High School Mathematics — Contents Advanced Resources AD 01 · 8 chapters + appendix

High School Mathematics— From Logic to Conic Sections: High School Mathematics You Can Derive Yourself —

Based on Yasuhide Mochizuki's handwritten notes, “Mathematics Notes: High School Mathematics,” this book is an educational resource that reorganizes the key points of the high school mathematics that forms the foundation for the mathematics, physics, and chemistry studied at university, with the goal that readers become able to derive formulas rather than memorize them. It contains all the definitions, theorems, proofs, and figures of the original notes; we have filled in the gaps between the lines so that high school students can read it on their own, and we have added examples to each section and exercises at the end of each chapter.

We begin with logic and sets, and then proceed to the properties of triangles, trigonometric ratios and the measurement of figures, coordinate geometry, the trigonometric addition formulas, logarithms, vectors, and conic sections. Each chapter of the companion volume “University Mathematics” (AD 02) refers to the corresponding chapter of this book as a prerequisite.

How to read this book
  • Read in order, starting from Chapter 1. Work out each example yourself before looking at its solution.
  • Solutions to the exercises at the end of each chapter are collected in Appendix A.
  • Readers who wish to go on to university mathematics should make sure, in particular, that they have a firm grasp of Chapter 1 (logic), Chapter 5 (addition formulas), and Chapter 7 (vectors).
  1. Chapter 1Logic and Sets Propositions; Necessary Conditions and Sufficient Conditions; Necessary and Sufficient Conditions (Equivalence); Converse, Inverse, and Contrapositive; Basics of Sets — Conditions and Truth Sets
  2. Chapter 2Properties of Triangles Angle Bisectors and Internal and External Division; The Five Centers of a Triangle (1) — Centroid, Median Theorem, and Orthocenter; The Five Centers of a Triangle (2) — Circumcenter, Incenter, and Centroid; Ceva's Theorem and Menelaus's Theorem
  3. Chapter 3Measurement of Figures — Trigonometric Ratios, the Law of Sines, and the Law of Cosines Basic Relations among the Trigonometric Ratios; The Law of Sines; The Law of Cosines — Proof by Cases: Acute, Right, and Obtuse Angles; Radian Measure
  4. Chapter 4Coordinate Geometry Internal and External Division Points on the Number Line; Internal and External Division Points in the Plane and the Centroid of a Triangle; Distance from a Point to a Line
  5. Chapter 5Trigonometric Addition Formulas and Related Formulas Building the Addition Formulas — Starting from Cosine; Addition Formulas for Sine and Tangent; Double-Angle and Half-Angle Formulas; Triple-Angle Formulas; Product-to-Sum Formulas; Sum-to-Product Formulas; Harmonic Addition (Combining Sine and Cosine)
  6. Chapter 6Properties of Logarithms Review of the Definition of Logarithms and the Change-of-Base Formula; Logarithms of Numbers with Exponents; Converting between Logarithms and Exponents
  7. Chapter 7Vectors Vector Operations — Equality, Addition, Subtraction, and Scalar Multiplication; Conditions for Parallelism, Unit Vectors, and Linear Independence; Component Representation of Vectors and Internal and External Division Points; The Inner Product of Vectors; Vector Equations of Lines and Normal Vectors (Plane Geometry); Equations of Planes, Lines, and Spheres in Space; Conditions for Collinearity, Concurrency, and Coplanarity, and the Area of a Triangle
  8. Chapter 8Conic Sections — Parabolas, Ellipses, and Hyperbolas Parabolas; Ellipses; Hyperbolas
Appendix
  1. Appendix ASolutions to the Exercises Complete solutions to the exercises at the end of each chapter

Source

Mochizuki Research Group, Department of Materials Science and Technology, Faculty of Advanced Engineering, Tokyo University of Science — Advanced Resources AD 01
← Back to Educational Resources