Math Foundations from Scratch

At first, your proofs will just be complete sentences that are a convincing argument to yourself. You don't need to have them verified or a solutions guide. The act of re-reading definitions, experimenting around with the examples, and trying to follow the logic is how you learn math. Remember when you learned to read and write, at first you just naively scribbled unaware your composition skills were terrible until you took more advanced grammar classes in school, and you began to see other people's writing to compare to your own. Then through experience you got better. This is same deal for proofs. You will naively scribble until we take enough logic (grammar) you can check your own proofs. You will read other people's more elegant proofs in books and online, and can at that point understand what they are doing because you have experience. We will see some proof theory and constructive logic that will help us verify our own arguments, and use proof assistants so we can run our proofs as code.

Presented here is a definition of a mathematical object called a natural number, as a string or sequence of strokes, which are successors of each other. ie: the successor of a number, is another number.

• YouTube - Arithmetic and Geometry Math Foundations 1 *note these videos eventually get better in quality

• YouTube - Arithmetic and Geometry Math Foundations 2

An interesting explanation of multiplication.

• Laws for addition
  • n + m = m + n (Commutative)
    • Changing the order of the elements.
  • (k + n) + m = k + (m + n) (Associative)
    • Changing the grouping of the elements.
  • n + 1 = s(n) (Successor)
• Laws for multiplication
  • nm = mn (Commutative)
  • (kn)m = k(nm) (Associative)
  • n \(\cdot\) 1 = n (Identity)
• Note the different notation for multiplication:
  • nm
  • n \(\cdot\) m
  • n \(\times\) m
• Distributive laws
  • k(n + m) = kn + km
  • (k + n)m = km + nm

Exercise: Prove why the multiplication and distributive laws work. Since we have no idea what a proof is yet, we can instead try going through some of our own examples.

Identity law: n \(\cdot\) 1 = n.

• We are creating n groups of 1. According to our definition of multiplication |||| \(\cdot\) | would be 4 groups of | or ||||. This is still 4.

Commutative law: nm = mn.

• ||| \(\cdot\) || = || || || or 6. Rearranging this, || \(\cdot\) ||| is ||| ||| which is again 6.

Associative law: (kn)m = k(nm).

• If k = |, and n = ||, and m = ||| and parens expressions are evaluated first:
  • Left side: (kn)m. We first evaluate (kn) as || (2 groups of 1, the identity law), and 2m is ||| ||| (2 groups of 3) for a total of 6.
  • Right side: k(nm): In the parens nm is 2 groups of 3, so ||| |||. k(6) is the identity law, one group of 6. Thus (kn)m = k(nm).

Distributive law: (k + n)m = km + nm.

• If k = ||||, and n = ||, and m = |:
  • Left side: (|||| + ||) is 6, and (6) x 1 is six groups of 1 or 6 according to the identity law.
  • Right side: km + nm: 4 groups of one is 4, plus two groups of 1 is 2 (identity law). Answer is again 6.

• YouTube - Arithmetic and Geometry Math Foundations 3

Wildberger shows another interpretation of multiplication, and walks through the proofs of the laws by rearranging according to the commutative law.

• YouTube - Arithmetic and Geometry Math Foundations 4

Subtraction is the inverse of addition

• If k + m = n then k = n - m
• Note: n - m is only defined if n > m as we are still in the realm of the Naturals and negative numbers/integers aren't yet defined.

+, -, are binary operations and can only do 2 operations at a time

• Laws of Subtraction:
  • n - (m + k) = (n - m) - k
  • n + (m - k) = (n + m) - k
  • n - (m - k) = (n - m) + k
• Distributive laws of subtraction:
  • n(m - k) = nm - nk
  • (n - m)k = nk - mk

Note to remove parens: https://www.themathpage.com/alg/parentheses.htm we can drop the grouping enforcement when using addition (associative law) but must distribute the negative sign, as subtraction is not associative.

• n - (m + k) = (n - m) - k
  • removing grouping: n - m - k = n - m - k

• YouTube - Arithmetic and Geometry Math Foundations 6

A simplified Roman numeral system is introduced to abstract our previous notation of one's so we can manipulate bigger numbers.

• Uses powers of 10
  • 327 = 3(100) + 2(10) + 7(1)

• YouTube - Arithmetic and Geometry Math Foundations 7

Multiplication with the Hindu-Arabic notation:

• Same laws as before:
  • (ab)c = a(bc)
  • a(b + c) = ab + ac
  • (a + b)(c + d) = ac + ad + bc + bd

A different way of hand multiplication is shown, taking advantage of the fact the notation is using the distributive law.

• YouTube - Arithmetic and Geometry Math Foundations 8

Division is repeated subtraction. Laws of Division:

• 1. \(\frac{k}{m}\) + \(\frac{n}{m}\) = \(\frac{k + n}{m}\)

• 2. \(\frac{k}{m}\) \(\cdot\) \(\frac{n}{l}\) = \(\frac{kn}{ml}\)

• 3. \(\frac{k/m}{n}\) = \(\frac{k}{mn}\)

• 4. \(\frac{k}{m/n}\) = \(\frac{kn}{m}\)

• YouTube - Arithmetic and Geometry Math Foundations 9

We're still using natural numbers.

• Def: A fraction, is an ordered pair (m,n) of natural numbers also written m/n.
  • Fractions m/n and k/l are equal precisely when ml=nk.

• YouTube - Arithmetic and Geometry Math Foundations 10

We're still in the natural numbers, which he defined as beginning with 1 so we aren't working with zeros yet (no division by zero corner cases). He defines the operations of addition and multiplication with fractions using the laws of division we already learned.

Note how he sets up the proof. The 'prime' notation (the tick, so c', a', b') represents another variable. You could rewrite this 'Suppose a/b = y/z so that a/z = b/y, and c/d = something/another, so that c/another = something/d does it then follow …'.

What we are trying to prove is ab' = ba' and cd' = dc'. He introduces some extra notation, multiplying both sides by (dd') to help do this.

• YouTube - Arithmetic and Geometry Math Foundations 11

Wildberger proves the distributive and commutative laws using fractions while still in the natural numbers.

• The 3rd edition of Analysis I from WorldCat or LibGen (or buy it, it's an inexpensive book if you get the international version from abebooks or Amazon India)
• The book How to Think Like a Mathematician by Kevin Houston. Most of this won't make sense until we do a few chapter's out of Tao's book, we will read it at the same time.
• The optional book An Infinite Descent into Pure Mathematics by Clive Newstead which provides another view of the same material if you're confused by something, look it up here (or search YouTube for a tutorial)

We begin like we did with Wildberger, basic counting but rigorously defined.

• The five Axioms
  • Axiom 2.1: 0 is a natural number
  • Axiom 2.2: If n is a natural number, then n++ is also a natural number
  • Axiom 2.3: 0 is not the successor of any natural number
  • Axiom 2.4: Different natural numbers must have different successors
  • Axiom 2.5: Let P(n) be any property pertaining to a natural number n. Suppose that P(0) is true, and suppose that whenever P(n) is true, P(n++) is also true. Then P(n) is true for every natural number n.
    • We make an assumption about a property P(n) being true meaning you do not prove P(n), you prove P(0) and P(n + 1) are true, and both these form a logical implication meaning that if they are both true, then your initial assumption of P(n) being true holds.

Remark 2.1.10 and the template in Proposition 2.1.11 is probably the clearest definition of proof by induction I've seen. Note the example where no natural number can be 'a half number', because of the increment step n + 1. The base case of 0 (as per Axiom 2.5, and Axiom 2.1) fails if we are using 'half numbers' instead of natural numbers because if zero is a natural number, and any n++ is another natural number, the only next successive number is 1 it can't be 0.5 or -3 or 3/4. The way you should view these axioms about the natural numbers is:

• 0 is an element of N
• if x is an element of N, then so is x + 1.
• nothing else is an element of N

Let's understand Axiom 2.5 by watching a lecture from MIT on proofs though you will end up writing so many induction style proofs on the natural numbers in Tao's book this will become second nature.

• 1. Induction lecture
  • From MIT's 6.042j Mathematics for Computer Scientists
• Read chapter 24 Techniques of proof IV: Induction in Kevin Houston's book

Proposition 2.1.16 (Recursive definitions) in the book Analysis I. Tao uses a function, it's definition can be read in chapter 3.3. Here \(a_0\) is function notation so \(a_0\) is \(f_0(0)\), \(a_1\) is \(f_1(1)\). This proposition is using a new function (a rule that maps from one input to one output) every time, I'm guessing to avoid sequences where they could repeat ie 0, 1, 1, 2, 3..

Let's do an example.

• \(a_0\) becomes 0 (for c = 0)
• \(a_{0++}\) becomes \(f_1(a_1)\) or 1
• \(a_{1++}\) becomes \(f_2(a_2)\) or 2

The proof notes Axiom 2.3 protects the value \(a_0\) from being redefined by one of the functions. His base case for this proof is \({a_0}\) (Axiom 2.3) and he is assuming P(n) which is that the procedure will always give a single value to \({a_n}\). Axiom 2.4 is the inductive step, that this procedure will also give a single value to \(a_{m++}\). All of the Axioms are used to prove this. If this didn't make sense it will when we come back here in chapter 3.5 for an exercise.

Let's understand his example for addition. 3 + 5 should be the same as (((5++)++)++) and adding 0 to 5 should result in 5.

Definition 2.2.1 notice he has a base case (0 + m is m), the hypotenuse "Suppose we inductively defined how to add n to m" which we saw at the beginning of 2.2 where 3 + 5 is defined as incrementing five three times, finally the inductive step, we can add n++ to m by defining (n++) + m to be (n + m)++.

Try writing your own examples as advised in the book How to Think Like a Mathematician

• 0 + m = m
• 1 + m = ((0++) + m) or m++
• 2 + m = ((1++) + m) or ((m++)++)
• 3 + m = ((2++) + m) or (((m++)++)++)
• 4 + m = ((3++) + m) or ((((m++)++)++)++)
• 2 + 3 = ((3++)++) or (4++) or 5.
• 3 + 2 = (((2++)++)++) or ((3++)++) or (4++) or 5.

Lemma 2.2.2 notice first how he is using the exact same induction template from 2.1.11, you will eventually get very familiar with this template. He's using prior established definitions to set up the proofs, notably definition 2.2.1 where 0 + m = m is defined. He assumed that n + 0 = n. Next the inductive step, where (n++) + 0 = n++. No arithmetic was done here, he pointed to the definition of addition from 2.2.1 that (n++) + 0 is (n + 0)++ or n++, so we have n++ = n++ and the proof is done. He used a different variable to not confuse it with the previously established definition of 0 + m = m, since we are now going the other way with n + 0 = n on our way to prove a + b = b + a.

Lemma 2.2.3 Prove: n + (m++) = (n + m)++. We are inducting on n, and keeping m fixed which means we apply the inductive step to n++ instead of m++. Replace all instances of n with 0 for the base case and show the inductive hypothesis holds that 0 plus the increment of some other number, equals that number. Try picking apart this proof yourself to follow it:

• base case:
  • 0 + (m++) = (0 + m)++
  • left side is m++ because of definition 2.2.1
  • right side is m++ because of definition 2.2.1
• inductive hypotenuse: 'Suppose n + m++ = (n + m)++'
• inductive step:
  • n++ + (m++) = (n++ + m)++
  • left side is ((n + m)++)++ because of 2.2.1
  • right side is ((n + m)++)++ because of 2.2.1

Corollary: Tao asks why n++ = n + 1 is a consequence from the two proved lemmas. Compare lemma 2.2.3 For any natural numbers n and m, n + (m++) = (n + m)++ to definition 2.2.1 where we were able to show m++ = m + 1. Compare lemma 2.2.2 to the base case in definition 2.2.1

Proposition 2.2.4 (Addition is commutative) the base case is straight forward. We use the definition of addition, and lemma 2.2.2 to rewrite the successor of n to be (n + m)++ = (m + n)++ and now it is in the form of the inductive hypothesis, so we can use our hypothesis of n + m = m + n to show that (m + n)++ is the same as (n + m)++. Proposition 2.2.6 is similar.

Proposition 2.2.6 (Cancellation Law) uses Axiom 2.4 which states 'Different natural numbers must have different successors; i.e., if n, m are natural numbers and n = m, then n++ = m++. Equivalently, if n++ = m++, then we must have n = m'. So (a + b)++ = (a + c)++ is = a + b = a + c by the line if n++ = m++, then n = m in axiom 2.4. We dropped down from the successor by using axiom 2.4 and can now use our inductive hypothesis of the cancellation law for a to show b = c and close the induction. .

Lemma 2.2.10 has an exercise where induction is suggested but this is actually much easier to prove using contradiction (See Kevin Houston book). Always read the comments on Tao's blog if you're stuck: "You can modify the statement of the exercise to an equivalent statement which makes sense for a=0. (The statement will probably look something like “For any natural number a, either a=0, or else there exists exactly one…”.)" If you see this stackexchange answer they use the peano axioms of 2.2 existence and 2.4 uniqueness. We can also try the chapter on contradiction proofs in Kevin Houston's book or the appendix of Analysis: Suppose for the sake of contradiction that that there exists two different natural numbers with the same successor. Then a = m++ and b = m++. But then by Axiom 2.4 we have a = b, a contradiction.

Definition 2.2.11 n is greater or equal to m if n = m + some other natural number. If that number is 0, then the equality holds, if that number is 0++ then the inequality holds.

Proposition 2.3.4 follow along with definition 2.3.1 of multiplication where (n++) x m = (n x m) + m also recall that n++ is n + 1: (n + 1)m so nm + m

• left side:
• a(b + c)++)) is a(b + c) + a
• because (m(n++)) = m(n) + m
• right side:
• ab + a(c++) = ab + ac + a
• ab + ac + a is in the form of the inductive hypothesis so a(b + c) + a

Remark 2.3.8 the contradiction is that ac = bc by corollary 2.3.7 when fed into the trichotomy of order can only result in a = b. This is a scheme to produce a temporary notion of division/cancellation.

• Pythagoras' Theorem 1 and Pythaoras' Theorem 2
  • A geometric proof of original Pythagoras theorem
  • A proof by contradiction why the square root of 2 is irrational
  • Part B shows classical cos/sin/tan trigonometry using chords

• Greek Geometry 1 and Greek Geometry 2
  • Euclid's book the Elements
  • Part B covers different planes, ellipse/conic geometry

This is the geometry we will be doing shortly that allows for computational Euclidean geometry. We don't have a definition of polynomials yet but you can probably understand most of this lecture anyway.

• Analytic Geometry
  • Cartesian coordinate plane explained
  • Part B covers periodic continued fractions method to estimate irrational numbers

Abstract geometry, project from one plane to another like 2D to 3D or into another field to reduce complexity.

• 3D Projective Plane
  • Shows you different perspectives when looking at objects
  • Lay the x,y cartesian plane flat, add a z dimension like a flag pole, add another flat cartesian plane in z, project lines from x,y into z

Even more abstract geometry, construct a new plane by slicing a sphere and studying a hyperbola. Hyperbolic geometry seems better suited for deep learning as it has 'more capacity'. Maybe we will end up back here projecting our data into a hyperbolic plane.

• Non-Euclidean/Hyperbolic Geometry
  • Multiple slicing planes/perspectives

Geometry of space curves, osculating plane

• Differential Geometry
  • Abstractions on top of abstractions
  • There's even more abstract geometry

Flip through sections 1.3.1, 1.3.2 and 1.3.3 Review, Redo, Renew from this draft book used in CMU's 15-151/21-28 course. The author has a list of errata on his page. You will get a crash course in some quick arithmetic, solving linear equations and polynomials. There's a few interesting methods here you may not know about like how to variables from equations. Wildberger covers much of this material too for example Problem 1.3.8, an infinite nest is also in a continuous fractions lecture from the History of Mathematics playlist: \(x = \sqrt{2 + x}\)

If you're interested there is a summary of Algebra here that was originally made for non-english speaking university students. Tao will also define everything here as we progress through his analysis texts.

Some optional mostly 10 minute lectures on algebra basics, such as how Pascal's array/Khayyam triangle/Yang Hui's triangle encodes the binomial theorem, this series sets up understanding algebra and later calculus and combinatorics.

• Introduction to Algebra
  • Wildberger notes there is a subtle change in variables as you get more sophisticated with algebra
• Baby Algebra
  • The basics of solving equations
• Quadratic equations I and part II
  • Completing the square
  • The quadratic formula
• How to find a square root
  • Manual square root algorithms
• Number patterns I
  • Table of differences, summation notation introduced
• Number patterns II
  • Table of differences geometric interpretation
• Euler and Pentagonal numbers
  • Recursive partition function
• Algebraic identities
  • Common identities
• The Binomial theorem
  • Coefficients are related to paths on a two dimensional array
• Binomial coefficients
  • N choose K intro to combinatorics
• Trinomial theorem
  • 3D Pascal's tetrah

Before we do trig, let's learn what \(\pi\) is. Wildberger talks about the history of pi such as Archimedes interval for pi, Newton's formula, a hexadecimal version of pi, his own formula. The end of this lecture he claims pi is some kind of complex object we still don't understand. At this stage we only need to know it's a constant, or the applied approximation of 22/7 or 355/113.

Now that we know trig functions, watch this basics of calculus crash course which was designed for non-english speaking students to learn math notation (with Chinese subtitles). Note in the last few examples how the derivative works d = \(nx^{n-1}\) so in the examples the derivative of \(y = \frac{-g}{2}t^2+v_0t+y_0\) is (\(2\cdot\frac{-g}{2}t^{2-1}+ 1\cdot v_0t^{1-1}\)) or \(-gt + v_0\) you take the exponent power, multiply the coefficient of the base, and subtract the power, so the derivative of \(-gt^1 + v_0\) is (\(1\cdot-gt^0 + v_0\)) which is -g, as we'll see later any terms without a variable (in this example t) cancels out or as Wildberger mentioned are so small so they are discarded.

Summary: if you know the acceleration, then you integrate to find the velocity, then integrate again to find the position. If you know the position, you take a derivative to find it's velocity, and take another derivative to find the acceleration. One function tells you how another function is changing. These are also higher order functions we learned in CS019, where a function produces another function.

Try reading the first chapter from Gilbert Strang's calculus and see if you understand it. His book was designed for self-learning, so that first chapter assumes little background and builds up this intuition of one function that tells you about another. There is also a lecture for this chapter and a self-study guide that acts as a recitation.

There is a lecture on the history of sets/functions, explaining ordinals, cardinals, various schools of logic, and a second lecture about the history of set theory with Godel and Turing.

This chapter on set theory in Analysis I relies on the fact that you've already read the appendix on logic and have an idea what implies means or disjunction. Now that we have done some proofs ourselves, and the basics of trigonometry we can finally understand some of the examples he uses in this appendix. Tao makes a point that this chapter is meant to be read and understood not studied and memorized, and thankfully he is not bombarding us with piles of confusing logic notation that you normally see in other 'how to do proofs' books. At the same time you read this appendix, read How to Think Like a Mathematician by K. Houston it will clear up any cryptic definitions like how implication works.

A.2 Implication seems confusing on first read, it's modern language use relating to casuality is different from the logician's use which is explained here. The only permutation where implication is false is if p implies q and p is true, but q is false. See the Kevin Houston book for more details.

An optional short series on the history of various schools of logic that led to our present state.

• Aristotle and deduction - Brief history of logic MF 251
  • Origins of modern logic
• Stoics and other thinkers - Brief history of logic MF 252
  • Implication discussed
• Islamic/Medieval logic - Brief history of logic MF 253
  • Modal logic origins
• Leibniz to Boole - Brief history of logic MF 254
  • From deduction to induction
  • Notation for relations
  • Boole's book Analysis of Logic
  • De Morgan's laws

Proposition A.2.3, confusing the hypotenuse "Suppose that…" by assuming the conclusion. We always assume the hypotenuse, and prove the conclusion. Proposition A.2.6, proof by contradiction, we know that when pi/2 (90deg) the sine function is 1, because remember it is the y axis of the unit circle. Tao also makes a note here to not use confusing logical notation in proofs.

Remark A.5.1 is in the above history of logic lectures about Aristotle logic.

Exercise A.5.1

• a) For every x++, and every y++, y*y = x or x is the square root of y. Is this true? Square root of two is the obvious candidate for claiming this is a false statement since what y * y = sqrt(2).
• Think about the rest, using the K. Houston book as you go to find a contrapositive

A.6 the point of the examples is to make a point about arguments that use outer variables depending on inner variables being false. We can come back later and reread this section when we learn more concepts like limits, indeed we will likely have to reread this entire appendix many times over while writing future proofs.

We are back to reading Tao's Analysis I and still in the realm of the natural numbers. Example 3.1.2 is similar to a nested list such as [list: 1, [list: 1, 2], 100] is length 3, even though there is a nested list overall it still only contains 3 elements. Remark 3.1.3 is in the above historical lectures on set theory. We are referred to Appendix A.7 Equality to read about the substitution axiom which you should have already read.

Lemma 3.1.6 if set A is equal to set (empty), then if x is in A it must also be in set empty, a contradiction, so A must be a non-empty set is how I interpret the lemma, which is a way to say that if an empty set exists, there must also be a possibility of a non-empty set. Remark 3.1.8 Cardinality if you watched the Wildberger History of Mathematics lecture linked above assigns a value.

Note the errata for Definition 3.1.4 from Tao's site: "Definition 3.1.4 has to be given the status of an axiom (the axiom of extensionality) rather than a definition, changing all references to this definition accordingly. This requires some changes to the text discussing this definition. Firstly, in the preceding paragraph, “define the notion of equality” will now be “seek to capture the notion of equality”, and “formalize this as a definition” should be “formalize this as an axiom”. For the paragraph after Example 3.1.5, delete the first two sentences, and remove the word “Thus” from the third sentence. Exercise 3.1.1 is now trivial and can be deleted." So this is now Axiom 3.1.4, but the definition remains the same.

Exercise 3.1.2 prove emptyset, {emptyset}, {{emptyset}}, and {emptyset{emptyset}} are all not equal.

• If \(\emptyset\) = {\(\emptyset\)} then by Def 3.1.4 \(\emptyset\) \(\in \emptyset\) a contradiction to Axiom 3.2
• Same reasoning as above for \(\emptyset\) is not equal to {{\(\emptyset\)}} or {\(\emptyset\){\(\emptyset\)}}
• {\(\emptyset\)} is not equal to {{\(\emptyset\)}} because [list: 1] is not equal to [list: [list: 1]]. The first has element 1 in it, the second has element [list: 1]. More precisely if {\(\emptyset\)} = {{\(\emptyset\)}} then by Def 3.1.4 every element of {\(\emptyset\)} (it's only element is \(\emptyset\)) must be in {{\(\emptyset\)}} (it's only element is {\(\emptyset\)}) and vice-versa, so claiming {\(\emptyset\)} \(\in\) {\(\emptyset\)} which contradicts Axiom 3.3, where y \(\in\) {a} iff y = a. In this case y must = \(\emptyset\) to be in {\(\emptyset\)}.

Remark 3.1.21 clarifies the above exercise how {2} is not in {1,2,3} but 2 is in {1,2,3}. The rest of this chapter is straight forward definitions of set operations, better than the usual treatment of venn diagrams. Remark 3.1.26 distinct is defined as there are elements of one set which are not elements of the other so \(\emptyset\) and \(\emptyset\) are not distinct because neither can have elements. In Proposition 3.1.28 notice how any operation on two sets results in a new set, like a higher order filter function over a list results in a new list.

Example 3.1.32 looks a lot like programming syntax. There seems to be an order of evaluation, applying the axiom of specification first to create a new set, then applying the axiom of replacement so working right to left in {n++ : n \(\in\) {3,5,9); n<6}

Example 3.3.3 if you forgot range notation. Remark 3.3.5 clears up some notation for functions notably a sequence subscript also denotes a function. Example 3.3.9 asks why 3.3.7 asserts that all functions from empty-set to X are equal, because they have the same domain (nothing) and range.

Our first proof in this chapter is in Lemma 3.3.12, and if we can go by all the previous chapters, this will be our template for the exercises. It also doesn't make sense, so I checked the errata page and sure enough there is errata for this lemma depending what version of the book you're using.

3.3.14 See this tutorial. Write your own examples for this definition. If two distinct elements of A map to the same element of B, this is not injective. However if those two A elements are equal to each other then it is injective:

• A = {a, b, a} (function input)
• B = {1, 2} (output)
  • f(a) = 1
  • f(b) = 2
  • f(a) = 1
    • but {a} = {a}, so a = a and this function is injective.
• A = {b, c}
• B = {2}
  • f(c) = 2
  • f(b) = 2
    • but {c} doesn't equal {b} so this not injective

3.3.17 Onto functions, another tutorial and for Remark 3.3.24 here is a tutorial for bijective inverse functions

Watch this starting @ 36:00, from 41:00+ he shows how he solves problems step by step, trying small cases to understand the problem better and poking at the problem enough to finally figure out a proof. You can check your own proofs too, going through Tao's mathematical logic appendix and seeing if the opposite of your argument holds, and double checking all your assumptions 'is there a definition of X or did I just assume it existed'.

So tl;dr relax and just keep writing because the practice of flipping back through the text and rereading definitions and theorems, tracing the logic and seeing how one definition helps define another, and putting this in the form of notes (a proof) is how you learn. This is why none of these books contain solutions because you're supposed to figure it out yourself.

Chapter 3.4 of Tao's Analysis I. Example 3.4.5 makes more sense if you read Example 3.4.6 too, f^-1({0,1,4}) is the set of everything that could result in {0,1,4} if f(x) = x², so {-2,-1,0,1,2} and in Example 3.4.5, the set of everything in the natural numbers that can result in {1,2,3} if f(x) = 2x, is only f({1}). Axiom 3.11 (Union) if A = {{2,3}, {3,4}, {4,5}} and Ua = {2,3,4,5} why? Because it's {2,3} U {3,4} U {4, 5} you are performing union on every set, this is notation to avoid having to write out an enormous page of union operations. Search youtube for indexed sets. The power set you can choose which subsets you want to add, again search for this on youtube.

The exercises, we've already done similar exercises in set theory there's nothing new here. Reminder that millions of discrete and applied math students have already done similar exercises so you can look up proofs of function inverse images anywhere. I did enough of the exercises so I was sure I understood these set and function operations otherwise move on.

Another topic with plenty of youtube tutorials. Definition 3.5.7 notation for product is introduced, it looks complicated but all it's doing is specifying an index (i is greater or equal to 1, and less than or equal to n) and repeated cross product. The exercises some of the solutions are walked through on youtube.

The cartesian plane itself is a cartesian product of R x R, you have tuples (x, y) or (x, y, z) so R x R x R. In programming the type int * int is the set Integers x Integers, anything that isn't a cross product of the integers with itself will not type check.

A good tutorial. Tao is using #(A) notation which means finite cardinality, while |A| cardinality notation means infinite. There exists errata for the exercises specifically 3.6.8. The last exercise the pigeonhole principle if set A has greater cardinality than B, then f(x) A -> B can't be injective, since there would be two elements in A that point to a single element in B.

If you want you can re-do what we've done so far to get better at this material by programming sets using the book Discrete Math and Functional Programming. Everything we covered such as peano axioms, inductive proofs, cardinality, function images and even converting math notation into programs is covered and highly recommended.

This is unique in that each linear algebra topic is setup with the problem it's trying to solve, how the special case of that problem can be solved, then how you would go about creating and proving a general case. The determinant is built up from scratch. The exercises are largely proofs of the lecture content presented at the end of each lecture. It frequently goes into much more advanced material you won't find in any other linear algebra course which is possible since we're just using a simple affine plane, and working with vectors. The idea is full understanding of the subject using visual geometry so later when we read piles of sums and notation for matrix manipulation we can understand what they are doing. This course teaches you to become 3blue1brown where you can make your own visualizations.

• Introduction - Wild Linear Algebra A1

He constructs the affine plane from scratch. The vector (8/3, 10,3) you need to full screen to better see exactly where they are. From B, you move right 2 lines, and 2/3 so \(2 + \frac{2}{3}\) which is 8/3. From there you move up 1/3, and 3 so \(3 + \frac{1}{3}\) which is 10/3. Reasons for using the affine plane are you do not impose anything on it artificially like the origin (0,0) in the Cartesian plane, you can do projections easily, and it's construction is simple, since only parallel lines.

The biggest a takeaway here is what vector direction is, what a basis vector is and how you can combine them to form larger vectors, and how to translate basis vectors to suit different perspectives. The exercises at the end the delta symbol will return in the lecture on area and the determinant.

In this optional related lecture he shows how ancient people made a ruler using just parallel line geometry, which leads to modelling the rational continuum. Some more optional (short) videos is this from 3Blue1Brown on what a vector is, and this presentation of basis vectors and what a span is. Change of basis is video 13 in the 3BlueBrown playlist but the explanation here by Wildberger in the first lecture I find is better.

• Geometry with Vectors - Wild Linear Algebra A2

All this vector arithmetic will come up later in other courses, at least now you will be somewhat familiar with it. Subtraction means going in the opposite direction, so if you're at the head of one vector and going to another vector's head, you are subtracting. The generalized affine combination where coefficients add up to 1 will come up again.

The proof for diagonals of a parallelogram bisect each other, note how quickly this can become a pile of symbols.

• let lambda = mu in: lambda = 1 - mu
  • if lambda = mu, then moving mu over to the other side means 2lambda
  • 2lambda = 1
  • Divide each side by 2 to isolate lambda
  • lambda = mu = 1/2

@38:00 or so solving linear eq with 2 unknown variables we already covered in the first lecture or review Crash course in solving equations & polynomials.

Since we just did Physics in the software workshop (CS019 Differential systems) let's learn about Archimedes principle of the lever, accelerations in the grid plane and Newton's laws. This will come up again, note the labs here for a course in applied math specifically Lab Manual vol 2, writing python fuctions that perform Barycentric Lagrange interpolation. This is used to study air pollution by approximating graphs.

The fourth lecture in the series to lead up to determinants. You probably recognize ad-bc from the first lecture, which was one of the exercises. He uses different notation for bi-vectors since it represents the union of vector a and b. The torque example you have probably seen somebody using a long pipe over a wrench in order to increase the torque without needing increased force, now you know why increasing the area of the wrench increases the torque. All 3 physics examples can be modelled with a parallegram/bi-vector. The bi-vector rules (ie: adding a scalar to one side of a bi-vector doesn't change the bi-vector) will make sense around 35:50 when he produces a picture proof showing how moving things doesn't change the bi-vector. At around 38:00 we once again get the formula ad-bc, the signed area is a scalar of two basis vectors. 45:50 introduces three-dimensional affine space.

Now that you've seen this model of area using bi-vectors, see this 10 minute presentation by 3Blue1Brown which will fill in some more details, like why the signed area can be negative if you didn't already figure it out. The exercises are conceptually easy, you don't have to do all of them but the whole point of linear algebra is to take these geometric concepts and do them in algebra.

Lecture 5. Now we get a full intro to matrix and vector notation, and matrix operations, this is normally where linear algebra courses start. I like the fact that he built up the change of coordinates first so we aren't just presented with a meaningless array of numbers.

Exer 5.1 if you make a B matrix with e, f, g, h and then multiply AB, take the resulting matrices determinant. The piles of letters should equal det(A) x det(B) after you do some commutative rearranging, this is a 10 minute exercise.

Lecture 6. I appreciate he starts all these lectures with a review, that makes them mostly self contained. This is something I haven't seen before, explaining all the arithmetic of 2x2 and higher matrices using just a 1x1 matrix. The inverse of a matrix is now simple to understand. He walks through some proofs of matrix laws like the associative law (AB)C = A(BC). The standard cartesian coord system is used here for examples of transformations like dilations and shear. A dilation is growing or shrinking, like your eye being dilated.

The end of this lecture: how do you visualize an arbitrary transformation? The next lecture expands on this intuition.

Lecture 7. This tells you how to visualize transformations T(e₁), T(e₂) are the columns of a matrix, so you can look at it and see how it is being transformed. He briefly introduces an alternate framework of rational trig so we don't have to use those sin/cos functions to figure out a rotation. There is a book and entire playlist for rational trig, if you want to learn it write your own software library for rational trig as you go.

End of the lecture introduces non-linear approximations, using differential calculus, to set up doing calculus in higher dimensions. The exercises you become 3Blue1Brown, able to visualize transformations yourself.

Lecture 8. 3 dimensional linear algebra. Pretty good setup explaining the inverse of a 3x3 matrix, most books just give you a formula. We find out why delta is 1/delta. Transpose is also explained better than I've seen before. Reminder that these really complicated introductions all have an off the shelf formula, but he's showing us how they are derived fully so you understand the formula. 3x3 matrix multiplication is explained and pretty simple, explains why B(A(vec)) is the same as (BA)vec. If you're also doing the AI workshop, in 15-388 it's claimed B(A(vec)) is preferable in terms of complexity of running time to (BA)vec.

• YouTube - Arithmetic and Geometry Math Foundations 12

Wildberger's definition is similar to Tao's, he walks through a proof of a-(b-c)=(a-b)+c

Chapter 4 of Analysis I by Tao. Definition 4.1.1 is exactly the definition from the above Wildberger foundations video. Lemma 4.1.3 proof you can finish most of it yourself without reading the whole proof. Lemma 4.1.5 Tao notes the same concerns Wildberger expressed why they both used the 'a less b' style of definition instead of just using Lemma 4.1.5 as the definition of integers, because of the many different cases that would be needed. Tao mentions this definition of the integers forms a commutative ring because xy = yx.

• YouTube - Arithmetic and Geometry Math Foundations 13

This definition is similar to 4.2 in Analysis I. The inverse law, write your own example, the rational number 4/1 inverse is 1/4 and 4/1 x 1/4 is 4/4 or 1. A rational number is (a-b)/(c-d) according to it's definition but we write them as p/q and in some cases just p if q = 1.

• YouTube - Arithmetic and Geometry Math Foundations 14

The rational number line sits on [0, 1], so you don't end up with a zero denominator which isn't defined. A rational number is a line through the origin [0,0], and an integer lying on point [m,n]. Ford circles are an example of the massive complexity of the rational numbers, which are expanded upon in the next lecture.

• YouTube - Real numbers and limits Math Foundations 95

This is about 35m long but worth the watch if you want to see what a convex combination is, learn about rational intervals, and a more complete description of Ford Circles set up with Farey sequences. This will help us understand what a sequence is later in Tao's book. The overall takeaway of this lecture is that the smaller the interval you take on the rational line, the more complex the numbers become with enormous denominators meaning it has a very deep structure.

The next videos, #96 and #97 about Stern-Brocot trees and wedges are highly recommended and can also be found in the book Concrete Mathematics. In Lecture #97, which we can now understand that we know basic linear algebra, he takes a visualization on the plane of the Stern-Brocot tree and shows it also can be used as a geometric interpretation for the rational numbers, it's kind of crazy when you see it. The 'fraction' 1/0 is also explained.

Tao's Analysis I chapter 4.2, definition 4.2.1: 3//4 = 6//8 = -3//-4 we know from Wildberger's visualization of the rational numbers, these numbers all lay on a line. Lemma 4.2.3, in the proof of addition getting rid of +bb'cd is a simple case of subtracting it from both sides, cancelling it out leaving ab'd² = a'bd² and he can divide both sides by d because d is known to be non-zero, since it's in the denominator of a/b + c/d.

To recap we defined the natural numbers, embedded them in the integers (a - b) to define integers, then embedded integers inside the rational numbers where the arithmetic of the integers is consistent with the arithmetic of the rationals. The rationals are a field because of the last identity in 4.2.4 the inverse algebra law. Tao's example proof is similar to the proofs we've seen Wildberger do, convert the letters into integer // integer form, apply law, commute the results together so that they are the same.

The next chapter in Tao's Analysis I uses decimals to represent tiny positive quantities (epsilon) so we should probably know what decimals are.

Wildberger has a basic introduction lecture using natural numbers that assumes you know nothing about exponents and reviews the hindu-arabic notation, shows how to convert a decimal to a fraction. The second lecture introduces rational number decimals and their visualization.

His definition is a decimal number is a rational number a/b, where b is a power of 10 (hence deca/decimal), then shows various examples how you can manipulate rational numbers like 1/2 to be 5/10, or 0.50, we also find out why 1/3 is not a decimal number. In binary, it's a base-2 system, so the only prime factor is 2. If you load up any language interpreter and try and add 0.1 + 0.2 you will get surprising results, typically this number: https://0.30000000000000004.com/

The examples are pretty good, showing how the hindu-arabic encoded powers of 10 drop down by the divisor and result in a decimal. The visualizations are for later lectures showcasing the complexity of bigger and bigger decimals. At 36:00 arithmetic w/decimals is covered, showing how the rational number and decimal number arithmetic methods are the same so you can just stay w/rational numbers if you want converting back to decimals at anytime.

Def 4.3.1 you apply that def to the rational number, so if x = -3, the absolute value is -x so -(-3) which of course makes it a positive rational number, as seen in the example of 4.3.2 where the absolute value of 3 - 5 = 2.

Def 4.3.4 that symbol is epsilon, meant to be a tiny quantity he defines as greater than 0. The example 4.3.6 if two numbers are equal, and their difference is zero, then they are 'epsilon close' for any positive epsilon, since their difference is not bigger than any positive epsilon.

Proposition 4.3.7 recall that all these definitions are for d(x, y) or d(z, w) so x - y or z - w. His proof he moves xz to one side, so it becomes negated (subtract from both sides). He then manipulates |az + bx + ab| according to previous definitions to get |a||z| etc., and then uses the various propositions in 4.3.7 to rewrite them as eps|z| + delta|x| + eps(delta). This will make sense after we attempt to prove the rest of the propositions.

Rest of this chapter covers exponents and their properties.

Proposition 4.4.1 there is an integer n, and a rational number x, such that n <= x < n + 1 so sandwiched between integers 2 and 3 are rational numbers such as 25/10 or 100/40 etc.

Proposition 4.4.3 there is always some rational you can squeeze in between two other rationals as we've already seen in Wildberger lectures. 1/800 < 1/500 < 1/100 etc. If you want put in examples for his generalized proof to better understand it like x = 2, y = 3 and z = (2 + 3)/2.

Proposition 4.4.4 famous proof there does not exist a rational number x for which x² = 2.

In chapter 4.3 we're already coming across the notion of epsilon that Tao is trying to set up for teaching us limits and the construction of the real numbers. It will be extremely helpful to watch parts of the next few lectures to see the architecture of what we will be doing so things like absolute value distance within epsilon makes sense.

• YouTube - Real numbers and limits Math Foundations 106

Watch the first 20 minutes of this lecture and you'll see what the notion of an epsilon displacement is. Note that Tao will define all these things for us later, but at least you'll have an idea of what epsilon is. The rest of the lecture is excellent covering a different definition of limits but optional.

• YouTube - Real numbers and limits Math Foundations 94

Watch from 2:20 to 16:00, he shows the 2D geometric interpretation of Cauchy sequences, and how they are interpreted as 1D on the real line, the idea of 'converging sequences'.

• YouTube - Real numbers and limits Math Foundations 111

He explains the triangle inequality around 8mins, we just read about this in chapter 4.3 of Analysis I (and is one of the exercises). He's not kidding about the huge and unweildly uncountable sets of sequences we will see these soon enough in Tao's book.

• YouTube - Real numbers and limits Math Foundations 112

An excellent lecture why we even need these complex limit and real definitions in the first place, showing how limits make many subjects in analysis much easier. Archimedes bracketed measurement is covered when he tried to figure out the approximation of pi. We also get an introduction to transcendental functions as a geometric series, something we will see in Analysis II by Tao in detail.

The next lecture in the series is optional but Wildberger presents an alternative definition of real numbers, using Archimedean style of nested intervals where equality, algebra laws, order etc are all trivial to prove.

We're watching this lecture: Three dimensional affine geometry | Wild Linear Algebra 9. He actually has a physical model of 3D affine space and shows how it projects to 2D. 3D (x, y, z) coordinates are explained with an interesting question 'how to model space?'. \(A^3\) space and \(V^3\) space are defined, insersecting and parallel planes are illustrated.

As typical with Wildberger lectures this goes far into more advanced material like Mobius bands to model all the lines in the plane, nobody ever showed me this in my linear algebra classes.

Lecture 10. Wildberger goes through different equations for lines: functions (calculus), cartesian and parametic. His example of cutting the Mobius band and not disconnecting all the lines reminds me of a lecture from Erik Demaine's advanced datastructures class and his fascination with origami. We get a new operation dmin(), when he finds the 'determinant of minors' you can see how easy this would be to write these operations as a program, and many optimized libraries already exist for doing so we won't be doing this by hand, this is to understand what those libraries do.

50:32 you can pause the lecture and easily calculate the dmin yourself for practice, in fact whenever he does some operation like finding the determinant see if you can do it yourself first. 51:39 the minus signs in 'the odd places', if you forgot this rewatch lecture 8. Add up the indices, for example the \(a_{1,2}\) index so 3 thus odd. The \(a_{2,3}\) index adds up to 5. Multiply those positions by -1 and any other odd indices.

Lecture 11 the algebra of 3x3 matrices. A diagonal matrix, with constant diagonal, is a model or copy of the field of rational numbers, every number has a constant diagonal matrix. He goes through various transformations like reflections, projections and rotations in 3D. He shows the standard cos/sin rotation then rational trigonometry rotations, which makes rotations easier to manipulate when we are just learning, the basic rotations using sin/cos are the same.todo

Lecture 12, change of coordinates.todo

The marketing indicates it's curriculum is primarily targeted AMC8 to AMC12 (depending on module) middle school competition difficulty emphasizing how to solve problems without requiring memorization. The idea is that everyday for a month you do a challenge problem and then has numerous follow up lectures explaining the challenge as each problem is designed to teach several topics at once. Even though this is directed at kids the content is sophisticated as it's math competition techniques. The pre-reqs are you know everything from pre-algebra and 'are already experienced in math competitions' so things like moems.org the math olympiads for elementary and middle schools.

Singing up for free demo tier requires all kinds of age verification bs like $0.33 non refundable deposit of proof of ID for various regulatory reasons since this is a jr highschool targeted site so I will save you the step and look around.

Module 3: Combinatorics Tools he specializes in this so should be a good module it covers graph theory, set theory, prime factors, rotations and reflections, factorials, problem complexity reduction, recursion, case analysis. First challenge is 'Ordered Arrangements' how many permutations of 4 objects can you make in a certain way, dropping the hint you can do it by brute force. The challenge explanation video is a good set up for n choose k, he describes factorials as diminishing choices. He models the same problem with a tree then describes how to just deduce the entire problem is 4!/2.

Module 1: Algebra Basics covers harmonic average, probability, recurrence relations, estimation and everything to do with understanding ratios and fractions. The challenges are all done with pen and paper. The first challenge is 'Fraction Gymnastics', we're asked to multiply 2.5 x 0.25 x 1.333..x 1.2 which since we've seen Wildberger decimal lectures we know how to convert these to fractions and that 1/3 is .333 repeating. After you convert these to fractions and solve the problem he has a series of tutorials explaining the challenge. He gives a short lesson on cancelling. He explains why these fractions were picked, any num/2 is a 0.5 decimal, num/4 is 0.25 decimal, num/3 will be a repeating infinite decimal, and num/5 will be a 0.2 decimal so you can quickly figure out their decimals in the future if confronted with an enormous decimal in a competition. Long division is used to example why 1/3 repeats forever. More challenge problems are given and the 'weekend' challenge problems are AMC10 level.

Module 4: Algebra Tools first challenge is 'Algebra Tricks' a percent question, a store adds 35% to wholesale value, if you apply a 35% off coupon how much do you save off the wholesale price, which boils down to what is 35% - 35%. He shows how to do this (1 + 0.35)(1 - 0.35) which is of course (x + y)(x - y), and tricks to quickly square any 2 digit number like 0.35. The setup for this is quite good, he really is teaching multiple concepts all with a single problem.

Module 5: Number Theory Tools contains mainly topics you find in a first year 'intro to proofs' style course. One challenge question is "How many squares less than 10,000 are congruent to 1 mod 10?" and this is easy to answer after you see his explanations. I don't remember learning any of this when I was in middle school.

Should you do this if you aren't in Grade 7? Maybe, some of those AMC10 problems are pretty difficult and I know plenty of adults who wouldn't be able to figure out the decimal to fraction challenge let alone adding up all the squares from 1 to 100 in the Number Theory Tools module. Tao's book Solving Mathematical Problems contains some similar competition tools or you could buy the Art of Problem Solving series books if you wanted but I find Poh's style here is superior in every way. There is of course the Don Knuth series The Art of Computer Programming if you want hard problems with solution explanations to try everyday.

The university level competition is the Putnam and he recommends the book Putnam and beyond Razvan Gelca, Titu Andreescu which flipping through it has a good crash course chapter in probability.

This is his free site however the original Expii looked like this https://v1.expii.com/grandmaster but you won't for some reason find that listed anywhere on their main page, and all the practice problems are gone too. The original concept was Expii is a personal tutor to fill missing gaps in the prereqs dependency chain though since this part of the site disappeared off the main page I'm assuming it's depreciated and being replaced with Expii solve.

Every Friday from 4:00pm – 5:00pm EST has a live session of the same content aimed at highschool competitive algebra and geometry where you can ask him anything, for example in 07/08 Wed stream he shows how to solve \(3^{x^2 - 2xy}\) = 1 where x² = y + 3, we can do this too because we've read Tao's book but he always has some interesting optimized method being a national olympiad coach, like solving nested absolute values with case analysis.

Here at 41:13 he answers even nonsense questions giving a tutorial on logical implication. Somebody after asks him what should students focus on breadth or depth? He answers that you get both by doing old problems from math competitions.

A list of questions asked and answered in all these live session are on the forum which you can't view without registration, and if you register you can't post unless you're in the Daily Challenge which is probably a good idea since it's all young kids you don't want to post there anyway.

This is an interesting method he found while we was working on Daily Challenge content trying to better explain quadratic equations without having to use the quadratic formula. The full version is here. Wildberger did something similar, rewriting the quadratic formula using Newton's method of approximation to make it easier for students.

Prof Loh recently put most of his PhD course 21-738 Extremal Combinatorics up on YouTube which uses similar teaching style of his live sessions. It's a theoretical course, if you've never seen one you can still follow along knowing only what we know now combinatorics has low barrier to entry because it's all finite math/sets. These courses aren't always linear they will on different days cover different theorems. Not that we expect to be able to solve hard problems in combinatorics or anything but it's worth watching to see how he plays around with structures and objects to test their properties in a theory course. This course has interesting content like Ramsey Theory where every large enough system no matter how chaotic it appears there must contain large structured subsystems. You can conjure up things with probability to show they exist, then use extremal techniques to 'derandomize' to reveal a deterministic solution.

You can get a crash course on graphs here and that same playlist has a beginner probability summary lecture good enough to follow along many of the lectures if you wanted.

This is a professional development seiminar for teachers given by Prof Loh on linear algebra specifically why is the formula for matrix multiplication constructed the way it is, and how to demystify the determinant. I audited them to see if we could use them here, the determinant lecture is definitely worth your time.