes and other geometers of the day used that Pappus referred to as the 'treasury of analysis'. These lost Porisms (corollaries) are speculated by Newton to be projective geometry.

Newton in his Lucasian chair lectures said that the ancients would never bother to introduce the algebra of curves with geometry because you lose the simplicity of working within geometry as its whole point was to escape the tediousness of calculations by simply drawing lines and circles. He also claims books like Pappus's Collectio deliberately hid the analysis, as it was considered an inelegant tool, and that ancient synthesis where they deduced a consequence from a given premise (a corollary) using visual means was a superior method as the analysis could not be reversed in steps like geometric synthesis could. He went further and claimed if you wanted to discover seemingly unrelated corollaries you had to use synthesis, describing the analysis of his day as a 'tedious pile of probabilities used by bunglers'. In other words if he didn't rely on geometry he could not have found most of the critical results of the Principia. People today claim he did this to avoid priority disputes over his analysis he kept hidden except for privately circulated manuscripts, but he used geometry to invent limits and plane transformations that were not formally developed until 200 years later so he was probably right. His limits are very similar to what everyone today is using in modern calculus courses.

He later wrote in a manuscript on geometry that mechanics of motion was what generated all geometry and that the ancients had understood this as well conceiving geometrical objects as generated by moving along a straight edge, circles/elipses via the movement of a compass, or via translation like in Proposition 4 of Book 1 of Euclid's Elements where one triangle form is moved to compare to another. This is where he demonstrated that rotation of rulers were in fact transformation of the plane. Here is how Newton used his rotating ruler to create a power series.

This story tells us a few things, one is that geometry is useful for when you need to see the big picture of something and be able to deduce the unexpected which you can't from analysis. Second is that hard work pays off, Newton refused to not know Descartes' book and in doing so developed advanced mathematics himself simply because of a misprint in translation turned out to be an unsolved problem he assumed was trivial so had the confidence that this problem was already solved thus he could figure it out and he did.

"Comparing today the texts of Newton with the comments of his successors, it is striking how Newton’s original presentation is more modern, more understandable and richer in ideas than the translation due to commentators of his geometrical ideas into the formal language of the calculus of Leibnitz" -Vladimir Arnold 1990


START HERE Elementary school redux

Problem Solving Introduction

Even though we don't know anything, try this single lecture from Poh-Shen Loh's discrete math course just to see you how you should be approaching math problems. He's the US national olympiad coach and if you look at their team results since he took over they won gold 4 times and in 2023 came in second so he knows what he's doing.

Try CMU Discrete Mathematics 1. This is combinatorics which only requires a high school background. The point in watching this is just to see how he throws out any tiny suggestion and then works on it like an experiment. This is how you should do every problem we come across it's an investigation and trial and error. I go through all of these lectures here in a The Art of Computer Programming workshop we won't do them here, instead we'll do his Putnam seminar when we learn calculus.

Construction of numbers

Welcome to 'advanced' elementary school. We start by seeing how number objects are constructed watching a few videos from NJ Wildberger and observing how he proves their operations.

Arithmetic with natural numbers

Wildberger originally made these lectures because his primary school kids were being taught trash mathematics in the 'new and improved' Australian school system (all western schools have this problem now).

  • YouTube - Arithmetic and Geometry Math Foundations 2

The successor function s(n) = n + 1 for natural numbers:

  • s(0) = 1
  • s(s(0)) = 2
  • s(s(s(0))) = 3

Let's prove the laws of multiplication using his definitions

Prove n * 1 = n

The definition of multiplication: mn is n + n repeated m times so n1 is 1 + 1 repeated n times so 11 + 12 + … + 1n = n

We will need this result for the other proofs.

Prove the distributive law: (k + n)m = km + nm

km is m + m repeated k times or m1 + … + mk and (k + n)m means m + m repeated (k + n) times. Distributing m in (k + n)m we get: ((m1 + m +…mk) + (m1 + m +…mn)) and now the left side matches the right side since km + nm is the same when expanded.

Prove the associative law: (kn)m = k(nm)

(kn)m is m + m repeated kn times and kn is n + n repeated k times:

(n1 + … + nk)m or (mn1 + … + mnk) using distributive law.

Right hand side: k(nm) is nm + nm repeated k times (nm1 + … + nmk) and factor out m, both sides of the equation are the same since we haven't proven the commutative law yet so can't claim nm sequence is the same as mn sequence.

Prove the commutative law: nm = mn

First let's prove n1 = 1n or 1n = n. We proved n1 = n already, so we can use substitution to replace n in 1n with 1(11 + .. + 1n) and using the distributive law this is (11 + .. + 1n) or n. Now we have n1 = 1n.

Looking at the right side: mn is (n1 + … + nm)

Factor out n: n(11 + .. + 1m) and a sum of 1's up to m is m, we have nm = nm.

Subtraction and division

  • YouTube - Arithmetic and Geometry Math Foundations 4

Prove the subtraction laws

  • n - (m + k) = (n - m) - k
    • The left side is n + (-1)(m + k)
    • (-1)(m + k) via distributive law is (-1)m + (-1)k
    • (-1)m is m + m groups at -1 times which makes no sense so we can commute
    • m(-1) now it's m copies of -1
    • n + (-11 + -12 .. + -1m) is -1 added up m times or -m
    • n - m + (-11 + -12 .. + -1k) is -1 added up k times or -k
      • n - m - k = n - m - k proved

Oops, he didn't define negative numbers yet (or even zero) since n - m is only defined if n > m (we are still in the natural numbers only) so we can't just move variables to the other side either. We will prove this anyway soon.

The Hindu-Arabic number system

  • YouTube - Arithmetic and Geometry Math Foundations 6

A simplified Roman numeral system is introduced

  • Uses powers of 10
    • 327 = 3 x 102 + 2 x 101 + 7 x 100

This will come up in many of the algebra books we'll do later.

Arithmetic with Hindu-Arabic notation

  • YouTube - Arithmetic and Geometry Math Foundations 7

A new way of hand multiplication is shown, taking advantage of the fact the notation is using the distributive law. "Borrowing" while doing hand subtraction now makes sense. Wildberger rants how his daughter was taught a ridiculous way and this is now true everywhere. Look at a kid's local school curriculum one day how utterly terrible it is. It's like they hired a clown college to rewrite all of primary math.

Laws of Division

  • YouTube - Arithmetic and Geometry Math Foundations 8

Make sure you follow the proofs and the substitution he's using. In every calculus course like the recorded lectures for MIT's 18.01 the prof repeatedly uses these laws without explaining them manipulating equations on the board.

Wildberger is ranting here about long division because schools removed it claiming it was 'too difficult'.

7)203

How many groups of 7 x 102 fit into 203? 0 so reduce to how many groups of 7 x 101 fit into 203? 20

   20  
7)203
 -140
   63  

How many groups of 7 x 101 fit into 63? 0 so reduce to how many groups of 7 x 100 fit into 63? 9

   29
7)203 
 -140
   63
  -63

This is the proper way of doing long division taking advantage of the encoding of the numbers as being multiplied by powers of 10.

Fractions

  • YouTube - Arithmetic and Geometry Math Foundations 9

Interesting how he defines all these as types.

Arithmetic with fractions

  • YouTube - Arithmetic and Geometry Math Foundations 10

The prime notation used in the proof represents another similar variable. Make sure you can figure out his proof write it out yourself if you have to.

Laws of arithmetic for fracions

  • YouTube - Arithmetic and Geometry Math Foundations 11

Follow all these proofs. As per Wildberger in a video in his playlist about primary school education the best possible outcome of a primary student is knowing what a fraction is and their laws.

Integer construction

  • YouTube - Arithmetic and Geometry Math Foundations 12

The point in watching these is to see how you can start with just the natural natural numbers and keep going defining more and more complexity with new numbers.

Rational numbers

  • YouTube - Arithmetic and Geometry Math Foundations 13

We learn what a field is. He says we should try proving all the field operations and we will. If you're curious in the introduction here he shows how the rational number line has been raised on the y-axis showing how all numbers on a line through the origin (0,0) are equal on the rational strip. It explains why a/1 is a and why a/0 is not defined.

Arithmetic full course

I'm going to do in it's entirety Arithmetic by Herb Gross. This is vocational school style math that was originally taught to adults at community colleges and in prisons in North Carolina. It is a seriously high quality resource designed for self-studying and contains all the things someone would expect you to know from elementary arithmetic like what is a rate. The book is here archived. The YouTube playlist has exercise explanation videos you don't have to watch unless needed but they're very good. The first video explains how the course works and I wish all courses had this kind of specific instructions. He uses a lot of Latin and English grammar terms in the videos if you want to relearn basic grammar then investigate the book Learn Latin from the Romans which was written by a linguist so explanations of grammar concepts in English are included.

Watching lecture 1 The Development of a Place Value System. We just learned some of this from Wildberger but he explains it in greater detail like the origins of the Roman X and V symbol, how a sand reckoner worked and became an abacus and how the same sand reckoner method is used today in our place value system. Interesting lecture worth watching if you've wondered where all this originated from. I skimmed the text and did the 'Self-test Form A and Form B' in the study guide which took only a few minutes.

Watching lecture 2 Addition and Subtraction of Whole Numbers. Interesting how all this is explained completely different from how I was taught using bounds and inequalities. This is as basic as it gets and of course the self test is trivial.

Watching lecture 3 Multiplication and Division of Whole Numbers. He explains the commutative property in a fool proof way. His multiplication explanation is pretty good, I've never seen it this way before I was just taught to memorize algorithms. End of lecture he introduces rates, and the division symbol I think I learned from his calculus course ages ago is a fraction line with two placeholders meaning a ratio or rational number. I didn't do any of the exercises they are trivial as I was drilled this countless times as a kid.

Watching lecture 4 Rational Numbers, Part 1. Once again it's amazing to me how he is able to explain all this in a 100% fool proof way. If you want motivation for watching this become the 3blue1brown of arithmetic using his style. This is what Khan Academy should have been but alas. The end of this lecture is one of the first problems in the algebra olympiad problem book we'll do. This lecture is only 52m long there's an editing problem at the end bleeding into the problem solving video.

Reminder there is a textbook for this course too in the archive of the course.

Watching lecture 5 Rational Numbers, Part 2. This is again elementary but presented in an interesting way to teach inverse rates. Doing the 'Self-Test 5, Form B' in the study module the first question asks which is bigger 3/4, 17/24 or 5/6. Make them all the same ratio. That question is very similar to one of Poh-shen Loh's 'daily challenge' vids for middle school olympiads where he says rates are the #1 thing students get wrong.

Second question which is cheapest: 25.20/40, 19.50/30 or 30.50/50 we can perform long division after converting 25.20 to 2520 cents or we can try estimation ignoring the decimals/rounding them and making them all the same ratio with common denominator 50x40x30 or 5x4 which is 20, and 20x3 which is 60 then add 3 zeros to get 60,000. First numerator is (50x30)25 or 5x3 is 15, add two zeros to get (1500)25 then 1000 x 25 is 25,000 and 5 x 25 is 125 plus 2 zeros total 37,500. You could instead scale everything by 10/1 then the common denominator is only (4x3x5) and the first numerator is (3x5)(25) or 3(5x5x5) or 3(125). Try these all in your head then round to make it easier turning 19 into 20 then later subtracting off 1 for how many times you multipled 20 so 20x20 is (400 - 20) if you want the original 19x20. This is how you should be doing these problem sets getting better at estimation and experimenting scaling/distributing to learn more how arithmetic with a place value system works.

  • Study guide for Module 6

    Watching lecture 6 Rational Numbers, Part 3. 'Mixed numbers' is simply addition of two fractions so 4 and 1/3 is 4/1 + 1/3 or A/B + C/D = AD + CB. This is once again trivial to us but try doing everything in your head a different way than an algorithm (after you know the algorithm he's teaching) using distribution, associative or commutative properties. For example @16m we know from Wildberger vids that's 2.5 groups of 4.5 so 4.5 + 4.5 + 2.25 and he shows this in the video too again using inequalities as a bound to check for correctness. @25:46 he shows us what you should have done already which is regroup the values being added to make it easier to estimate. ~@32mins (8 + 1/4 - (2 + 1/3) what I did was distribute -1 to get (8 + 1/4 -2 - 1/3) then regroup (6 - 1/3 + 1/4) and 1/3 of 6 is obviously 5 and 2/3 since 5 + 3/3 is 6. Now we have (5 + 2/3 + 1/4) or 5 and 11/12. @33mins that is how many times can we subtract 2 and 1/4 from 8 and 1/3? 3 times would be 3(2 + 1/4) or 6 and 3/4 then the question is how many 2 and 1/4 can we subtract from 1 and 7/12 which is what is left over. 19/12 divided by 9/4 which is 19/12 x 4/9 or ((20 x 4) - 4)/108 or 76/108 = 19/27. If we were doing algebra we could figure out 9/4 * x = 25/3 by dividing each side by 9/4.

This lecture is to set up the last half which is all about percents and he tells us where the percent notation comes from.

Watching lecture 7 Rational Numbers, Part 4 - Decimals.

More rates, why because calculus is the study of rates of change so if we don't know rates we won't get calculus.

  • Study guide for Module 8 Rational Numbers, Part 5 - Decimal Division

This lecture ends the techniques of arithmetic, the rest of the course is practical applications and this geometry primer.

Proving arithmetic

Obtain Terence Tao's Analysis I I'm using the 4th ed, 2023 Springer version you can get on library genesis if needed. Everything we do here will be point-to reasoning. Start reading at chapter 2 Starting at the Beginning. Notice the natural numbers are similar to how a list type is defined. A list is either empty or link + another list. Then empty is length 0 (base case), link(1, empty) is length 1 (the n-th case), link(1, link(1, empty)) is length 2 (the n++ case) etc. Basic proof by induction is proving a property holds for a base case, assume it holds for n more cases, then prove it holds for the successor or n++ case. It will make sense when you write your own proofs.

Induction A little bit more about inductive reasoning it means you observe specific facts then make an 'inductive leap' to a generalized case. Recall the Isaac Newton story in the beginning of this page, he always tried to generalize everything he came across in texts so he was constantly practicing inductive reasoning.

Proposition 2.1.16 (recursive definitions) which is also in Robert Harper's book Programming in SML where you assign a fixed constant to some base value of the function like f(0) = c then every other value f(x+1) = f(x) or successor inputs rely on previous inputs. Since every number must have a different successor as per Axiom 2.4 this function then defines uniquely all natural numbers. I don't know why Tao includes this here before defining what functions are but whatever it's just a scheme to use a recursive function to enumerate the natural numbers with all unique values.

2.2 Addition. Here is proposition 2.1.16 and definition 2.1.3 in action: 1 + m is defined as a previous input (0++) + m and 2 + m is (1++) + m. Notice that (1++) + m is (1 + m)++ is (m++)++ letting us know the double plus increment is 1 since (1 + m) is equal to (m++). 2 + 3 is ((1++) + 3)++ is (3++)++ is (4)++ or 5. Tao asks how can we prove 'easily' that that the sum of two natural numbers is a natural number and if you look at the form (n + m)++ where both n and m are defined inductively as natural numbers, then we have (number)++ which is Axiom 2.2.

The Lemma 2.2.2 proof, the reasoning is pointing to prior definitions. First for the base case n + 0 = 0 substitute 0 for n, now it is in the form 0 + m = m. Tao then assumes that n + 0 = n so anytime that form comes up in the proof you can point to it as it was declared true. He rewrites using more point-to reasoning arriving at (n + 0)++ and it is in the form of the assumption so can substitute (n + 0)++ for (n)++ because we declared that n + 0 = n. We proved the base case, assumed the n case, and proved the n++ case that in each step a natural number n held the property that n + 0 = n.

The Lemma 2.2.3 proof, more pointing to prior definitions. 0 + (m++) is really 0 + thing = thing or 0 + m = m. Note that m++ is 'fixed' so consider it just another constant value m. He then substitutes (expression)++ for ((n + m)++)++ or (new exp)++ pointing to the assumption. The corollary why is n++ = n + 1 because n++ = n + 0++ equals n++ = (n + 0)++ or n++ = n++

Proposition 2.2.5 try and prove it. Following Tao's examples, we induct on c and keep a and b fixed. First we do the base case c = 0 and show (a + b) + 0 = a + (b + 0). Since we proved addition is commutative, this is 0 + (a + b) = a + (0 + b). The left side, by the definition of addition, 0 + m = m so 0 + (a + b) = (a + b). The right side, a + (0 + b) by the definition of addition is a + b and we have (a + b) = a + b. Now suppose inductively that (a + b) + c = a + (b + c) then (a + b) + c++ = a + (b + c++). Try the rest yourself. You go through the prior definitions and substitute around trying to get it into the form of the assumption. The left side (a + b) + c++ is (expression) + c++ or n + c++ so by Lemma 2.2.3 ((a + b) + c)++. Now it's in the form of the assumption/hypothesis and we can substitute (a + (b + c))++ and you keep doing this until both sides are the same.

The basics of mathematical logic

To complete the next proposition in chapter 2 of Tao's Analysis I we have to read some of the appendix on logic. Skim over it now and as you do the rest of the exercises go back here and read it again. A.1 Mathematical Statements reads like a programming languages text. Why does a negation convert and into or? Because of De Morgan's laws with inclusive or which we haven't seen yet. Tao's example of negation being impossible with an integer x where 'x is neither even nor odd' it's because 0 has an even parity. Reminder just skim this for now and come back later to try the exercises we are just beginning math. A.2 Implication is what we are after. X implies Y is read as 'X is only true if Y is true' and the truth of X never matters in fact the falsity of a hypothesis (if X) makes an implication (then Y) automatically true. He writes you assume X is true, then deduce Y is true. Look up implication on YouTube if you're still confused it takes a while to get used to it. You can come back to this later after you reach A.3 The Structure of Proofs because once we start writing a lot of proofs in analysis this section will be extremely helpful.

2.2 Addition

We can finish the chapter now. 2.2.6 proof if a + b = a + c then b = c Tao uses axiom 2.4 that if n++ = m++ then n = m. Note how you aren't doing any algebra at all, you're just pointing at definitions and rewriting them claiming they are equal. You could do these first few chapters simply cutting out the text and rearranging. Try to prove the rest of the chapter which I already did long ago. If you get stuck refer to online solutions. 2.2.14 'strong' induction you assume every number up to n that all previous numbers hold this property then by implication n++ also has the property. The solutions are all over the internet but at this point don't care about the exact solution, just think why this is true and write an argument to an invisible person trying to convince them that if every natural number from 1 to 50 holds this property, then 50++ will also hold this property.

Read at your leisure through chapters on multiplication, integers and rationals. They are almost exactly like Wildberger showed us in the beginning of this workshop. There exists solutions online everywhere for the problems this is a famous book but you don't have to do the problems, just read his proofs for now. Let the math wash over you at first and we will be doing impossible proofs soon enough.

Algebra TODO

Lean proof assistant

I'm doing the workbook The Mechanics of Proof here.

Calculus

I'm doing Calculus by Herb Gross here.


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