Sample · 00-The Real Number System
Chapter 0 Overview
Pre-Calculus · CHAPTER 0: The Real Number System
Everything in this book is built on the real numbers, and this chapter makes sure the foundation is solid before the building starts. You will name the sets of numbers and place any number in the right ones, use the properties that make algebra legal, read and draw the number line, write sets of numbers as intervals, treat absolute value as distance, work with the integer part of a number, and compare and simplify exact values such as radicals without reaching for a calculator.
The topics run 0.1 through 0.7. Every topic opens with a short Math at Work story about a real career that runs on exactly that mathematics, continues with a full lesson and three windows on business, technology and science, and closes with an exercise set in Parts A to F that climbs from Easy to Hard — every question multiple-choice, each with a full solution, a trap analysis, a Teacher comment and an EduCoach comment. The last part of every set is textbook-style examples rewritten as MCQ.
The chapter finishes with two more sections: SAT - ACT Questions, mixing every topic in the format those tests actually use, and AP Precalculus - Math Competition and Math Olympiad Questions for the AMC, AIME and olympiad circuit. Three mascots travel with you: the Honey Badger warns you at every trap, the Snow Owl keeps the facts straight, and the Lion shows up whenever a question needs real power.
📖 Topics in this chapter
Work through the topics in order; each one uses the ones before it. The two closing sections mix everything in exam and competition format.
| Topic | What you will learn | Questions |
|---|---|---|
| 0.1 Sets of Numbers | Name the sets of numbers (natural, whole, integer, rational, irrational, real), classify any number by its decimal expansion, and decide when a root is rational | 31 |
| 0.2 Properties of Real Numbers | Use the commutative, associative, distributive, identity and inverse properties, apply the order of operations, and recognise which properties fail for subtraction and division | 30 |
| 0.3 Order and the Number Line | Compare and order real numbers, plot and read them on the number line, use the rules for adding and multiplying inequalities, and reverse the sign when multiplying by a negative | 32 |
| 0.4 Intervals and Set Notation | Write sets of real numbers in set-builder and interval notation, graph them on the number line, and combine them with unions and intersections | 30 |
| 0.5 Absolute Value and Distance | Read absolute value as distance on the number line, evaluate and simplify expressions with absolute value, solve |x - a| = c, |x - a| < c and |x - a| > c, and model tolerances | 32 |
| 0.6 The Integer Part: Floor, Ceiling and Rounding | Evaluate the greatest integer (floor), ceiling and fractional part of a real number, graph the staircase, count how many numbers share a value, and model billing, packing and rounding | 32 |
| 0.7 Exact Values, Estimation and Comparison | Simplify radicals to exact form, locate and estimate them on the number line, compare numbers without a calculator by squaring, and decide signs inside absolute value bars | 32 |
| SAT - ACT Questions | Number sets, properties, order, intervals, absolute value, floor and ceiling, radicals and estimation | 22 |
| AP Precalculus - Math Competition and Math Olympiad Questions | Chapter 0 stretched: AP-style modelling with number sets, tolerances and step costs; AMC-style sums of distances, floor staircases and radicals; AIME and olympiad ideas rewritten as MCQ | 25 |
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