Math Foundations with Norman Wildberger

Video Lectures

Displaying all 127 video lectures.
Lecture 1
What is a number?
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What is a number?
The 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.
Lecture 2
Arithmetic with numbers
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Arithmetic with numbers
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.
Lecture 3
Laws of Arithmetic
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Laws of Arithmetic
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.
Lecture 4
Subtraction and Division
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Subtraction and Division
Subtraction and division are inverse operations to addition and multiplication. Here we work with a very simple, even naive, approach to numbers, pre-dating the Hindu-Arabic number system. However we can still discuss arithmetical operations, and do computations!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.
Lecture 5
Arithmetic and Math education
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Arithmetic and Math education
A one page summary of the contents of K-12 mathematics is followed by some basic principles that may be useful in mathematics education. For example---calculators 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.
Lecture 6
The Hindu-Arabic number system
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The Hindu-Arabic number system
This foundational talk introduces the most important development in the history of mathematics and science--the Hindu-Arabic 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 Hindu-Arabic system.
Lecture 7
Arithmetic with Hindu-Arabic numbers
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Arithmetic with Hindu-Arabic numbers
The Hindu-Arabic number system allows us to perform addition, subtraction and multiplication smoothly. We also connect these to primary school education.
Lecture 8
Division
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Division
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 division---something every student should learn!
Lecture 9
Fractions
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Fractions
Fractions can be introduced in many different ways. We give a definition depending only on natural numbers, not geometry.
Lecture 10
Arithmetic with fractions
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Arithmetic with fractions
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 well-defined, that is that they respect the notion of equality built into the definition of a fraction.
Lecture 11
Laws of arithmetic for fractions
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Laws of arithmetic for fractions
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 book-keeping, keeping always in mind that the form of a fraction is not unique.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.
Lecture 12
Introducing the integers
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Introducing the integers
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.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.
Lecture 13
Rational numbers
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Rational numbers
Rational numbers are obtained from the integers the same way fractions are obtained from natural numbers---by 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.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.
Lecture 14
Rational numbers and Ford Circles
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Rational numbers and Ford Circles
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.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.
Lecture 15
Primary school maths education
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Primary school maths education
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.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.
Lecture 16
Why infinite sets don't exist
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Why infinite sets don't exist
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 that---talk, not mathematics. The paradoxes discovered a hundred years ago are still with us, even if we ignore them.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.
Lecture 17
Extremely big numbers
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Extremely big numbers
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.
Lecture 18
Geometry
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Geometry
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.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.
Lecture 19
Euclid's Elements
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Euclid's Elements
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 set-up of Definitions, Axioms and Postulates, and some of the highlights from Books I,II and III.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.
Lecture 20
Euclid and proportions
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Euclid and proportions
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.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.
Lecture 21
Euclid's Books VI--XIII
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Euclid's Books VI--XIII
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.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.
Lecture 22
Difficulties with Euclid
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Difficulties with Euclid
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 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.
Lecture 23
The Basic Framework for Geometry I
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The Basic Framework for Geometry I
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.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.
Lecture 24
The Basic Framework for Geometry II
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The Basic Framework for Geometry II
We discuss parallel and perpendicular lines, and basic notions relating to triangles, including the notion of a side and a vertex of a triangle.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.
Lecture 25
The Basic Framework for Geometry III
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The Basic Framework for Geometry III
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 mathematics---with purely algebraic proofs. 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.
Lecture 26
The Basic Framework for Geometry IV
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The Basic Framework for Geometry IV
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.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.
Lecture 27
Trigonometry with rational numbers
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Trigonometry with rational numbers
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.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.
Lecture 28
What exactly is a circle?
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What exactly is a circle?
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.'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.
Lecture 29
Parametrizing circles
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Parametrizing circles
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.
Lecture 30
What exactly is a vector?
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What exactly is a vector?
The notion of vector is here made completely explicit. Vectors arise in physics as forces, positions, velocities, accelerations, torques, displacements. It is useful to distinguish between points and vectors; they are different types of mathematical objects. In particular the position of a vector is not fixed, as it is determined by a pair of points.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.
Lecture 31
Parallelograms and affine combinations
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Parallelograms and affine combinations
We use vectors to introduce parallelograms, the parametric representation of a line, and affine combinations, such as midpoints.This video belongs to Wildberger's MathFoundations series, which sets out a coherent and logical framework for modern mathematics.
Lecture 32
Geometry in primary school
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Geometry in primary school
Some comments on the teaching of geometry in primary schools (K-6). I emphasize the importance of the grid plane, as well as constructions and drawing, and give examples of important topics.This video belongs to Wildberger's MathFoundations series, which sets out a coherent and logical framework for modern mathematics.
Lecture 33
What exactly is an area?
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What exactly is an area?
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.This video belongs to Wildberger's MathFoundations series, which sets out a coherent and logical framework for modern mathematics.
Lecture 34
Areas of polygons
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Areas of polygons
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.This video belongs to Wildberger's MathFoundations series, which sets out a coherent and logical framework for modern mathematics.
Lecture 35
Translations, rotations and reflections I
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Translations, rotations and reflections I
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.This video belongs to Wildberger's MathFoundations series, which sets out a coherent and logical framework for modern mathematics.
Lecture 36
Translations, rotations and reflections II
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Translations, rotations and reflections II
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.This video belongs to Wildberger's MathFoundations series, which sets out a coherent and logical framework for modern mathematics.
Lecture 37
Translations, rotations and reflections III
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Translations, rotations and reflections III
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.This video belongs to Wildberger's MathFoundations series, which sets out a coherent and logical framework for modern mathematics.
Lecture 38
Why angles don't really work I
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Why angles don't really work I
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 cheat---calculus is required to make it work correctly.This video belongs to Wildberger's MathFoundations series, which sets out a coherent and logical framework for modern mathematics.
Lecture 39
Why angles don't really work II
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Why angles don't really work II
This video continues to discuss difficulties with angles. It takes a historical approach, and emphasizes that with angles, imprecision is unavoidable for most non-trivial geometrical problems. It ends with a challenge problem.This video belongs to Wildberger's MathFoundations series, which sets out a coherent and logical framework for modern mathematics.
Lecture 40
Correctness in geometrical problem solving
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Correctness in geometrical problem solving
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.This video belongs to Wildberger's MathFoundations series, which sets out a coherent and logical framework for modern mathematics.
Lecture 41
Why angles don't really work III
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Why angles don't really work III
Although angles appear to be simple, mostly due to the linearity they impose on an essentially non-linear 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.This video belongs to Wildberger's MathFoundations series, which sets out a coherent and logical framework for modern mathematics.
Lecture 42
Deflating Modern Mathematics: the problem with 'functions' - Part 1
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Deflating Modern Mathematics: the problem with 'functions' - Part 1
[First of two parts] Here we address a core logical problem with modern mathematics--the 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.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.
Lecture 43
Deflating Modern Mathematics: the problem with 'functions' - Part 2
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Deflating Modern Mathematics: the problem with 'functions' - Part 2
[Second of two parts] We address a core logical problem with modern mathematics--the 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.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.
Lecture 44
Reconsidering `functions' in modern mathematics
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Reconsidering `functions' in modern mathematics
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'.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.
Lecture 45
Definitions, specification and interpretation
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Definitions, specification and interpretation
We discuss important meta-issues 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. 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.
Lecture 46
Quadrilaterals, quadrangles and n-gons
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Quadrilaterals, quadrangles and n-gons
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 n-gons are good examples.This video belongs to Wildberger's MathFoundations series, which sets out a coherent and logical framework for modern mathematics.
Lecture 47
Introduction to Algebra
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Introduction to Algebra
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.This video belongs to Wildberger's MathFoundations series, which sets out a coherent and logical framework for modern mathematics.
Lecture 48
Baby Algebra
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Baby 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 manipulations--that go back to Hindu and Arab mathematicians.
Lecture 49
Solving a quadratic equation
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Solving a quadratic equation
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.
Lecture 50
Solving a quadratic equation
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Solving a quadratic equation
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!This video belongs to Wildberger's MathFoundations series, which sets out a coherent and logical framework for modern mathematics.
Lecture 51
How to find a square root
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How to find a square root
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.This video belongs to Wildberger's MathFoundations series, which sets out a coherent and logical framework for modern mathematics.
Lecture 52
Algebra and number patterns
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Algebra and number patterns
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.This video belongs to Wildberger's MathFoundations series, which sets out a coherent and logical framework for modern mathematics.
Lecture 53
More patterns with algebra
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More patterns with algebra
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.This video belongs to Wildberger's MathFoundations series, which attempts to create a coherent and logical framework for modern mathematics.
Lecture 54
Leonhard Euler and Pentagonal numbers
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Leonhard Euler and Pentagonal numbers
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.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.
Lecture 55
Algebraic identities
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Algebraic identities
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.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.
Lecture 56
The Binomial theorem
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The Binomial theorem
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.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.
Lecture 57
Binomial coefficients and related functions
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Binomial coefficients and related functions
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.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.
Lecture 58
The Trinomial theorem
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The Trinomial theorem
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.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.
Lecture 59
Polynomials and polynumbers
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Polynomials and polynumbers
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.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.
Lecture 60
Arithmetic with positive polynumbers
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Arithmetic with positive polynumbers
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!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.
Lecture 61
More arithmetic with polynumbers
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More arithmetic with polynumbers
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 Hindu-Arabic system, and sketch the proof of associativity of multiplication of polynumbers.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.
Lecture 62
What exactly is a polynomial?
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What exactly is a polynomial?
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. 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. 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.
Lecture 63
Factoring polynomials and polynumbers
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Factoring polynomials and polynumbers
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.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.
Lecture 64
Arithmetic with integral polynumbers
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Arithmetic with integral polynumbers
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.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.
Lecture 65
The Factor theorem and polynumber evaluation
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The Factor theorem and polynumber evaluation
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.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.
Lecture 66
The Division algorithm for polynumbers
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The Division algorithm for polynumbers
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 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.
Lecture 67
Row and column polynumbers
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Row and column polynumbers
This video introduces a two-dimensional 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.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.
Lecture 68
Decimal numbers
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Decimal numbers
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 Hindu-Arabic to negative powers of 10, such as one-tenth, on-hundredth etc.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.
Lecture 69
Visualizing decimal numbers and their arithmetic
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Visualizing decimal numbers and their arithmetic
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 Hindu-Arabic 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.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.
Lecture 70
Laurent polynumbers (the New Years Day lecture)
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Laurent polynumbers (the New Years Day lecture)
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. This is the first lecture in this series in 2012, and we prepare for a mighty battle this year, replacing the current ambiguous and flawed foundations of the subject with clear thinking and precise definitions. You are invited to join the fray!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.
Lecture 71
Translating polynumbers and the Derivative
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Translating polynumbers and the Derivative
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 sub-derivatives, denoted D_2, D_3 etc. This lecture should be disorienting but exciting to those who have been indoctrinated into thinking that calculus is analysis, resting on limit notions and so called `real numbers'. A dramatical shift of the mathematical landscape is at our doorstep: a shift going back to Euler and Lagrange.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.
Lecture 72
Calculus with integral polynumbers
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Calculus with integral polynumbers
We introduce calculus in the context of integral polynumbers: first by reviewing Pascal's Array and binomial coefficients, and then explaining why the Taylor bipolynumber of a polynumbers can be obtained by suitably multiplying diagonals of this array by the coefficients. The derivative and the higher subderivatives can be then directly read off. In particular we recover the standard form for the derivative of a polynomial, and also a lesser known formula for the second subderivative. Then we find the important rules for finding the subderivatives of a sum and a product of polynumbers. This fundamental lecture (along with the ones immediately preceeding and following) should be studied carefully by anyone teaching (or learning) calculus. There are real alternatives to the tired status quo that pervades mathematics education and saddles students and teachers with waffly concepts and dubious logic!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.
Lecture 73
Tangent lines and conics of polynumbers
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Tangent lines and conics of polynumbers
This video introduces tangent lines and tangents conics of polynomials, using the very simple high school approach through polynumbers and bipolynumbers. We first define constant, linear, quadratic, cubic, quartic and quintic polynumbers in terms of the degree. Then we make some subtle shifts in the Taylor bipolynumber to find the Taylor expansion of a polynomial at a point c, and then using that to define the 1st tangent, 2nd tangent and so on. We will see how these are useful in computing approximations to the values of a polynomial.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.
Lecture 74
Graphing polynomials
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Graphing polynomials
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.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.
Lecture 75
Lines and Parabolas I
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Lines and Parabolas I
We study lines and parabolas using elementary calculus derived only from algebraic manipulations with polynomials, or polynumbers in our setting. We discuss y-intercepts, slopes and x-intercepts 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.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.
Lecture 76
Lines and Parabolas II
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Lines and Parabolas II
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.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.
Lecture 77
Cubics and the prettiest theorem in calculus
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Cubics and the prettiest theorem in calculus
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 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.
Lecture 78
An introduction to algebraic curves
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An introduction to algebraic curves
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.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.
Lecture 79
Object-oriented versus expression-oriented mathematics
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Object-oriented versus expression-oriented mathematics
Twentieth century mathematics has been object oriented. Twenty-first 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.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.
Lecture 80
Calculus on the unit circles
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Calculus on the unit circles
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 three-dimensional 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 summer---sorry about that. I will tell them to keep it down next time.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.
Lecture 81
Calculus on a cubic: the Folium of Descartes
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Calculus on a cubic: the Folium of Descartes
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.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.
Lecture 82
Inconvenient truths about Square Root of 2
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Inconvenient truths about Square Root of 2
This video begins a discussion on the role of irrationality in mathematics, starting with the "square root of 2". The difficulties with this concept go back to the ancient Greeks, as the Pythagoreans realized that the side and diagonal of a square were incommensurable. The Greeks realized that there was no rational number whose square was exactly two, a result which historically appeared in Euclid. In the modern age this idea that there were "irrational numbers" that could be incorporated into the Hindu-Arabic decimal number system was introduced by Stevin in 1585.There are these days three approaches to "sqrt(2)": an applied one dealing with approximations, an algebraic one involving a finite field extension of the rational numbers, and an analytic one which attempts to apply the square root algorithm to assign to sqrt(2) an infinite decimal. It is this last approach which does not work, leading to serious logical problems with modern analysis.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. Those who are intrigued but not fearful of the non-standard approach of this video might also like the four videos FMP19, FMP19b, FMP19c and especially FMP19d. Not for the faint hearted!
Lecture 83
Measurement, approximation and interval arithmetic I
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Measurement, approximation and interval arithmetic I
This video introduces interval arithmetic, first in the context of natural numbers, and then for integers. We start with some remarks from the previous video on the difficulties with irrational numbers, sqrt(2), pi and e. Then we give some general results about order (less than, greater than) for natural numbers and recall the constructions of integers and rational numbers from natural numbers. We introduce intervals for natural numbers and discuss various interpretations, and explain how to do arithmetic with them. The video ends with some interesting challenges to figure out how to extend this arithmetic to the case of integral intervals.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.
Lecture 84
Measurement, approximation and interval arithmetic II
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Measurement, approximation and interval arithmetic II
We continue on with a short intro to interval arithmetic, noting the difference between the laws of arithmetic over the natural numbers and the integers. The case of rational number intervals is also briefly discussed. We end the lecture with some remarks on the vagueness of ``real number'' arithmetic.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.
Lecture 85
Newton's method for finding zeroes
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Newton's method for finding zeroes
Newton, the towering scientific figure of the 17th century, discovered a lovely method for finding approximate solutions to equations, involving iterated constructions of tangent lines and their intersections. We describe this method in general and then apply it to the simplest and most familiar example; the standard quadratic polynomial x^2. To calculate tangent lines we use the algebraic calculus, and focus on approximating sqrt(2). This gives, remarkably, the same algorithm as the ancient Babylonians had for approximating sqrt(2), and so another alternative to the Vedic procedure.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.
Lecture 86
Newton's method for approximating cube roots
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Newton's method for approximating cube roots
We discuss cubes, cube roots and the impossibility of finding a cube root of 5 exactly. However Newton's method applies to allow us to find numbers whose cubes are approximately equal to 5. In this case there is also a higher variant which uses tangent conics instead of tangent lines.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.
Lecture 87
Solving quadratics and cubics approximately
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Solving quadratics and cubics approximately
We review the standard formulas for solving quadratic and cubic equations, the latter going back to work in the 1500's by del Ferro, Tartaglia and Cardano, and pointing out that in reality these formulas only generate approximate solutions, at least in general. We also connect both formulas with Newton's method to solve an example quadratic and an example cubic, at least approximately.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.
Lecture 88
Newton's method and algebraic curves
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Newton's method and algebraic curves
Newton's method can be extended to meets of algebraic curves. We show how, using the examples of the Fermat curve and the Lemniscate of Bernoulli. We start by finding the Taylor expansions of the associated polynomials (polynumbers) at a fixed point (r,s) in the plane.The first tangents are viewed as tangent planes to the associated surfaces in three dimensional space (Interpretation B in the language of MF78).This is a more advanced lecture, and is the final lecture in this first part of this course. After this, we will turn our attention to the logical weaknesses of modern pure mathematics--of which there are many!
Lecture 89
Logical weakness in modern pure mathematics
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Logical weakness in modern pure mathematics
We begin PART II of this video course: "Mathematics on trial - why modern pure mathematics doesn't work". This video outlines the case for the prosecution: that modern pure mathematics suffers from: 1. Inconsistent rigour 2. Problematic definitions 3. Reliance on `axioms' 4. Computational weakness 5. Impoverished examples We give some initial orientation to the first two claims, suggesting that contrary to popular opinion, rigour in mathematics has been on a consistent downward trend in the last few centuries, and give an explicit list of problematic definitions, cutting across many areas of modern pure mathematics. In subsequent videos in this series, we will be substantiating the claims made here. We will also be inviting comments and a wide discussion of these highly contentious, but vitally important, issues.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.
Lecture 90
The decline of rigour in modern mathematics
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The decline of rigour in modern mathematics
Rigour means logical validity or accuracy. In this lecture we look at this concept in some detail, describe the important role of Euclid's Elements, talk about proof, and examine a useful diagram suggesting the hierarchy of mathematics. We give some explanation for why rigour has declined during the 20th century (there are other reasons too, that we will discuss later in this course).Critical in this picture is the existence of key problematic topics at the high school / beginning undergrad level, which form a major obstacle to the logical consistent development of mathematics. We list some of these topics explicitly, and they will play a major role in subsequent videos in this series.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.
Lecture 91
Fractions and repeating decimals
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Fractions and repeating decimals
We introduce some basic orientation towards the difficulties with real numbers. In particular the differences between computable and uncomputable irrational numbers is significant. Then we discuss the relation between fractions and repeating decimals, giving the algorithms for converting back and forth, familiar from high school. The operations of addition and multiplication for repeating decimals are more subtle, and involve some lovely number theoretical aspects.The current theory of `real numbers' is logically deeply flawed. Essentially this theory is awol---everyone refers to it, but no one can tell us where it it is actually written down properly and completely. We are moving here towards the realization that mathematics is really about rational numbers, and theories that can be built from them in a finite and completely precise way. Hello future mathematics!
Lecture 92
Fractions and p-adic numbers
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Fractions and p-adic numbers
This video is an exploratory video in which we loosely introduce an interesting variant on repeating decimals: namely 10-adic numbers, or repeating reversimals (this is our own name). These are decimal-like objects whose digits carry on to the left, rather than the right, but still in a periodical fashion. Surprisingly, the arithmetic with these "repeating reversimals" parallels that of repeating decimals, but is generally simpler, due to the way that carrying works in the Hindu-Arabic framework (namely it proceeds to the left).We look at addition, multiplication, negatives and converting fractions to reversimals and back again. This is not at all a rigorous lecture, but just introduces a playful, yet curiously powerful, alternative to arithmetic with decimal numbers. It is a fun topic to explore if you have an interest in number theory or just patterns with numbers.
Lecture 93
Difficulties with real numbers as infinite decimals I
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Difficulties with real numbers as infinite decimals I
There are three quite different approaches to the idea of a real number as an infinite decimal. In this lecture we look carefully at the first and most popular idea: that an infinite decimal can be defined in terms of an infinite sequence of digits appearing to the right of a decimal point, each digit chosen arbitrary and independently. There are several very attractive reasons for taking this position, which we outline. Primary among these is that the definition allows us to turn a process into a number (ostensibly!). But ultimately the idea founders on the rocks of reality: the impossibility of specifiying competely a general such number, the impossibility of defining the addition (and multiplication) of such numbers via finite algorithms, and the resulting problematic aspects of the laws of arithmetic.We look also briefly at the role of the Axiom of Choice in trying to provide an axiomatic framework for real numbers as such `infinite choice decimals'. 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.
Lecture 94
Difficulties with real numbers as infinite decimals II
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Difficulties with real numbers as infinite decimals II
This lecture introduces some painful realities which cast a long shadow over the foundations of modern analysis. We study the problem of trying to define real numbers via infinite decimals from an algorithmic/constructive/computational point of view. There are many advantages of trying to do this: historically this was the point of view towards decimals like sqrt(2) or pi or e, and this provides us with tools to define and evaluate infinite series, functions and integrals. However in reality the idea bumps against seemingly unsurmountable technical obstacles: the difficulties in defining algorithms and implementing arithmetical operations at this level, non-uniqueness of algorithms and corresponding ambiguity in recognizing equality of real numbers, and a vagueness or tautological aspect to arithmetic with these objects. 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.
Lecture 95
The magic and mystery of π
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The magic and mystery of π
The number "pi" has been a fascinating object for thousands of years. Intimately connected with a circle, it is not an easy object to get hold of completely rigourously. In fact the two main theorems associated to it--the formulas for the area and circumference of a circle of radius pi--are usually simply assumed to be true, on the basis of some rather loose geometrical arguments in high school which are rarely carefully spelt out.Here we give an introduction to some historically important formulas for pi, going back to Archimedes, Tsu Chung-Chi, Madhava, Viete, Wallis, Newton, Euler, Gauss and Legendre, Ramanujan, the Chudnovsky brothers and S. Plouffe, and culminating in the modern record of ten trillion digits of Yee and Kondo. And I also throw in a formula of my own, obtained from applying Rational Trigonometry to Archimedes' inscribed regular polygons.It should be emphasized that the formulas here presented are not ones that can easily be rigorously justified, relying as they do on a prior theory of real numbers and often Euclidean geometry. The lecture ends with some speculations about the future role that "pi" might play in our understanding of the continuum--a huge problem which is not properly appreciated today.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.
Lecture 96
Problems with limits and Cauchy sequences
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Problems with limits and Cauchy sequences
One of the standard ways of trying to establish `real numbers' is as Cauchy sequences of rational numbers, or rather as equivalence classes of such. In the next few videos we will be discussing why this attempt does NOT in fact work!In this lecture we provide an introduction to these ideas in an informal and descriptive way. In particular we visualize sequences of points in the plane, and discuss two different notions of when two sequences converge; including Cauchy convergence.We also outline the challenges that lie in wait for anyone who tries to set up arithmetic with `real numbers' in this way. It is not a surprise that this is nowhere properly done, although many students are under the mistaken impression that they have `covered' this material at some point in their analysis courses.
Lecture 97
The deep structure of the rational numbers
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The deep structure of the rational numbers
The rational numbers deserve a lot of attention, as they are the heart of mathematics. I am hopeful that modern mathematics will (slowly) swing around to the crucial realization that a lot of things which are currently framed in terms of "real numbers" are more properly understood in terms of the rationals-- in which case richer number theoretical/combinatorial aspects start to become more visible. After a review of basic definitions including the idea of average and mediant of two rational numbers, we focus on the interval [0,1], and discuss how convex combinations allow us to match up any two intervals. We introduce the idea of the level of a rational number, and the famous Farey sequences. These are connected with the notion of Ford circles which we talked about in MF14. A key principle is that even in [0,1], the uniformity of the rational numbers is an illusion; rather they are a layered strata which we can delve into deeper and deeper, yielding more and more complicated numbers.The layered structure of the rationals will play an important role when we start to discuss sequences of rational numbers in a few more videos.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.
Lecture 98
Fractions and the Stern-Brocot tree
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Fractions and the Stern-Brocot tree
Here we introduce the Stern-Brocot tree, a remarkable representation of fractions by means of a binary tree, discovered around 1860 by a German mathematician and French clockmaker. The mediant operation is used to generate the tree. We describe a number of pleasant properties of the tree, including one discovered by a Canadian musical theorist, Pierre Lamothe.We also connect the Stern-Brocot tree with the Ford circles discussed in the last video as well as MF14, and the Farey sequences. This is a good example of the deep structure of the rational numbers!
Lecture 99
The Stern-Brocot tree, matrices and wedges
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The Stern-Brocot tree, matrices and wedges
We discuss further manifestations of the Stern-Brocot tree, which we discussed in the previous lecture. Here we introduce the matrix form of the tree, which sheds new light on how to move smoothly down the tree. To do this we introduce some basic definitions on 2x2 matrices and their determinants, including the operations of addition and multiplication, and the three special matrices I,L and K.Then we exhibit a somewhat novel planar, geometrical representation of the Stern-Brocot tree, identifying fractions with visible integral points in the positive quadrant. This gives an explanation of the curious role of the pseudo-fraction 1/0 which we used in the last lecture to generate the tree. Finally we introduce a new geometrical structure that results from combining the matrix and geometrical forms of the Stern-Brocot tree; the theory of wedges. A wedge is a particular triangle associated to a fraction, or to a matrix in the matrix form of the S-B tree, and these provide a curious tesselation of the visible positive quadrant, rather challenging to visualize!
Lecture 100
What exactly is a sequence?
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What exactly is a sequence?
The term `sequence' is so familiar from daily life that it is easy to dismiss the need for a precise mathematical definition. In this lecture we start by looking at finite sequences, of a particularly pleasant kind, namely sequences of natural numbers. The distinction between the specification of such a sequence and a description of it is emphasized. We must also maintain some respect for sequences that get large---ultimately they can have qualitatively quite different properties. We also mention N. Sloane's On-line Encyclopedia of Integer Sequences, a valuable resource for mathematically minded people.
Lecture 101

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"Infinite sequences": what are they?
This lecture tries to clarify the big gap between the (finite) sequences we introduced in the last lecture, and "infinite" or "ongoing sequences" (we introduce the term "on-sequence") as are found in Sloane's Online Encyclopedia of Integer Sequences. We concentrate discussion on three such: the basic on-sequence of natural numbers, the on-sequence of prime numbers, and the on-sequence of Catalan numbers. We lay out Euclid's well-known proof that given any finite list of primes we can find a prime not in it, and suggest some computational snags that perhaps we might worry about. The Catalan sequence of combinatorics fame has many possible definitions and formulas, illustrating the crucial distinction: finite sequences are specified by the elements of the sequence themselves, while "on-sequences'' can only be specified more indirectly, and generally non-uniquely.Along the way we point out that the grammatical closeness of the terms finite and infinite suggest a dichotomy which is not really born out in experience.
Lecture 102
Slouching towards infinity: building up on-sequences
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Slouching towards infinity: building up on-sequences
While finite sequences are specified by listing all elements, this approach does not work for "infinite sequences" which carry on indefinitely. We prefer the more modest term of 'on-sequence', and stress the importance of building up such a theory one step at a time. This lecture begins such a theory by first defining constant on-sequences, which turn out to be finite expressions such as [3) (actually a pointy right bracket) with the possibility of an index to define the m-th element, all of which are 3 in this case. The arithmetic of such constant on-sequences parallels, in an obvious way, the arithmetic of natural numbers.The next step is to define polynumber (or polynomial) on-sequences. We will use the letter n here to characterize such objects, for example the on-sequence [n), enclosed in a square and a pointed bracket, is the basic natural number sequence which we usually would write 1,2,3,.... Again the arithmetic of such polynumber on-sequences parallels that of polynumbers.We give some examples of particular polynumber on-sequences that arise in Sloane's Online Encyclopedia of Integer Sequences (OEIS). In particular we have a look at the frog-and-toad-hopping-interchange sequence :). Is there a convenient way to access all of the polynumber on-sequences in OEIS??We also take this opportunity to dedicate this series to the Australian taxpayers!
Lecture 103
Challenges with higher on-sequences
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Challenges with higher on-sequences
In our last video we introduced polynumber (or polynomial) on-sequences. Today we consider how we might go beyond this, to introduce a wider range of sequences, to deal with more of the examples in OEIS, for example. The first attempt will involve 'exponomials": arithmetical expressions extending polynomials by allowing also exponentials, such as n^n for example. While tempting, this extension is rather problematic logically, and I explain why.Another important kind of object in this area are "recursive sequences". There are many familiar examples of such; primarily the Fibonacci numbers 1,1,2,3,5,8,13,21,... or the closely related Lucase numbers 1,2,4,7,11,18,29,...In OEIS we find many more recursively defined sequences, such as the Catalan numbers, or the Euler numbers (also called Slyvester's sequence). But what exactly is the definition of a ``recursive sequence"?? This is an embarrassing question for many mathematicians, who naively regard the term as almost self-evident, or at least only requiring a few examples and a bit of hand-waving. No, no, no! The idea is far from clear.Conclusion: without a lot of work, neither exponomials or general recursive on-sequences are available to us at this point. More work needs to be done first! In our next video we will move to a much more solid extension of polynumber on-sequences.
Lecture 104
Limits and rational poly on-sequences
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Limits and rational poly on-sequences
We introduce more general ``infinite sequences'', or on-sequences, generated by rational polynumbers, otherwise often known as rational functions: ratios of one polynomial over another. The association of a sequence to such an expression is surprisingly delicate, and requires us to look at factorizations. In particular we are naturally led to introduce a new symbol for ``infinity", corresponding to a ratio of integers with zero in the denominator, and associated with a limiting behaviour getting larger and larger as we approach a particular value.
Lecture 105
MF103: Extending arithmetic to infinity!
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MF103: Extending arithmetic to infinity!
We are interested in investigating how to rigorously and carefully extend arithmetic with rational numbers to a wider domain involving the symbol 1/0, represented by a ``sideways 8''. First we have a look at the simpler case of natural number arithmetic, where extending to infinity is relatively simple. For the rational number case, which is quite separate from the natural number case, we look to connect arithmetic with the geometry of the plane, and the projective geometry of lines through the origin. Our strategy is to focus on integral points [a,b] in the plane, and define suitable notions of rational addition and rational multiplication of these. By associating to the integral point [a,b], where b is not zero, the rational number a/b, we can make rational number arithmetic follow from arithmetic with integral points. In this way, the point [1,0] plays a role analogous to infinity. This connects also with earlier videos on the Stern-Brocot tree.
Lecture 106
Rational number arithmetic with infinity and more
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Rational number arithmetic with infinity and more
We systematically introduce the four arithmetical operations on integral points of the plane, state some of the main arithmetical laws satisfied, and then show how to obtain the extended rational numbers by suitably identitying integer points, in a similar way to the introduction of rational numbers. Notably this allows a uniform and completely unambiguous introduction of 1/0 into arithmetic.Everything is motivated by projective geometry--the idea that a line through the origin in two dimensional space can be specified by homogenous coordinates, and more or less gives us an extended rational number. However we must pay a price: the admission of the strange new object 0/0, which is separate from all the other extended rationals, and plays a curious but central role. I also suggest a contest: what is a good name for this object 0/0??Dear Viewers-- After pondering our notational challenge, I am leaning towards one of the first suggestions, made by teavea10: that 0/0 by called ``zoz'', short for ``zero over zero''. Can anyone think of any compelling reason to not adopt this somewhat novel but appealing nomenclature??
Lecture 107
The extended rational numbers in practice
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The extended rational numbers in practice
We review the extended rational numbers, which extend the rational numbers to all expressions of the form a/b, where a and b are integers---even b=0. Then we give some examples of how these strange beasts might prove useful in mathematics. But first we give one example of where they are unlikely to be useful---in economics, where there is a big difference between a very big positive number and a very big negative number!When we graph rational polynumbers, we see situations where the graph seems to want to ``go up to infinity'' and then immediately ``come up from infinity''. This suggests to us that the mathematics wants us to connect these seemingly divergent arms of a graph, by introducing a point at infinity. We give some initial suggestions that this might allow us to compactly visualize the entire graph of a rational polynumber on a finite square with opposite sides identified. Topologically this is a torus, so perhaps studying infinity will naturally lead to calculus being more naturally visualized, at least for some problems, on a donut!The extended rational numbers also naturally arise in projective geometry, particularly the one dimensional projective line, which may be viewed as an affine line with one point (infinity) attached at both ends, or, as Mobius and Plucker realized, as the space of lines through the origin in a two dimensional affine space. This naturally connects with our earlier picture of extended rationals as arising from grouping integer points together into teams, along lines.Finally we show that the extended rationals can be used to parametrize a circle, or more general conic, as long as it passes through the origin. In fact we have seen this parametrization before (with axes interchanged!) but now infinity can be also used to close up and unify the parametrization. I end with two very interesting challenges: how to geometrically describe addition and multiplication on points on the circle.
Lecture 108
What exactly is a limit?
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What exactly is a limit?
In this video we aim to give a precise and simpler definition for what it means to say that: a rational polynumber on-sequence p(n) has a limit A, for some rational number A. Our definition is both much simpler and more logical than the usual epsilon -delta definition found in calculus texts. What is required is that we need to find two natural numbers: k called the scale, and m called the start that allow us to bound in a pretty simple way the difference between p(n) and A. The epsilon-delta definition of a limit is usually considered a high point of logical rigour. Not so. It is also considered too logically involving to be taken seriously as a pedagogical pillar for most undergrads. Hence students may be told about the definition, but are not required to seriously understand it, or be able to use it--unless they are prospective maths majors. There is a subtle ambiguity in the definition: given an epsilon we are supposed to demonstrate there is a delta (with certain properties) but how are we to do this, since an potential infinity of epsilons are involved? In practice what is required is a correspondence (function/relation etc) between epsilon and delta but the nature of this required correspondence is not clear. We return to our familiar conundrum of using the work``function'' without a proper definition of it.The key point that makes our simpler more intuitive notion of limit of a sequence work is that we are dealing with very particular and clearly defined on-sequences: those generated by a rationl polynumber. A good example of the benefits of being careful rather than casual when dealing with the foundations of analysis!
Lecture 109
Inequalities and more limits
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Inequalities and more limits
The epsilon-delta definition of a limit of a sequence, going back to Cauchy and Weierstrass, is here dramatically simplified by restricting attention to the basic objects of calculus: rational polynumbers (or ``rational functions''). We review the basic definition and give a visual interpretation: instead of an infinite number of nested epsilon neighborhoods, we have a single hyperbolic envelope of the limit. It means that a limit can be certified by exhibiting just two natural numbers: the start m and the scale k.We illustrate the concept by going carefully over the 3 Exercises from the last video.Then we begin a quick review/exposition of inequalities, which are important tools when dealing with limits (and analysis more generally). Starting with inequalities for natural numbers, we progressively move to inequalities for integers, and then rational numbers. Notions of positivity are of course crucial here.
Lecture 110
Limits to Infinity
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Limits to Infinity
We carry on with our study of the definition of a limit, concentrating on particularly pleasant and amenable kinds of sequences, associated to rational polynumbers (or rational functions) and now going to infinity. Again we use a simpler and more elegant variant on the classical definition which is well suited to this situation, and makes verifying limits into finite tasks!
Lecture 111
Logical difficulties with the modern theory of limits I
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Logical difficulties with the modern theory of limits I
This is the first of two videos that will look at the official formal definition of a limit of a sequence, as initiated by Bolzano, Cauchy and Weierstrass. Although commonly regarded as a pillar of modern analysis, in fact this definition has serious logical problems. We state what these problems are, and then start to try to explain them. This lecture will involve some more complicated notions, often only introduced in introductory analysis courses at the 2nd or 3rd year undergraduate level. However by looking carefully at some examples, I hope to show you what the definition is trying to say, and then ultimately why it doesn't work.
Lecture 112
Logical difficulties with the modern theory of limits II
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Logical difficulties with the modern theory of limits II
This is the second of two videos that look at the official formal definition of a limit of a sequence, as initiated by Bolzano, Cauchy and Weierstrass. Although commonly regarded as a pillar of modern analysis, in fact this definition has serious logical problems. We state what these problems are, and then start to try to explain them. In this lecture, we go beyond the baby examples usually found in calculus and analysis texts, revealing more of the typical generality and inaccessibility of a lot of the discussion of general limits. In particular we examine an extended harmonic sequence formed by playing the Collatz game (also called the 3n+1 problem, and described at some length in my video Famous Math Problems 2: The Collatz Conjecture FMP2). We will see the essential problem is that the definition actually requires us to make an infinite number of computations to assert that a general `sequence' has a limit. Is it possible to make an infinite number of computations? No, it is not. Are we allowed to pretend that we can make an infinite number of computations? Well I suppose so, but let's not call it mathematics.
Lecture 113
Real numbers and Cauchy sequences of rationals I
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Real numbers and Cauchy sequences of rationals I
We introduce the idea of a `Cauchy sequence of rational numbers'. The notion is in fact logically problematic. It involves epsilons and N's, much as does the notion of a limit, and suffers from similiar issues: how to guarantee that we can find an infinite number of N's for an infinite number of epsilons (making the very generous assumption that the term `sequence' does not really have to be defined properly!)Built on top of this idea is the most familiar story for the `construction' of real numbers: to imagine that the limit of a Cauchy sequence of rationals can be defined to be essentially the sequence itself! The `essentially' refers to the fact that different Cauchy sequences can head in the same direction: so it means that we must introduce a complicated notion of equivalence into the story (more infinite numbers of checks, repeated an uncountably infinite number of times!). This crude and dubious attempt at pulling oneself up by one's bootstraps renders most computations with real numbers essentially vacuous. Of course this is not what current pure mathematicians want to hear! We want to believe in real numbers as Cauchy sequences, for reasons that will become clearer in the next video.
Lecture 114
Real numbers and Cauchy sequences of rationals II
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Real numbers and Cauchy sequences of rationals II
There is a good reason why pure mathematicians cling so tenaciously to the idea of real numbers. They provide us with the ostensible `values' that lengths, areas, values of functions and solutions to equations seem to require. But is this all really just a dream??In this video we have a new look at these notions, with a view of examining whether they really do support `real values', or whether perhaps they are intrinsically approximate notions. To motivate the discussion, we go back to a crucial calculation of Archimedes.
Lecture 115
Real numbers and Cauchy sequences of rationals III
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Real numbers and Cauchy sequences of rationals III
Motivated by Archimedes calculation of an approximate ratio of circumference to diameter of a circle, we introduce an Archimedean view on `real numbers": nested sequences of intervals whose sizes go to zero. If you are going to waffle about the existence of real numbers, this is at least an excellent way to do it! That's because it actually corresponds to what we want to do in many applied situations: get increasingly tighter bounds on an intrinsically approximate quantity. With this introduction, we then show how Cauchy sequences are an attempt to minimize the data involved in the representation of a real number, but this is a source of one of the serious problems with the idea: a lack of any kind of control in general about how close to the `limit' we are as we go down a Cauchy sequence.
Lecture 116
Real numbers as Cauchy sequences don't work!
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Real numbers as Cauchy sequences don't work!
This longish video lays out the various reasons why Cauchy sequences---as a basis for the theory of real numbers---don't work. Necessary viewing for all maths students!
Lecture 117
The mostly absent theory of real numbers
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The mostly absent theory of real numbers
In this video we ask the question: how do standard treatments of calculus and analysis deal with the vexatious issue of defining real numbers and their supposed arithmetic??We pull out a selection of popular Calculus and Analysis texts, and go through them with a view of finding out: what exactly is a real number? All the books I examine are excellent books---aside from their treatment of foundational issues, where we see that they mostly fall clearly short.We look at Calculus texts by Steward, Sallas Hille and Etgen, Rogawski, Courant, Spivak, Caunt, Apostol, Keisler and Adams, and Analysis texts by Spiegel, Apostol, Royden, Kolmogorov and Fomin, and Rudin.This video really should be an eye opener to students of mathematics. Yes, it is possible to challenge the standard thinking, and the mathematical world need not collapse. Admitting current weaknesses, and the lack of acknowledgement of them by the Academy, is an important step in moving forward to a newer, better mathematics.
Lecture 118
Difficulties with Dedekind cuts
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Difficulties with Dedekind cuts
Richard Dedekind around 1870 introduced a new way of thinking about what a real number `was'. By analyzing the case of sqrt(2), he concluded that we could associated to a real number a partition of the rational numbers into two subsets A and B, where all the elements of A were less than all the elements of B, and where A had no greatest element. Such partitions are now called Dedekind cuts, and purport to give a logical and substantial foundation for the theory of real numbers.Does this actually work? Can we really create an arithmetic of real numbers this way? No and no. It does not really work. In this video we raise the difficult issues that believers like to avoid.
Lecture 119
The continuum, Zeno's paradox and the price we pay for coordinates
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The continuum, Zeno's paradox and the price we pay for coordinates
In this video we venture into a range of topics, from the nature of the continuum, to the paradoxes of Zeno, to an understanding of some of the consequences for mathematics in the shift from geometry to arithmetic that flowed from the Cartesian revolution. As we let go of the real numbers, we must prepare ourselves to appreciate that the arithmetical view of geometry does not always exactly fit with our physical intuition. Our biology constrains us in important ways, and so guides our thinking down certain paths, for better or worse.If we demand that mathematics adheres to our biological orientation and physical intuitions, then we can be led astray. Such is the reason why modern pure mathematics has lost its way logically, with its unreasonable insistence that there is a theory of real numbers.
Lecture 120
Real fish, real numbers, real jobs
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Real fish, real numbers, real jobs
In 1999, I wrote a paper called `Real fish, real numbers, real jobs', which appeared in the Mathematical Intelligencer. In this video, I read this paper in full. It was meant as a somewhat humorous take on an important issue. The reception of this paper is also briefly discussed; I feel this raises some interesting additional issues about the attitude that modern mathematics has towards any kind of self-criticism.I hope you enjoy this somewhat different video!
Lecture 121
Mathematics without real numbers
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Mathematics without real numbers
The mathematics of the coming century is going to look dramatically different. Real numbers will go the way of toaster fish; claims of infinite operations and limits will be recognized as the balder dash they often are; and finite, concrete, write-downable mathematics will enter centre stage. In this overview video, we look at some of the directions that mathematics will take, once the real number dream is abandoned. You can think of this as a bird's eye view of a lot of the rest of this video series. It is an exciting time. Mathematics has not had a proper revolution; surely it is long overdue!
Lecture 122
Axiomatics and the least upper bound property I
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Axiomatics and the least upper bound property I
The role of axiomatics in mathematics is a highly contentious one. Originally the term always referred to Euclid, and his use of the term to mean `a self-evident truth that requires no proof '. However in modern times the meaning of the term has shifted dramatically, to the idea that an Axiom is `a convenient fact that we assume'. This casts considerable doubt on the validity of the usual claim that `Mathematics is built on Axioms", which these days appears more and more as a religious position rather than a scientific one. Is that what we want our subject to be?? Your belief system as opposed to my belief system??In this video we discuss this shift in meaning and its consequences when trying to set up a theory of real numbers. We will be discussing this important issue further when we get around to critiquing `modern set theory'.
Lecture 123
Axiomatics and the least upper bound property II
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Axiomatics and the least upper bound property II
Here we continue explaining why the current use of `axiomatics' to try to formulate a theory of `real numbers' is fundamentally flawed. We also clarify the layered structure of the rational numbers: we have seen these several times already in prior discussion of the Stern- Brocot tree, here we view rationals in [0,1] in terms of increasing denominators. This allows us to explain why, if you believe the usual sad story of being able to `do an infinite number of operations', we can `create' nested interval sequences (or equivalently Cauchy sequences) with no rational limit.This paves the way for introducing the much vaunted ``least upper bound property'' of `real numbers', which is a mainstay of classical analysis, and props up the modern theories of areas, integrals, infinite sums, transcendental functions, infinite products, lengths of curves and much more. The reality is that much of modern mathematics, sadly, is an elaborate dream system. Here we are slowly, slowly....waking up. Feel free to join us, it's actually quite invigorating!
Lecture 124
Mathematical space and a basic duality in geometry
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Mathematical space and a basic duality in geometry
Happy New Year everyone, and I wish you all the best for 2015!In this video we introduce some basic orientation to the problem of how we represent, and think about, space in mathematics. One key idea is the fundamental duality between the affine and projective views: two sides of the same coin. We explain how the Cartesian revolution of the 17th century built geometry from a prior theory of arithmetic: for us that of the rational numbers- of course!And we introduce some useful notations for points and proportions, and give a geometrical view of the relation between the affine line and the projective line.
Lecture 125
Affine one-dimensional geometry and the Triple Quad Formula
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Affine one-dimensional geometry and the Triple Quad Formula
In this video we introduce the second most important theorem in all of mathematics (excluding the laws of arithmetic)!It is certainly remarkable that the recognition of the importance of this result (called the Triple quad formula) is only now taking place at the beginning of the 21st century. How does this formula arise? By asking the deceptively simple questions: how do we measure the separation between two points, and then what is the relation between three collinear points?In this video, at a certain point I invite you to discover this crucial formula for yourself. It is highly recommended, especially if you are a serious student of mathematics, that you spend an hour or so to deduce the required result before I tell you what it is. This occurs around 14:30 of the video.
Lecture 126
Heron's formula, Archimedes' function, and the TQF
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Heron's formula, Archimedes' function, and the TQF
Remarkably, the Triple Quad Formula (TQF), even though it is purely a one-dimensional result, contains in it the seeds of higher dimensional results. In this video we look at a classical result from ancient Greece, called Heron's formula for the area of a triangle in terms of the side lengths, and reformulate it purely in terms of rational quantities: the quadrances of the sides, not the lengths!We call this reformulation Archimedes' theorem, since it is known from Arab sources that Archimedes had Heron's formula several hundred years before Heron did.
Lecture 127
Brahmagupta's formula and the Quadruple Quad Formula I
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Brahmagupta's formula and the Quadruple Quad Formula I
In this video we introduce Brahmagupta's celebrated formula for the area of a cyclic quadrilateral in terms of the four sides. This is an obvious extension of Heron's formula. We are interested in finding a rational variant of it, that will be independent of a prior theory of `real numbers', `square roots' and `lengths'. For motivation we look at the situation of four affine 1-points. Is there an analog of the Triple Quad Formula? Yes there is: it is the more mysterious and complicated Quadruple Quad Formula. To get it, we will examine some important manipulations for a pair of quadratic equations which are of independent interest.This lecture has some more serious algebra in it: a great place to practice your manipulation and organizational skills. If you are teaching college mathematics, please consider doing yourself and your students a favour: teach them some of the material of this lecture carefully and explicitly!