math FOundations A : videos MF 1 79
The Math Foundations series is probably the most important Playlist on NJW's Channel. Its aim is to encourage a shift in pure mathematics towards a more explicit, computational, and hence logical foundation. We want a mathematics unencumbered by philosophical theory, wishful thinking, appeals to authority, or rote allegiance to enshrined texts. We want people to actually understand and appreciate the logical structure of the subject directly, not from a distance or great height.
Currently we have more than 200 videos in the series, so we've organized it into smaller playlists, namely Math Foundations A, Math Foundations B and Math Foundations C. Here are the videos in Math Foundations A. All 79 videos are covered by two screenshot pdfs, which are available here at a very modest price. 
This is first of a series that will discuss foundations of mathematics. Contains a general introduction to the series, and then the beginnings of arithmetic with natural numbers. This series will methodically develop a lot of basic mathematics, starting with arithmetic, then geometry, then algebra, then analysis (calculus) and will also treat so called set theory.
It will have a lot of critical things to say once we get around to facing squarely up to the many logical weaknesses of modern pure mathematics. The series is meant to be viewed sequentially. We spend a lot more time and effort than usual on fundamental issues with number systems. If you are a more advanced student, or a fellow mathematician, then the first few dozen videos might be a bit slow. But they are nonetheless important! 

We introduce the two basic operations on natural numbers: addition and multiplication. Then we state the main laws that they satisfy. This is a basic and fundamental fact about natural numbers; that we can combine them in these two different ways. A lot of arithmetic, and later algebra, comes down to the interaction between addition and multiplication!
This lecture is part of the MathFoundations series, which tries to lay out proper foundations for mathematics, and will not shy away from discussing the serious logical difficulties entwined in modern pure mathematics. The full playlist is at http://www.youtube.com/playlist?list=PL5A714C94D40392AB&feature=view_all 

We explain why the basic laws for addition and multiplication hold, using a model of natural numbers as strings of ones. These are the basic operations, and all students should have some understanding that these operations actually satisfy laws, that are then tools we can use to make calculations faster and more economically. These laws are also the basis for algebra later on.
This lecture is part of the MathFoundations series, which tries to lay out proper foundations for mathematics, and will not shy away from discussing the serious logical difficulties entwined in modern pure mathematics. 

Subtraction and division are inverse operations to addition and multiplication. Here we work with a very simple, even naive, approach to numbers, predating the HinduArabic number system. However we can still discuss arithmetical operations, and do computations!


A one page summary of the contents of K12 mathematics is followed by some basic principles that may be useful in mathematics education. For examplecalculators are unnecessary. After that, some tips on how the foundations so far on arithmetic with natural numbers can guide primary school education.
This lecture is part of the MathFoundations series, which tries to lay out proper foundations for mathematics, and will not shy away from discussing the serious logical difficulties entwined in modern pure mathematics. 

This foundational talk introduces the most important development in the history of mathematics and sciencethe HinduArabic number system. To motivate it, we start by reviewing natural numbers as strings of ones, then introduce the Roman numerals in a simplified form, then the HinduArabic system.


The HinduArabic number system allows us to perform addition, subtraction and multiplication smoothly. We also connect these to primary school education.
It is hard to over emphasize how important this numerical system is for the history of science, technology and the modern world. 

The most challenging of the four basic operations, division is a source of confusion for millions of students. Here we explain why division is really repeated subtraction. Then we prove some basic division rules, and give a simplified form of long divisionsomething every student should learn!


Fractions can be introduced in many different ways. This is a very important topic for mathematics education!
We give a definition depending only on natural numbers, not geometry. 

We define addition and multiplication of fraction to parallel the operations for natural number quotients. A crucial step is to check that these operations are actually welldefined, that is that they respect the notion of equality built into the definition of a fraction.


We explain addition and multiplication for fractions, and the basic laws these operations satisfy. These reduce to the corresponding laws for natural numbers.
There operations are the cornerstone of algebra, and invariably cause young students difficulty. There is no getting around the fact that the operations are somewhat sophisticated, and proving the various laws requires some careful bookkeeping, keeping always in mind that the form of a fraction is not unique. 

The integers are introduced as pairs of natural numbers, representing differences. The standard arithmetical operations are also defined. Often these important mathematical objects are defined only loosely, by `negating' somehow the usual natural numbers. However this makes proving the laws of arithmetic more painful, as then we need to worry about different cases. This procedure here, developed in the 19th century, provides a more uniform approach with distinct theoretical advantages.


Rational numbers are obtained from the integers the same way fractions are obtained from natural numbersby taking pairs of them. The main operations are defined. The rational numbers form a `field', an important technical term in mathematics whose definition we give precisely.


How to visualize rational numbers using lines in the plane through the origin and the rational number strip. We connect this with the lovely theory of Ford circles.


What do foundational issues tell us about teaching mathematics at the primary school level? Here we give some insights into arithmetic with different kinds of numbers. We also introduce a two dimensional, geometrical, view of rational numbers.


Historically mathematicians have been careful to avoid treating `infinite sets'. After G. Cantor's work in the late 1800's, the position changed dramatically. Here I start the uphill battle to convince you that talking about`infinite sets' is just thattalk, not mathematics. The paradoxes discovered a hundred years ago are still with us, even if we ignore them.


We look at extremely big numbers. This is the best way to get a feel for the immensity and complexity in the sequence of natural numbers. And why we have no right to talk about `all' of them as a completed `infinite set'. Our main tool is a cool inductive way of defining higher and higher operations, going beyond multiplication and exponentiation.


How to begin geometry? What is the correct framework? How to define point, line, circle etc etc?
These are some of the issues we will be addressing in this first look at the logical foundations of geometry. 

Euclid's book `The Elements' is the most famous and important mathematics book of all time. To begin to lay the foundations of geometry properly, we first have to make contact with Euclid's thinking. Here we look at the basic setup of Definitions, Axioms and Postulates, and some of the highlights from Books I,II and III.


The ancient Greeks considered magnitudes independently of numbers, and they needed a way to compare proportions between magnitudes. Eudoxus developed such a theory, and it is the content of Book V of Euclid's Elements. This video describes this important idea.


A very brief outline of the contents of the later books in Euclid's Elements dealing with geometry. This includes the work on three dimensional, or solid, geometry, culminating in the construction of the five Platonic solids.


There are logical ambiguities with Euclid's Elements, despite its being the most important mathematical work of all time. Here we discuss some of these, as well as Hilbert's attempt at an alternative formulation. We prepare the ground for a new and more modern approach to the foundations of geometry.


This video begins to lay out proper foundations for planar Euclidean geometry, based on arithmetic. We follow Descartes and Fermat in working in a coordinate plane, but a novel feature is that we use only rational numbers.
Points and lines are the basic objects which need to be defined.. 

We discuss parallel and perpendicular lines, and basic notions relating to triangles, including the notion of a side and a vertex of a triangle.


Distance is not the best way to measure the separation of two points, as Euclid knew. The better way is using the square of the distance, called quadrance. Here we introduce this concept, and the two most important theorems in mathematicswith purely algebraic proofs.


Angles don't make sense in the rational number system. The proper notion of the separation of two lines is the `spread' between them, which is a purely algebraic quantity and can be calculated easily using rational arithmetic only. This video highlights some of the advantages in replacing `angle' with `spread'. It also gives an explicit formula for the `inverse cosine' function, which rarely appears in trigonometry texts, despite the universal reliance on this function via calculators.


Rational trigonometry works over the rational numbers, and allows us a more elementary and logical approach to the basics of trigonometry. This video illustrates the Spread law, the Cross law and the Triple spread formula. These are among the most important formulas in geometry, indeed in all of mathematics, and they allow us to recast trigonometry into a simpler and more computationally elegant subject. See the WildTrig YouTube series for lots of applications of these laws.


Moving beyond points and lines, circles are the next geometrical objects we encounter. Here we address the question of how best to introduce this important notion, strictly in the setting of rational numbers, and without metaphysical waffling about `infinite sets.'


Moving beyond points and lines, circles are the next geometrical objects we encounter. Here we address the question of how best to introduce this important notion, strictly in the setting of rational numbers, and without metaphysical waffling about `infinite sets.'


How to describe all the points on a circle, using a rational parametrization. This is a major improvement on the usual transcendental parametrization with circular functions.
Also some interesting number theory arises when we ask which lines through the center of a circle meet that circle. 

We use vectors to introduce parallelograms, the parametric representation of a line, and affine combinations, such as midpoints.


Some comments on the teaching of geometry in primary schools (K6). I emphasize the importance of the grid plane, as well as constructions and drawing, and give examples of important topics.


While there is a naive idea of area in terms of number of unit squares that can fit inside a region, this is not the best definition. It is better to work with oriented triangles and maintain linearity.


How to define the area of a polygon? The right way is to consider signed areas of oriented polygons. This leads to natural formulas that are important for calculus.


We introduce translations in the rational plane. However we do not assume the conventional understanding of functions and mappings, which actually has some logical difficulties and conceptual disadvantages. We prefer a cleaner and more flexible understanding emphasizing the use of expressions.


We introduce rotations acting on vectors, not points. Angles are not used, but the rational parametrization of the unit circle is important. An interesting formula for the product of two rotations is given.


We introduce reflections acting on vectors, not points, in a similar way to rotations in the last video. Now the product of two reflections is a rotation.


We begin to address the many logical difficulties arising from the reliance on angles in modern mathematics. The main issue is one of precise definitions: what exactly is an angle?? We give three different answers, including the modern one met by most high school students, and explain why it is really a cheatcalculus is required to make it work correctly.


This video continues to discuss difficulties with angles. It takes a historical approach, and emphasizes that with angles, imprecision is unavoidable for most nontrivial geometrical problems. It ends with a challenge problem.


The current technology for solving geometrical problems means that answers are typically in an approximate decimal form, and so strictly speaking incorrect. The problem arises with the reliance on angles, which are inherently imprecise.


Although angles appear to be simple, mostly due to the linearity they impose on an essentially nonlinear problem, they are really full of difficulty. A key reason is that most calculations involving angles also require the transcendental circular functions such as cosx, sinx etc. Here we discuss some of the difficulties and confusions surrounding these functions.


[First of two parts] Here we address a core logical problem with modern mathematicsthe usual definition of a `function' does not contain precise enough bounds on the nature of the rules or procedures (or computer programs) allowed.
Here we discuss the difficulty in the context of functions from natural numbers to natural numbers, giving lots of explicit examples. WARNING: this video and the next destabilizes much of the mathematics taught in universities. 

[Second of two parts] We address a core logical problem with modern mathematicsthe usual definition of a `function' does not contain precise enough bounds on the nature of the rules or procedures (or computer programs) allowed.
Here we discuss the difficulty in the context of functions from natural numbers to natural numbers, giving lots of explicit examples. WARNING: this video and the last one destabilizes much of the mathematics taught in universities. 

The general notion of `function' does not work in mathematics, just as the general notions of `number' or `sequence' don't work.
This video explains the distinction between `closed' and `open' systems, and suggests that mathematical definitions should respect the open aspect of mathematics. So while we may well define `constant functions', or `linear functions' etc, we cannot at once capture the idea of a `general function'. 

We discuss important metaissues regarding definitions and specification in mathematics. We also introduce the idea that mathematical definitions, expressions, formulas or theorems may support a variety of possible interpretations.
Examples use our previous definitions from elementary geometry. 

Precise definitions are important! Especially in geometry, where traditional texts too often just assume that the meanings of the main terms are obvious. Quadrilaterals, quadrangles and ngons are good examples.


There are three main branches of mathematics: arithmetic, geometry and algebra. This is the correct order, both in terms of importance and of historical development. Here we introduce our program for setting out foundations of algebra.


Algebra starts with the natural and simple problem of trying to solve an equation containing an unknown number, or `variable'. Here we start with simple examples familiar to public school students.
This video belongs to Wildberger's MathFoundations series, which sets out a coherent and logical framework for modern mathematics. The idea is to transform an equation with a variable into a simpler but equivalent equation, which can be more easily solved. We review examples of such manipulationsthat go back to Hindu and Arab mathematicians. 

We introduce the algorithm for solving a quadratic equation known as `completing the square'. This technique was known since ancient times, and students should know the derivation, not just the formula.
This lecture has a second part, and belongs to Wildberger's MathFoundations series, which sets out a coherent and logical framework for modern mathematics. 

We introduce the algorithm for solving a quadratic equation known as `completing the square'. This important technique was known since ancient times, and students should know the derivation, not just the formula!


We consider three methods, or algorithms, for finding the square root of a natural number we know to be a square. One is trial and error estimation, the other is the Babylonian method equivalent to Newton's method, and the third we call the Vedic method, since it goes back to the Hindus. It is completely feasible to do by hand.


One important use of letters in algebra is to describe patterns in a quantitative and general way. We look at the `sequences' of square numbers and triangular numbers, and derive formulas for the nth terms. A table of differences shed light on these and other number patterns.


The patterns formed by triangular numbers and square numbers have generalizations in different directions. One is to three dimensional tetrahedral numbers, and three dimensional pyramidal numbers. Another is to pentagonal numbers, which have a number of interesting features.


Leonhard Euler was the greatest mathematician of modern times. His work on pentagonal numbers shows that they connect naturally to sums of divisors of numbers, and also to the partition functions. These are both really surprising facts.


Algebraic identities are at the heart of a lot of mathematics, especially geometry and analysis. Here we have a look at some simple and familiar identities, such as the difference of squares, the geometric series, and identities that go back to Pythagoras and Fibonacci.


The Binomial theorem is a key result in elementary algebra, arising naturally from the Distributive law. We connect Pascal's triangle to the difference table of triangular numbers. The entries are related to paths in a two dimensional array using only two types of steps.


Binomial coefficients are the numbers that appear in the Binomial theorem, and also in Pasal's triangle. They are also naturally related to paths in Pascal's array, essentially the difference table associated to the triangular numbers. We also relate binomial coefficients to the rising and falling powers notation introduced by Knuth.


The Binomial theorem has extensions to more than two variables. The next interesting case is the Trinomial theorem, which connects naturally to triangular numbers and whose coefficients related to three dimensional paths. There is a lovely three dimensional analog of Pascal's array.


We begin the important task of defining the fundamental objects of modern algebra. First we review different roles played by polynomials. We are going to base polynomials on something more fundamental called polynumbers, whose arithmetic parallels but is richer than that of the natural numbers and rational numbers.


Polynumbers are extensions of the positive numbers 0,1,2,3... and have an arithmetic which is the same as that of polynomials. In fact polynumbers present us with a more logical and fundamental approach to polynomial arithmetic.
This video presents some basic definitions, such as the degree of a polynumber, and then explains addition and multiplication of polynumbers. The latter is not much different from the way you multiply ordinary numbers! 

Polynumbers are extensions of numbers, but with a richer arithmetic. We will use them to provide a more solid foundation for the study of polynomials.
Here we look at multiplying a positive polynumber by a scalar or number, connecting the multiplication of polynumbers with ordinary multiplication in the HinduArabic system, and sketch the proof of associativity of multiplication of polynumbers. 

Polynomials are fundamental objects in algebra, but unfortunately most accounts of them skimp on giving a proper definition. Here we base polynomials on the more basic objects of polynumbers.
We introduce the particular positive polynumber alpha, and show that any polynumber can be written as a linear combination of powers of alpha. Then we define a positive polynomial to be a positive polynumber written in this standard alpha form. 

We discuss multiples and factoring, first for natural numbers, and then for polynumbers. This motivates us to extend our consideration to integral polynomials involving negative numbers.


We introduce basic arithmetic with integral polynumbers; the operations of addition, multiplication and subtraction. Simple examples relate to the Binomial theorem and other interesting identities.


We introduce the idea of evaluating a polynumber p at an integer c. This evaluation respects the additive and multiplicative structures of arithmetic. Then we state and prove the important Factor theorem of Descartes and its important Corollory relating a zero of a polynumber and a linear factor of the polynumber.


We review our approach to natural numbers, integers, fractions and rational numbers. Then we consider the analogous objects for polynumbers. Division of integral polynumbers is similiar to long division of ordinary numbers. There are two approaches: one starting with lower order terms, the other starting with higher order terms.


This video introduces a twodimensional aspect to arithmetic by considering both polynumbers written as columns and as rows, and then putting those two ideas together to define bipolynumbers, whose coefficients form a rectangle. The associated polynomials are written in two variables, alpha and beta, which now have a distinguished meaning as special polynumbers.


Decimal numbers are a source of confusion in primary school, high school, university and research level mathematics. Here we begin a rather careful study of these objects, starting with extending the HinduArabic to negative powers of 10, such as onetenth, onhundredth etc.


This video gives a precise definition of a decimal number as a special kind of rational number; one for which there is an expression a/b where a and b are integers, with b a power of ten. For such a number we can extend the HinduArabic notation for integers by introducing the decimal form, with additional digits to the right of the decimal point. Visualizing decimal numbers requires a notion of successive magnifications on the number line.


We introduce Laurent polynumbers, the analogs of decimal numbers in the polynumber framework. We first review arithmetic of rational polynumbers/ polynomials, and say some derogatory things about the beliefs of modern pure mathematicians regarding `real numbers' and `complex numbers'. In fact the two most important fields are the rational numbers and the rational polynumbers. Laurent polynumbers are introduced and their arithmetic is described.


We begin moving towards calculus with polynumbers/polynomials by introducing the Derivative D=D_1 in a simple algebraic way. First we discuss composition of integral polynumbers, and the translation of a polynumber by an integer. This leads to the Taylor (bi) polynumber/polynomial of a polynumber/polynomial, which contains not only the usual derivative, but also higher analogs called subderivatives, denoted D_2, D_3 etc.


This video introduces the graphs of polynumbers of polynomials. We plot both integer and rational points, and use straightedge lines between those. We look at a quartic polynomial in some detail.


We study lines and parabolas using elementary calculus derived only from algebraic manipulations with polynomials, or polynumbers in our setting. We discuss yintercepts, slopes and xintercepts of lines, along with another look at the meets of two lines. Then we study parabolas in the context of quadratic polynomials, using elementary algebraic calculus to find Taylor expansions, derivatives and tangent lines.


We continue our study of parabolas as quadratic polynomials using elementary algebraic calculus. We compute subderivatives, tangent lines and Taylor expansions. We apply completing the square to studying the shape and behaviour of parabolas, and derive some interesting geometrical relations related to addition and multiplication.


We introduce cubic polynomials, and the basic algebraic calculus for them, involving their Taylor expansions, subderivatives and tangent lines and tangent conics. The tangent conics are particularly interesting, and lead to the (arguably!) prettiest theorem in calculus. This is a result due to Etienne Ghys, described in his video lectures on Osculating Curves. We give here an elementary and elegant proof.


This is a gentle introduction to curves and more specifically algebraic curves. We look at historical aspects of curves, going back to the ancient Greeks, then on the 17th century work of Descartes.
We point out some of the difficulties with Jordan's notion of curve, and move to the polynumber approach to algebraic curves. The aim is to set the stage to generalize the algebraic calculus of the previous few lectures to algebraic curves. 

Twentieth century mathematics has been object oriented. Twentyfirst century mathematics, if it gets its act together, will be much more expression oriented. Here we describe the distinction by studying the key example of the unit circle.


We illustrate algebraic calculus on the simplest algebraic curves: the unit circle and its imaginary counterpart. Starting with a polynumber/polynomial of two variables, the derivation of the Taylor polynumber, subderivatives, Taylor expansion around a point [r,s] and various tangents are analogous to the case of a polynumber/polynomial of one variable. We get tangent planes and tangent lines both corresponding to the first tangent.
The algebraic derivation is illustrated with threedimensional diagrams involving the associated elliptic paraboloid to the unit circle. The background noise in parts of the video is due to crickets, common in the Australian summersorry about that. I will tell them to keep it down next time. 

We investigate the Folium of Descartes, viewing it both as a cubic curve in the plane, and as a surface in three dimensional space. This is an extended exercise in algebraic calculus.
We parametrize it, calculate Taylor polynumbers and expansions and tangents, and interpret them both as tangent planes of tangent lines, or as tangent quadrics or tangent conics. We finish with a pleasant duality satisfied by the tangent conics. 
