GRE Mathematics Subject Test
Study notes, worked examples, formula recall, and practice questions across the GRE Mathematics domains, with sources cited.
The syllabus
GRE Math preparation, organized into the content areas below.
- Calculus
- Linear algebra
- Abstract algebra
- Real analysis
- Topology
- Differential equations
Representative topics
- Multivariable calculus
- Vector calculus
- Group theory
- Linear operators
- Sequences and series
- ODEs
Study notes
Free to read, no account needed. Every exam fact and concept links to its official or open source.
- Limits and ContinuityA free GRE Mathematics note on limit laws, the squeeze theorem, L'Hopital's rule, the three-condition definition of continuity, and the classic limits every calculus question assumes you know.
- Series Convergence TestsA free GRE Mathematics note on the convergence toolkit: geometric and p-series benchmarks, comparison, ratio, and root tests, the alternating series test, and the difference between absolute and conditional convergence.
- Eigenvalues and DeterminantsA free GRE Mathematics linear algebra note on determinant properties, the eigenvalue equation, trace and determinant as eigenvalue invariants, triangular matrices, and the basics of diagonalizability.
- Group Theory EssentialsA free GRE Mathematics note on group and subgroup axioms, element order, cyclic groups, Lagrange's theorem and its corollaries, direct products, and how to tell two small groups apart.
- Metric Spaces and CompactnessA free GRE Mathematics note on metrics, open and closed sets, the Heine-Borel characterisation of compactness, what compactness buys for continuous functions, and how connectedness is tested.
- First- and Second-Order Differential EquationsA free GRE Mathematics note on classifying an ODE, separable equations, the integrating factor for first-order linear equations, and the three characteristic-root cases for constant-coefficient second-order equations.
- Analytic Functions and ResiduesA free GRE Mathematics note on the Cauchy-Riemann test for analyticity, Cauchy's theorem and integral formula, classifying singularities from a Laurent series, and evaluating contour integrals with residues.
- Calculus Domain GuideThe whole GRE Mathematics calculus domain in one place: derivatives as linear approximation, the fundamental theorem, the series toolkit, gradients and Lagrange multipliers, and the Green/Stokes/divergence family.
- Linear Algebra Domain GuideVector spaces, rank and nullity, determinants, eigenvalues, diagonalisation, and linear operators for the GRE Mathematics Test — with the invertibility equivalences and the trace/determinant checks that settle most items in seconds.
- Abstract Algebra Domain GuideGroups, rings, fields, and the number theory the GRE Mathematics Test actually uses: Lagrange's theorem, cyclic structure, quotients and the first isomorphism theorem, zero divisors, and modular arithmetic.
- Real Analysis Domain GuideCompleteness, sequences, uniform versus pointwise convergence, continuity theorems, and the counting and probability rules the GRE Mathematics Test groups with them — taught through the counterexamples that decide the questions.
- Topology Domain GuidePoint-set topology for the GRE Mathematics Test: the open-set axioms, continuity by preimages, closure and boundary, and which properties — compactness and connectedness — actually survive a continuous map.
- Differential Equations Domain GuideHow the GRE Mathematics differential-equations domain fits together: existence and uniqueness, the homogeneous-plus-particular structure, the Wronskian, linear systems solved by eigenvalues, and long-run behaviour from equilibria.
- Complex Analysis Domain GuideFrom polar form and roots of unity to Cauchy–Riemann, Cauchy's theorem, Laurent series, and residues — the complex-analysis material the GRE Mathematics Test expects, with the rigidity theorems that make its questions answerable at a glance.
Worked examples
Full solutions with the correct answer, the reasoning, and why each option is right or wrong. No account needed.
- A limit at a missing pointFactor before cancelling, retain the domain restriction, and distinguish a limit from a function value.
- Eigenvalues of a triangular matrixUse a characteristic polynomial and a trace check rather than assuming all matrix entries are eigenvalues.
- Limits of Rational FunctionsA public limits question that resolves a $0/0$ indeterminate form by factoring and canceling a common factor before substituting.
- Geometric Series ConvergenceA public series question that checks the common-ratio convergence condition and applies the closed-form sum of an infinite geometric series.
- Determinants of Scaled, Transposed, and Inverted MatricesA public linear algebra question that chains three determinant properties — scalar multiples in dimension n, transposes, and inverses — into a single evaluation.
- Counting Elements of a Given OrderA public group theory question that counts the elements of order 6 in a cyclic group by using the order formula n/gcd(k, n) rather than by listing elements.
- Is the Set Open, Compact or Connected? A Planar AnnulusA public topology question that runs the open, closed, bounded, and connected checks on a half-open annulus in the plane and reads off compactness from Heine-Borel.
- A Second-Order Initial-Value Problem with Complex RootsA public differential equations question that solves a constant-coefficient initial-value problem whose characteristic roots are complex, including the product rule needed for the second condition.
- A Contour Integral at a Third-Order PoleA public complex analysis question that evaluates a contour integral with a pole of order three, where the factorial in Cauchy's formula for derivatives decides the answer.
Guides
Free articles for GRE Math test takers, each checked against official sources.
What these questions are built to catch
Every wrong answer in the GRE Math bank was written to catch one specific mistake, and each is recorded by name. The taxonomy sets out the named mistakes for this exam, grouped by the part of the syllabus each turns up in, with what to do differently. It describes how the questions were written, not how students perform.
Open the GRE Math section of the error taxonomySources
- GRE Subject Test Content and Structure — ETS. Accessed 2026-07-06. Use as a cited source for exam facts; do not imply affiliation or reproduce protected test material.
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- 2026-07-06
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- Next review: 2026-10-06
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