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Chapter 1 — Fundamental Concepts

In this first chapter we lay the formal foundations on which all of mathematical analysis and university algebra are built.

Throughout these sections we move from operational intuition to the logical precision that abstract mathematics requires, studying the primitive objects of the discipline and the laws that govern their behavior.

Contents of the chapter

  • 1.1 — Real Numbers: Construction in layers (\(\mathbb{N}\), \(\mathbb{Z}\), \(\mathbb{Q}\), \(\mathbb{I}\), \(\mathbb{R}\)), proof of irrationality, algebraic and order axioms, intervals, absolute value and completeness of \(\mathbb{R}\) (the Supremum Axiom).
  • 1.2 — Exponents and Radicals: Integer and rational powers, laws of exponents, existence of \(n\)th roots, simplification and rationalization.
  • 1.3 — Algebraic Expressions: Polynomials, degree, special products and factoring algorithms over \(\mathbb{R}\).
  • 1.4 — Fractional Expressions: Rational expressions, domain of definition, simplification, operations and complex fractions.
  • 1.5 — Equations: Linear, quadratic, radical and rational equations. The zero-product property and extraneous solutions.
  • 1.6 — Problem Solving with Equations: A systematic method for translating word problems into algebraic equations: mixtures, joint work, motion and geometry.
  • 1.7 — Inequalities: Linear, quadratic, absolute-value and rational inequalities, solved by sign analysis on \(\mathbb{R}\).
  • 1.8 — Analytic Geometry: The Cartesian plane, the distance formula, the midpoint, the circle and the symmetry of curves.
  • 1.9 — Graphing Calculators and Computers: The role of technology as a tool for verification and exploration, without replacing algebraic understanding.
  • 1.10 — Lines: Slope, equations of a line, conditions for parallelism and perpendicularity, and the general form of the linear equation.
  • 1.11 — Chapter 1 Closing: Number, axiom and certainty: epistemological reflections on the continuum, deductive form and formalization.

"Every exact science begins by naming what it is going to speak about."