### Lecture Description

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.

### Course Index

- What is a number?
- Arithmetic with numbers
- Laws of Arithmetic
- Subtraction and Division
- Arithmetic and Math education
- The Hindu-Arabic number system
- Arithmetic with Hindu-Arabic numbers
- Division
- Fractions
- Arithmetic with fractions
- Laws of arithmetic for fractions
- Introducing the integers
- Rational numbers
- Rational numbers and Ford Circles
- Primary school maths education
- Why infinite sets don't exist
- Extremely big numbers
- Geometry
- Euclid's Elements
- Euclid and proportions
- Euclid's Books VI--XIII
- Difficulties with Euclid
- The Basic Framework for Geometry I
- The Basic Framework for Geometry II
- The Basic Framework for Geometry III
- The Basic Framework for Geometry IV
- Trigonometry with rational numbers
- What exactly is a circle?
- Parametrizing circles
- What exactly is a vector?
- Parallelograms and affine combinations
- Geometry in primary school
- What exactly is an area?
- Areas of polygons
- Translations, rotations and reflections I
- Translations, rotations and reflections II
- Translations, rotations and reflections III
- Why angles don't really work I
- Why angles don't really work II
- Correctness in geometrical problem solving
- Why angles don't really work III
- Deflating Modern Mathematics: the problem with 'functions' - Part 1
- Deflating Modern Mathematics: the problem with 'functions' - Part 2
- Reconsidering `functions' in modern mathematics
- Definitions, specification and interpretation
- Quadrilaterals, quadrangles and n-gons
- Introduction to Algebra
- Baby Algebra
- Solving a quadratic equation
- Solving a quadratic equation
- How to find a square root
- Algebra and number patterns
- More patterns with algebra
- Leonhard Euler and Pentagonal numbers
- Algebraic identities
- The Binomial theorem
- Binomial coefficients and related functions
- The Trinomial theorem
- Polynomials and polynumbers
- Arithmetic with positive polynumbers
- More arithmetic with polynumbers
- What exactly is a polynomial?
- Factoring polynomials and polynumbers
- Arithmetic with integral polynumbers
- The Factor theorem and polynumber evaluation
- The Division algorithm for polynumbers
- Row and column polynumbers
- Decimal numbers
- Visualizing decimal numbers and their arithmetic
- Laurent polynumbers (the New Years Day lecture)
- Translating polynumbers and the Derivative
- Calculus with integral polynumbers
- Tangent lines and conics of polynumbers
- Graphing polynomials
- Lines and Parabolas I
- Lines and Parabolas II
- Cubics and the prettiest theorem in calculus
- An introduction to algebraic curves
- Object-oriented versus expression-oriented mathematics
- Calculus on the unit circles
- Calculus on a cubic: the Folium of Descartes
- Inconvenient truths about Square Root of 2
- Measurement, approximation and interval arithmetic I
- Measurement, approximation and interval arithmetic II
- Newton's method for finding zeroes
- Newton's method for approximating cube roots
- Solving quadratics and cubics approximately
- Newton's method and algebraic curves
- Logical weakness in modern pure mathematics
- The decline of rigour in modern mathematics
- Fractions and repeating decimals
- Fractions and p-adic numbers
- Difficulties with real numbers as infinite decimals I
- Difficulties with real numbers as infinite decimals II
- The magic and mystery of π
- Problems with limits and Cauchy sequences
- The deep structure of the rational numbers
- Fractions and the Stern-Brocot tree
- The Stern-Brocot tree, matrices and wedges
- What exactly is a sequence?
- "Infinite sequences": what are they?
- Slouching towards infinity: building up on-sequences
- Challenges with higher on-sequences
- Limits and rational poly on-sequences
- MF103: Extending arithmetic to infinity!
- Rational number arithmetic with infinity and more
- The extended rational numbers in practice
- What exactly is a limit?
- Inequalities and more limits
- Limits to Infinity
- Logical difficulties with the modern theory of limits I
- Logical difficulties with the modern theory of limits II
- Real numbers and Cauchy sequences of rationals I
- Real numbers and Cauchy sequences of rationals II
- Real numbers and Cauchy sequences of rationals III
- Real numbers as Cauchy sequences don't work!
- The mostly absent theory of real numbers
- Difficulties with Dedekind cuts
- The continuum, Zeno's paradox and the price we pay for coordinates
- Real fish, real numbers, real jobs
- Mathematics without real numbers
- Axiomatics and the least upper bound property I
- Axiomatics and the least upper bound property II
- Mathematical space and a basic duality in geometry
- Affine one-dimensional geometry and the Triple Quad Formula
- Heron's formula, Archimedes' function, and the TQF
- Brahmagupta's formula and the Quadruple Quad Formula I

### Course Description

Does mathematics make logical sense? No, it does not. Foundational issues have been finessed by modern mathematicians, and this series aims to turn things around. And it will have interesting things to say also about mathematics education---especially at the primary and high school level. The plan is to start right from the beginning, and to define all the really important concepts of basic mathematics without any waffling or appeals to authority. Roughly we discuss first arithmetic, then geometry, then algebra, then analysis, then set theory. This course is aimed for a general audience, interested in mathematics, or willing to learn.