Is Basic Mathematics by Serge Lang the Right Math Book for You?

[Greg Bernhardt]

• Author: Serge Lang
• Title: Basic Mathematics
• Amazon Link: https://www.amazon.com/dp/0387967877/?tag=pfamazon01-20
• Prerequisities: Mathematics before high school
• Level: Motivated high-school students, college students
• Contents: Real numbers, solving equations, logic, geometry, trigonometry, functions, complex numbers, induction, determinants

Table of Contents:

[LIST]
[*] Algebra
[LIST]
[*] Numbers
[LIST]
[*] The integers
[*] Rules for addition
[*] Rules for multiplication
[*] Even and odd integers; Divisibility
[*] Rational numbers
[*] Multiplicative inverse
[/LIST]
[*] Linear equations
[LIST]
[*] Equations in two unknowns
[*] Equations in three unknowns
[/LIST]
[*] Real numbers
[LIST]
[*] Additions and multiplication
[*] Real numbers: positivity
[*] Powers and roots
[*] Inequalities
[/LIST]
[*] Quadratic equations
[*] On logic and mathematical expressions
[LIST]
[*] On reading books
[*] Logic
[*] Sets and elements
[*] Notation
[/LIST]
[/LIST]
[*] Intuitive Geometry
[LIST]
[*] Distance and Angles
[LIST]
[*] Distance
[*] Angles
[*] The Pythagoras Theorem
[/LIST]
[*] Isometries
[LIST]
[*] Some standard mappings of the plane
[*] Isometries
[*] Composition of Isometries
[*] Inverse of isometries
[*] Characterization of isometries
[*] Congruences
[/LIST]
[*] Area and application
[LIST]
[*] Area of a disc of radius [itex]r[/itex]
[*] Circumference of a circle of radius [itex]r[/itex]
[/LIST]
[/LIST]
[*] Coordinate Geometry
[LIST]
[*] Coordinates and geometry
[LIST]
[*] Coordinate systems
[*] Distance between points
[*] Equation of a circle
[*] Rational points on a circle
[/LIST]
[*] Operations on points
[LIST]
[*] Dilatations and reflections
[*] Addition, subtraction, and the parallelogram law
[/LIST]
[*] Segments, Rays, and Lines
[LIST]
[*] Segments
[*] Rays
[*] Lines
[*] Ordinary Equation for a Line
[/LIST]
[*] Trigonometry
[LIST]
[*] Radian measure
[*] Sine and cosine
[*] The graphs
[*] The tangent
[*] Addition formulas
[*] Rotations
[/LIST]
[*] Some Analytic Geometry
[LIST]
[*] The straight line again
[*] The parabola
[*] The ellipse
[*] The hyperbola
[*] Rotation of hyperbolas
[/LIST]
[/LIST]
[*] Miscellaneous
[LIST]
[*] Functions
[LIST]
[*] Definition of a function
[*] Polynomial functions
[*] Graphs of functions
[*] Exponential function
[*] Logarithms
[/LIST]
[*] Mappings
[LIST]
[*] Definition
[*] Formalism of mappings
[*] Permutations
[/LIST]
[*] Complex Numbers
[LIST]
[*] The complex plane
[*] Polar form
[/LIST]
[*] Induction and Summations
[LIST]
[*] Induction
[*] Summations
[*] Geometric Series
[/LIST]
[*] Determinants
[LIST]
[*] Matrices
[*] Determinants of order [itex]2[/itex]
[*] Properties of [itex]2\times 2[/itex] determinants
[*] Determinants of order [itex]3[/itex]
[*] Properties of [itex]3\times 3[/itex] determinants
[*] Cramer's rule
[/LIST]
[/LIST]
[*] Index
[/LIST]

 

[micromass]

This book is truly one of a kind. This is a book on basic mathematics, written from the point-of-view of an advanced mathematician. Do not expect many plug-and-chug exercises that just drill the method into you. Do not expect silly notions such as "it is important to rationalize the denominator", it is not. What you should expect is a book that teaches what mathematics really is about. It teaches proofs and logic as a foundation of mathematics.
However, Lang's style of writing is a bit weird, it takes some time to get used to it. The book does contain some (typographical) errors, but if you are aware of this then this shouldn't bother much.

 

[AnTiFreeze3]

I went through most of this book due to MicroMass's recommendation a little while ago, and really enjoyed my first introduction to real mathematics, and not the garbage I was/am being taught in middle/high school.

The basic topics covered doesn't necessarily mean that the book will be easy; this was my first introduction to proofs, and to trying to create my own (which I often failed at doing). However, even if I knew most of the material already (like that a negative number multiplied by a negative number results in a positive number), I had never known the reasoning behind why this was true. The logical approach presented taught me a different, more effective way to learn mathematics. Lang even has an entire section (somewhere after chapter 3, I think) where he simply covers logic and some notation, to help the reader get a better grasp of the idea.

I highly recommend this book for either someone who has not learned any of the material covered yet, but wishes for a great introduction, or to someone who has covered most of this material already (like I had), but recognizes that their understanding of that material is shoddy at best, and would like to learn them with more rigor.

 

[theoristo]

Just great!

 

[theoristo]

A good and different alternative would be Numbers and Geometry by John Stillwell https://www.amazon.com/dp/0387982892/?tag=pfamazon01-20