Famous Math problems : videos 1 19
The Algebraic Topology series
This is first of a series that will discuss foundations of mathematics. Contains a general introduction to the series, and then the beginnings of arithmetic with natural numbers. This series will methodically develop a lot of basic mathematics, starting with arithmetic, then geometry, then algebra, then analysis (calculus) and will also treat so called set theory.
It will have a lot of critical things to say once we get around to facing squarely up to the many logical weaknesses of modern pure mathematics. The series is meant to be viewed sequentially. We spend a lot more time and effort than usual on fundamental issues with number systems. If you are a more advanced student, or a fellow mathematician, then the first few dozen videos might be a bit slow. But they are nonetheless important! 

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

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

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


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

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


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

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


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

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


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

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


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


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


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


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


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


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

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


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


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


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


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

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


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

