初等数学解题法
Author: Zhiming Ou
Publisher: 3265 Public Way
ISBN: 978-1-0677109-7-2
Summary
When we encounter a problem, either given by school teachers for educational purpose, or proposed by some industrial workers or researching professors, we usually follow these 5 steps to solve it (Grass):
- Understand the problem: what area does it belong to? Stem or Steam? What condition(s) is given? What is required to find?
- What Knowledge or tool does the problem require? Do I have the knowledge/tool?
- Build a link between the conditions and the requirements. In mathematics, this is usually to set up the equations, or to find the quantitative relation.
- Carry out the plan. Solve the equations.
- Check for perfectness. Make a statement.
For problems in high school math, which is also called elementary math, they are usually classified into 5 categories:
- Algebra, include elementary number theory, sequence and series;
- Geometry, both Euclidean and Cartesian;
- Function or relation, the combination of algebra and geometry;
- Counting, not just count by listing, but very skillful. Eventually, counting is thinking.
- Logical reasoning, include proposition logic, mathematical games, probability analysis.
In this book, for each category, I summarized
- the formulas and laws;
- common types of problems, and inspiring examples, and
- Methods and skills to solve them.
Content
Chapter 1 What is Mathematics?
Objects of study
Structure of mathematics
What does mathematics do?
General mathematical methods
Steps for Problem Solving
Chapter 2 Algebra
List of formulas
Typical Problems
-- algebraic operations 13
--Equations 17
--Inequalities 19
--Continued fractions 21
/Methods/skills to solve]
Chapter 3 Sequence and Series 25
List of formulas 25
Typical Problems
/Methods/skills to solve]
Chapter 4 Geometry and Trigonometry 34
List of formulas
Typical Problems
--Measuring Angles, perimeters, areas, and volumes
--Shape quantifying
--Geometric Proofs
/Methods/skills to solve]
Chapter 5 Functions 66
List of formulas
Typical Problems
--Function notation (domain, range, graph)
--Functional Equations
/Methods/skills to solve]
Chapter 6 Combinatorics 70
List of principles and formulas
Typical Problems
--count by mapping, recursion and generating function
--Combinatorial Construction
/Methods/skills to solve]
Chapter 7 Logic Reasoning 96
List of formulas
Typical Problems
--symbolizing
--set modelling
--math gaming
/Methods/skills to solve]
Chapter 8 Number Theory 101
List of formulas
Typical Problems
--Factors/multiples, primes, congruences
--rational numbers
--Specific integers
--arithmetic functions
--Diophantine Equations
--Existence and approximation
/Methods/skills to solve]