Self-Study High School Math: Best Books & Order to Learn

To self-study high school mathematics in a logical sequence, most learners should progress through basic algebra, then synthetic geometry, then trigonometry, then analytic geometry, before consolidating everything with a comprehensive review text. Israel Gelfand’s algebra and trigonometry books, Andrei Kiselev’s two-volume geometry series, and Serge Lang’s geometry and consolidation texts form a widely recommended path, supplemented by Schaum’s Outlines for extra practice problems.

Key Takeaways

• Gelfand’s “Algebra” explains the reasoning behind rules such as why a negative number times a negative number produces a positive result, rather than just stating the rule.
• Kiselev’s geometry series is split into two volumes: Book I covers planimetry (plane geometry) and Book II covers stereometry (solid geometry).
• Serge Lang’s “Geometry” is designed to be studied after learners already have some background in Euclidean geometry, algebra, and trigonometry.
• Serge Lang’s “Basic Mathematics” is positioned as a consolidation text for reviewing high school material before moving on to calculus, not as a first introduction to the subject.
• Schaum’s Outlines are recommended only as supplementary problem sets, not as primary textbooks for first-time learners.

What basic math topics should a self-study curriculum cover?

Basic algebra

Basic algebra is the study of solving equations and applying mathematical reasoning to real-world situations. The only prerequisite is being able to count well. Core topics include the different number systems (naturals, integers, rationals, reals, and complex numbers), rules for manipulating fractions and other numbers, solving first- and second-degree equations, solving systems of linear equations, working with powers and roots, solving inequalities, understanding functions and their graphs, logarithms and their manipulation rules, methods for solving special polynomial equations such as substitution and the rational root theorem, and basic logic and proof techniques.

Basic geometry

Basic geometry requires algebra as a prerequisite and is best approached by starting with synthetic geometry (axioms and basic theorems) before moving to more algebraic viewpoints. Key topics include axioms for points, lines, and planes; notions of circles, polygons, and parallelism; congruence; triangle theorems including the Pythagorean theorem and its converse; theorems for polygons and circles, such as the fact that a tangent line to a circle is perpendicular to the radius at the point of tangency; area and length calculations; transformations such as projection and rotation; coordinates and vectors; equations representing lines, planes, circles, and parabolas; conic sections; and the dot product.

Trigonometry

Trigonometry introduces functions such as sine, cosine, and tangent, and requires both basic algebra and basic geometry as prerequisites. It supports the study of triangles and solves many real-world problems. Important topics include oriented angles and radians; the sine, cosine, and tangent functions visualized on the unit circle and as graphs; solving triangle problems using the sine and cosine rules; trigonometric identities such as product-to-sum formulas; and inverse trigonometric functions (arcsine, arccosine, and arctangent).

Which books are recommended for learning basic algebra?

Israel Gelfand and Alexander Shen’s “Algebra” is recommended as the entry point for basic algebra. Gelfand is a well-known mathematician, and the book explains not just what algebra is but why algebraic facts hold true, including why a negative number times a negative number equals a positive number. It includes many worked examples and exercises, though some readers may want a supplementary problem book for additional practice.

The book covers the basics of addition and multiplication, negative numbers and fractions, powers and roots, special formulas such as expansions of (a+b)^2 and more general identities, polynomial factoring and roots, division and interpolation, arithmetic and geometric progressions, quadratic and biquadratic equations, symmetric equations, inequalities, and arithmetic, geometric, and harmonic means. This book suits both newcomers to algebra and readers already comfortable with the subject who want a deeper conceptual understanding.

Which books are recommended for synthetic geometry?

Andrei Kiselev’s two-volume “Geometry I and II” is recommended after algebra. Book I covers planimetry, or plane geometry, including straight lines, circles, similarity, regular polygons, circumference, and areas. Book II covers stereometry, or spatial geometry, including lines and planes, polyhedra, round solids, and a brief introduction to vectors and foundational concepts. The series also includes a brief introduction to non-Euclidean geometry.

This two-volume set suits readers who have never taken a geometry class or who want to revisit the subject, but it does not cover analytic geometry, meaning it does not address representing lines and circles through equations.

Which book is recommended for trigonometry?

Israel Gelfand and Mark Saul’s “Trigonometry” is recommended after geometry. The book emphasizes understanding the reasoning behind trigonometric formulas rather than memorization, though it contains relatively few exercises, so pairing it with a separate problem collection is worthwhile.

Topics covered include trigonometric ratios in a triangle, relations among trigonometric ratios, relationships within triangles, angles and rotations, radian measure, angle addition formulas, trigonometric identities, graphs of trigonometric functions, and inverse functions and trigonometric equations. This book suits first-time learners as well as those needing a refresher.

Which book is recommended for analytic geometry?

Serge Lang and Gene Murrow’s “Geometry” covers similar ground to Kiselev’s series but emphasizes coordinates and algebraic methods rather than synthetic proofs. It is best used after some prior exposure to Euclidean geometry, algebra, and trigonometry, rather than as a first introduction to geometry.

Topics include distance and angles, coordinates, area and the Pythagorean theorem, the distance formula, polygons, congruent triangles, dilations and similarities, volumes, vectors and the dot product, and transformations and isometries. This book suits readers who are new to analytic geometry specifically, or who need a refresher on it.

What book ties everything together before calculus?

Serge Lang’s “Basic Mathematics” covers everything needed for high school mathematics and is recommended before advancing to topics such as calculus. It is described as excellent for consolidation rather than as a first exposure to these topics.

Topics covered include integers, rational numbers, real numbers, and 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; and determinants. This book is recommended for anyone who wants to solidify their basic knowledge or revisit high school mathematics in a structured, comprehensive way.

Recommended book sequence at a glance

Recommended self-study sequence for basic high school mathematics, listing each book, its author(s), and its primary subject area

Order	Book Title	Author(s)	Primary Subject
1	Algebra	Israel Gelfand, Alexander Shen	Basic algebra
2	Geometry I and II	Andrei Kiselev	Synthetic geometry (plane and solid)
3	Trigonometry	Israel Gelfand, Mark Saul	Trigonometry
4	Geometry	Serge Lang, Gene Murrow	Analytic geometry
5	Basic Mathematics	Serge Lang	Consolidation of all high school topics

What are Schaum’s Outlines used for in this curriculum?

Schaum’s Outlines are recommended strictly as supplementary problem sets for extra practice, not as primary textbooks for learning a topic for the first time. Since solving many exercises is described as key to understanding mathematics, these outlines fill the gap left by textbooks such as Gelfand’s, which contain relatively few problems. Separate Schaum’s Outline titles are available for elementary algebra, geometry, trigonometry, and precalculus.

Frequently Asked Questions

What is the recommended order for self-studying basic high school math?

The recommended order is basic algebra first, followed by synthetic geometry, then trigonometry, then analytic geometry, and finally a consolidation text covering all of high school mathematics. This sequence follows the prerequisite structure, since geometry requires algebra, and trigonometry requires both algebra and geometry.

Do I need a separate problem book if I use Gelfand’s textbooks?

Yes. Gelfand’s “Algebra” and “Trigonometry” books focus on explaining the reasoning behind mathematical facts and contain relatively few exercises. Supplementing them with a dedicated problem book, such as a Schaum’s Outline, is recommended for additional practice.

What is the difference between Kiselev’s geometry and Lang’s geometry?

Kiselev’s two-volume “Geometry I and II” covers synthetic geometry using axioms and theorems, without equations for lines and circles. Lang and Murrow’s “Geometry” covers similar topics but emphasizes coordinates and algebraic methods, and is best studied after some exposure to Euclidean geometry, algebra, and trigonometry.

Is Serge Lang’s “Basic Mathematics” a good starting point for a beginner?

No. It is described as excellent for consolidating and revisiting high school mathematics rather than as a first exposure to the material. It is recommended after working through algebra, geometry, and trigonometry, as a way to solidify and connect that knowledge before moving toward calculus.

What prerequisites does trigonometry require?

Trigonometry requires both basic algebra and basic geometry as prerequisites. It builds on algebraic manipulation and geometric concepts like triangles and angles to introduce functions such as sine, cosine, and tangent.

What topics does basic algebra cover before moving to geometry?

Basic algebra covers number systems, manipulation of fractions and other numbers, first- and second-degree equations, systems of linear equations, powers and roots, inequalities, functions and graphs, logarithms, methods for solving polynomial equations, and basic logic and proofs. Counting ability is the only prerequisite.