Math Self-Study Roadmap: Topics & Book Recommendations

Introduction

We often get questions here from people self-studying mathematics. One common question is: what mathematics should I study, and in what order? To answer that question I have made a list of topics a mathematician should ideally know and what prerequisites those topics require.

Basic stuff

Of course, we have the basic (high school) material. This includes:

1. Basic algebra
   This is the art of solving equations and applying mathematical knowledge to some real-world situations.

   Prerequisites: Being able to count well.

   Important topics:

   • a) Different number systems: naturals, integers, rationals, reals, and complex numbers.
   • b) Basic rules for manipulating numbers (such as fractions).
   • c) Solutions of equations of the first degree and second degree.
   • d) Solutions of systems of linear equations.
   • e) Manipulating powers and roots and solving equations involving powers and roots.
   • f) Solutions to inequalities.
   • g) Functions, mappings and their graphs.
   • h) Logarithms, rules of manipulation of logarithms and equations involving logarithms.
   • i) Various methods of solving (special) polynomial equations such as substitution and the rational root theorem.
   • j) Basic logic and proofs.
2. Basic geometry
   Geometry is important in mathematics. A useful approach is to start with synthetic geometry (axioms and basic theorems) and later move to more algebraic viewpoints.

   Prerequisites: Algebra.

   Important topics:

   • a) Axioms for points, lines, and planes. Notions of circles, polygons, parallelism, etc.
   • b) Notions of congruence.
   • c) Theorems for triangles including the Pythagorean theorem (and its converse), criteria for triangle congruence, special points in triangles such as the incenter, etc.
   • d) Theorems for polygons and circles (for example, the tangent line to a circle is perpendicular to the radius).
   • f) Area and length of geometric figures.
   • g) Special transformations such as projection and rotation.
   • h) Coordinates and vectors.
   • i) Representing lines, planes, circles, parabolas, etc. by equations.
   • j) Conic sections.
   • k) Dot product and the related algebra.
3. Trigonometry
   Trigonometry introduces special functions such as sine, cosine, and tangent. It helps the study of triangles and solves many real-world problems.

   Prerequisites: Basic algebra and basic geometry.

   Important topics:

   • a) Oriented angles and radians.
   • b) Sine, cosine, and tangent functions: their visualization on the unit circle and their graphs.
   • c) Solving triangle problems with sine, cosine, and tangent functions, including the sine and cosine rules.
   • d) Trigonometric identities, such as product-to-sum formulas.
   • e) Inverse trigonometric functions: arcsine, arccosine, and arctangent.

My book recommendations for basic stuff

If you’re new to abstract high-school mathematics, there is no better place to begin than the following books.

Algebra by Gelfand, Shen

http://www.amazon.com/Algebra-Israel-M-Gelfand/dp/0817636773

Gelfand is a top mathematician, and this book is excellent. It explains not only what algebra is but why algebraic facts hold. For example, it explains why a negative times a negative is positive. The book includes many worked examples and exercises, though some readers may wish to supplement it with a problem book for additional practice.

The topics covered in Gelfand include:

• Basics of addition and multiplication
• Negative numbers and fractions
• Powers and roots
• Special formulas such as for (a+b)^2 (and more general identities)
• Polynomials: factoring, roots, division, interpolation
• Arithmetic and geometric progressions
• Quadratic and biquadratic equations; symmetric equations
• Inequalities
• Arithmetic, geometric, and harmonic means

This book is ideal for newcomers to algebra or for those who wish to revisit the subject; it is also enjoyable for readers already comfortable with algebra.

After algebra, it is a good time to study geometry. For that, I recommend:

Geometry I and II by Kiselev

http://www.amazon.com/Kiselevs-Geometry-Book-I-Planimetry/dp/0977985202

http://www.amazon.com/Kiselevs-Geometry-Book-II-Stereometry/dp/0977985210

Kiselev’s two-volume work covers synthetic geometry: plane geometry in the first volume and spatial geometry in the second. It includes a brief introduction to vectors and non-Euclidean geometry.

The first book covers:

• Straight lines
• Circles
• Similarity
• Regular polygons and circumference
• Areas

The second book covers:

• Lines and planes
• Polyhedra
• Round solids
• Vectors and foundations

This work is suitable for those who have never had a geometry class or who wish to revisit it; it does not cover analytic geometry (equations of lines and circles).

After geometry, study trigonometry. Recommended:

Trigonometry by Gelfand and Saul

http://www.amazon.com/Trigonometry-I-M-Gelfand/dp/0817639144

Gelfand’s trigonometry book emphasizes understanding the reasons behind formulas. It contains valuable insights but relatively few exercises, so supplement it with problem collections.

The book covers:

• Trigonometric ratios in the triangle
• Relations among trigonometric ratios
• Relationships in the triangle
• Angles and rotations
• Radian measure
• The addition formulas
• Trigonometric identities
• Graphs of trigonometric functions
• Inverse functions and trigonometric equations

This book is ideal for first-time learners or for those needing a refresher.

So far you have seen synthetic geometry; analytic geometry is also important.

Geometry by Lang, Murrow

http://www.amazon.com/Geometry-School-Course-Serge-Lang/dp/0387966544

Lang’s book covers similar topics to Kiselev but emphasizes coordinates and algebra. It is better used after some exposure to Euclidean geometry, algebra, and trigonometry.

The book covers:

• Distance and angles
• Coordinates
• Area and the Pythagorean theorem
• The distance formula
• Polygons
• Congruent triangles
• Dilations and similarities
• Volumes
• Vectors and dot product
• Transformations and isometries

This book suits those new to analytic geometry or needing a refresher.

Finally, some topics not covered above (or worth revisiting in a structured way) can be found in the following:

Basic Mathematics by Lang

http://www.amazon.com/Basic-Mathematics-Serge-Lang/dp/0387967877

This book covers everything needed for high-school mathematics and is recommended before advancing to topics such as calculus. It is not necessarily the best first exposure, but it is excellent for consolidation.

The book covers:

• Integers, rational numbers, real numbers, complex numbers
• Linear equations
• Logic and mathematical expressions
• Distance and angles
• Isometries
• Areas
• Coordinates and geometry
• Operations on points
• Segments, rays, and lines
• Trigonometry
• Analytic geometry
• Functions and mappings
• Induction and summations
• Determinants

I recommend this book to anyone who wants to solidify their basic knowledge or revisit high-school mathematics in detail.

Exercise books

The key to understanding mathematics is solving many exercises. For problem practice, Schaum’s Outlines are useful as supplements (not as primary introductions).

• Elementary algebra: http://www.amazon.com/Schaums-Outline-Elementary-Algebra-3ed/dp/0071611630
• Geometry: http://www.amazon.com/Schaums-Outline-Geometry-5th-Problems/dp/0071795405
• Trigonometry: http://www.amazon.com/Schaums-Outline-Trigonometry-5th-Problems/dp/0071795359
• Precalculus: http://www.amazon.com/Schaums-Outline-Precalculus-3rd-Problems/dp/0071795596

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