A cornucopia of mathematics

Mathematics is one of the most important subjects we ever learn in school. It underpins everything in our everyday life, as well as the whole of science and engineering. We would have no technology without it.

So it's a pity many of us struggle to grasp it, and miss the world of wonder it contains. Banished from a proper understanding, we can only participate blindly in decisions made by those supposedly more knowledgeable.

"Mathucopia", by Bill Spears - demystifies mathematics, showing how it links to our everyday experience, and is controlled with some rather straightforward rules we can quickly grasp. That way you will not be baffled by supposed experts. Armed with its knowhow, you'll be able to approach the world around you in the confidence that you can understand it to the extent you want to. The many areas touched in passing led to the title, and hopefully you too will discover a cornucopia of rich mathematical concepts.

This book is not for the purist. It keeps explanations as short and as relevant as possible. It does not delve into all the interesting corners of each topic, nor does it prove or provide all the formulae you might eventually need. It gives as complete an overview as possible of all the important everyday mathematics, without asking the reader to take too much for granted, providing a springboard for the interested reader to go further with other texts and online. Ideally, it is meant to be read in order, so that earlier sections introduce concepts which underpin later ones. But it can be skipped a bit if you get bogged down, especially later on.

I apologise in advance to those who will find it in some parts trivial, and those who will stumble along the way over some unclear explanation I have failed at. It is aimed at all ages, but I believe it is going to be especially useful for the young in mind. I have not shied away from introducing you to the proper notation and shorthand, and I make no apology for that. While it can look like hieroglyphics at first, I give you the tools in due time to decipher it. And that capability brings you increasingly into the realm of wider understanding, and appreciation that here is something beautiful. If a picture is worth a thousand words, then an equation is certainly worth the several hundred used to avoid it.

This book is available as an iBook, for you to keep as a mobile and ever-present reference, for approximately €10, but if you would like the printed paperback copy, more useful and restful on the eye for intensive study, please fill in this form and send it to me at the e-mail address indicated. With proof of purchase in the iBook store, the paperback copy costs an additional €7, or €17 without. When I receive your payment, the book will be on its way to you (normally by book post, so please allow sufficient time).

I look forward to hearing from you, and I do hope you enjoy the book.

Bill Spears
April 2019


Table of Contents

  1. Numbers and sets
    1. Natural numbers
    2. The concept of zero
    3. Bases
    4. First operations on numbers
    5. Prime numbers
    6. Introducing powers
    7. Subtraction, introducing integers
    8. Rational numbers
    9. Operations on rationals
    10. Decimal notation and long division
    11. Percent and permille
    12. Parentheses and collective laws
    13. Working with equations
    14. Introducing algebra
    15. Binomial theorem and Pascal's triangle
    16. Logarithms
    17. Introducing e
    18. Square roots
    19. Real numbers
    20. Countability
    21. Complex numbers
  2. Geometry
    1. Planar Euclidian geometry
    2. Basic triangle theorems, and parallel lines
    3. Useful circle-related theorems
    4. Pythagoras' theorem and coordinate systems
    5. Sine, cosine and tangents and their relatives
    6. Introducing π, and radians
    7. Vectors
    8. Projections and Polyhedra
  3. Introduction to series and limits
    1. Finite series - geometrical and arithmetic
    2. Infinite series - convergence and limits
    3. Infinite sequences - e again
  4. Functions and Equations
    1. Straight line
    2. Linear Algebra - the Gaussian method - and matrices
    3. Quadratic functions and equations
    4. Conic sections
    5. Polynomials
    6. Non-polynomial functions and functional classification
  5. Calculus
    1. Differentiation
    2. Differentiating sums, products and quotients
    3. Applications of differentiation
    4. Integration
    5. Applications of Integration
    6. Integrating sums, products and quotients
    7. Differential equations in one variable
  6. Series of infinite functions
    1. Taylor series
    2. ex and remainders
    3. Series for the other trig functions
    4. Recursive sequences
  7. Partial differential equations
    1. The wave equation
    2. Fourier analysis
  8. Complex numbers revisited
    1. Polar and exponential representation, and roots
    2. Mandelbrot and Julia sets
  9. Probability and Statistics
    1. Permutations and Combinations
    2. Probability Distributions
    3. Testing significance
    4. Poisson distribution
    5. Correlation
    6. Conditional probability
  10. Numerical methods
    1. Roots of equations
    2. Least squares fit
    3. Euler's method
    4. Simpson's rule


Mathucopia © Bill Spears 2018