Questing Vole Press
Essential Properties of Numbers

Essential Properties of Numbers: A Self-Teaching Guide
by Tim Hill


Knowledge of the basic properties of numbers is crucial for solving equations, simplifying expressions, and multiplying polynomials. Yet many precalculus and calculus students are taught only simple arithmetic tools and techniques, leaving them with a superficial understanding of problem-solving.

To solve a “one-step” equation like x + 4 = 12, for example, many students are taught to subtract 4 from both sides to isolate the variable x, perhaps crossing out (“cancelling”) 4s as they proceed. Unfortunately, this method suppresses a variety of algebraic properties. Justifying each step formally requires equivalence, associativity of addition, additive inverse, additive identity, and closure under addition. Similarly, a proper solution to an equation like 4x = 12 requires the analogous properties for multiplication.

This book teaches students and self-learners how to use the basic properties of real numbers to derive proper algebraic solutions. The collective power of these individual properties reveals subtle issues, makes clear underlying structures, and shows just how much is hidden from view in the standard cancellation process.


1. Basic Properties of Numbers
2. Absolute Value
3. Natural Numbers and Induction
4. Summation
5. Integers, Rationals, and Reals
6. Problems
7. Solutions


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