Preparatory Review of Number Theory, Planar Geometry, and Basic Algebra

Virtually all of mathematical theory is built on the foundations of the theory of numbers. This may be difficult to grasp for many of us because we sometimes view arithmetic as being an inconvenience to learn. Besides, now that most of us have calculating machines of one kind or another to do that for us, is it really necessary to understand it at all? In fact, it is precisely the understanding of the various number systems (and there are lots of them) and knowing their operations, rules, and properties which deepens our comprehension of algebraic symbolism and why its adoption became an absolute necessity before we collectively could proceed to understand the laws of our universe. It was only when visual planar geometry became numerically embedded into René Déscartes’ (1596–1650) coordinatized plane (by the way, that took almost 2,000 years to accomplish) that the three major mathematical disciplines—Number Theory, Euclidean Geometry, and Basic Algebra—merged into a very powerful mathematical theory which propelled scientific discovery to the complexity and relative accuracy it has today. The spinoffs of these insights have resulted in the inventions of a myriad of machines, architectural designs, medical diagnostic tools, and other useful devices which have made our modern world what it is today.

This is why we should begin all foundational training in mathematical thought with these three subject disciplines—Number Theory, Euclidean Planar Geometry, and Introductory Algebra. It is important to know the past in order to understand better the present and how we got here—not to mention having some insights into ways of proceeding in the future.

The following list of tutorials from the AU Math Centre is intended to provide you with a top-down summary of important terminology, methods and concepts which are foundational to the mathematics covered in this course. Treat them as a quick reference to these skills just in case you have forgotten some details or have never covered these ideas before.

Online Tutorials—the Overview

Pertinent to “Preparatory Review”

AU Math Centre: welcome and instructions on how to use the site to your best advantage

Basic Number Theory

Numbers and Arithmetic: an introduction
Arithmetic of the whole numbers
Arithmetic of fractions—multiplication and division
Arithmetic of fractions—addition and subtraction
The order of arithmetic operations—just so that we are all on the same page
The Real Numbers—and a gateway to higher-level algebra
Napier’s Bones—using chess moves to multiply large numbers
Exponentials: An introduction
Exponentials with natural number exponents
Numerical Equality, Equivalence, and Order: An introduction
Equality of numerical expressions
Equivalence of numerical equations
Numerical Inequality: leading to an order on the real number line
Absolute Numerical Value: numbers without orientation on the real line
Comparing and Contrasting—mathematically: an introduction
Comparison by Division: using ratios
Comparison by Division: using percentages
Comparison by Division: using odds
Comparison by Division: using proportions
Real World Applications using Ratios: from Finance to Biology to Meteorology
Rates of Change as Ratios: with real-world applications

Introductory Algebra

Basic Algebra: an introduction
Algebraic expressions
Algebraic operations: The arithmetic of algebraic expressions
Basic Algebraic equations and how to solve them
Basic Algebraic inequalities and how to solve them

Euclidean Planar Geometry

Geometry: an introduction
Planar Geometry: Lines in Euclid’s plane
Planar Geometry: Descartes’ coordinatized plane
Slope of a line as a ratio
Slope of a line as a trigonometric ratio
Planar Geometry: lines in the Cartesian plane leading to linear equations
Planar Geometry: measurement of angles in the Cartesian plane
Trigonometry of Angles in the Plane: an introduction

Online Maple TA Practice

See links under the top menu ‘Assessment’ tab of the particular topic page.