FREE GRE Math Subject Test Study Guide 2026: Calculus, Algebra & Additional Topics

This free GRE Math study guide teaches to the GRE Mathematics Subject Test — the ETS exam for prospective mathematics and quantitative-field graduate students.^[1] It is not the quantitative section of the general GRE: this Subject Test covers about three years of an undergraduate math major, organized exactly the way ETS describes the content.

The exam is about 66 multiple-choice questions in 2 hours 50 minutes, reported on a 200–990 scaled score.^[1] Roughly half is calculus, a quarter is algebra, and a quarter is additional topics. This guide is interactive, not a wall of text: every module has a built-in checkpoint quiz, hover-able glossary terms, worked theorems and formulas, and concept questions, so you learn by doing.

Read it module by module, test yourself at each checkpoint, then round out your free GRE Math prep with our practice questions and flashcards.

GRE Math Subject Test Snapshot

ETS reports the content as three broad areas with approximate shares, so the exact mix shifts slightly each form.^[1] Because calculus is about half the exam, fluency with limits, derivatives, integrals, and series is the single highest-leverage thing you can build:

This guide teaches all three areas as six study modules: two calculus modules (single-variable, then multivariable plus sequences and series), one linear-algebra module, one abstract-algebra and number-theory module, and two modules covering the additional topics — analysis, topology, and complex variables, then probability, statistics, and discrete math.

1 · Single-Variable Calculus

Part of the ~50% calculus content area — the heart of the exam. Single-variable calculus is limits, derivatives, and integrals, and the theorems that connect them. Master the toolkit below and you can answer the most common question type quickly.^[1]

Limits & Continuity

A limit describes the value a function approaches. Evaluate one by direct substitution first; if that gives an indeterminate form like $0/0$, factor and cancel, or apply L’Hôpital’s rule: $\lim \frac{f}{g} = \lim \frac{f'}{g'}$ for $0/0$ or $\infty/\infty$.

Know the standard limits $\lim_{x\to0}\frac{\sin x}{x} = 1$ and $\lim_{x\to\infty}(1+\tfrac{a}{x})^x = e^{a}$. A function is continuous at $a$ when $\lim_{x\to a} f(x) = f(a)$.

Derivatives & Differentiation Rules

The derivative is an instantaneous rate of change. Know every rule cold — the test rewards speed, not first-principles:

The chain rule is the most-used. For $f(x) = e^{2x}\sin x$, the product rule gives $2e^{2x}\sin x + e^{2x}\cos x$. Use implicit differentiation when $y$ is defined implicitly — it is just the chain rule applied to $y$.

Applications of the Derivative

Derivatives find extrema, describe shape, and link related rates. A critical point has $f'(x) = 0$ (or undefined); the second-derivative test classifies it ($f'' > 0$ min, $f'' < 0$ max). The Mean Value Theorem guarantees a point where the tangent slope equals the average rate of change; Rolle’s theorem is the case $f(a) = f(b)$.

Integrals & Techniques

Integration is antidifferentiation. The Fundamental Theorem of Calculus ties it to the derivative: $\int_a^b f(x)\,dx = F(b) - F(a)$. The four core techniques are the power rule, u-substitution (the chain rule in reverse), integration by parts ($\int u\,dv = uv - \int v\,du$), and partial fractions.

Applications of the Integral

Definite integrals accumulate. Know the standard applications:

An improper integral (infinite limit or unbounded integrand) is evaluated as a limit — for example, $\int_1^{\infty}\frac{1}{x^2}\,dx = 1$ converges, mirroring the p-series rule.

2 · Multivariable Calculus, Sequences & Series

The rest of the ~50% calculus area. This module covers infinite series, Taylor expansions, partial derivatives and the gradient, and the vector-calculus theorems. Series questions in particular are common and fast once you recognize the right test.^[1]

Sequences & Series Convergence

A series converges if its partial sums approach a finite limit. The skill is choosing the right test. Start with the nth-term test(if the terms don’t go to 0, it diverges), then work down:

Anchor everything on two known series: the geometric series $\sum ar^n = \frac{a}{1-r}$ for $|r| < 1$, and the p-series $\sum 1/n^p$, which converges iff $p > 1$.

Power & Taylor Series

A Taylor series represents a function as $\sum \frac{f^{(n)}(a)}{n!}(x-a)^n$; with $a = 0$ it is a Maclaurin series. Memorize the staples:

The radius of convergence $R$ (found by the ratio test) gives the interval where the series equals the function; $e^x$ has $R = \infty$.

Partial Derivatives & Gradients

For a function of several variables, a partial derivative $\partial f/\partial x$ differentiates with respect to one variable, holding the others constant. The gradient $\nabla f$ collects them and points in the direction of steepest ascent. Lagrange multipliers ($\nabla f = \lambda \nabla g$) solve constrained optimization, and the Hessian (second-partials) test classifies critical points.

Multiple & Line Integrals

A double integral $\iint_R f\,dA$is evaluated as an iterated integral (Fubini’s theorem). The big vector-calculus theorems convert between dimensions:

Green’s theorem is the most-tested — recognize the form $\oint P\,dx + Q\,dy$ and switch to the easier double integral.

3 · Linear Algebra

The largest part of the ~25% algebra content area. Matrices, determinants, vector spaces, rank, and — most tested of all — eigenvalues and eigenvectors.^[1]

Matrices & Determinants

The determinant of a square matrix is a single scalar: $\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc$. It is nonzero exactly when the matrix is invertible, and it equals the product of the eigenvalues. An $n\times n$ matrix is invertible iff $\det A \ne 0$ iff its rank is $n$ iff its columns are independent.

Vector Spaces, Rank & Independence

A vector space is closed under addition and scalar multiplication. A basis is an independent spanning set; its size is the dimension. The rank of a matrix is the dimension of its column space, and the rank–nullity theorem ties it to the kernel: $\operatorname{rank}(T) + \operatorname{nullity}(T) = \dim V$.

Eigenvalues & Eigenvectors

An eigenvector is a nonzero vector whose direction $A$ preserves: $A\mathbf{v} = \lambda \mathbf{v}$, with eigenvalue $\lambda$. Find eigenvalues from the characteristic equation $\det(A - \lambda I) = 0$, then solve $(A - \lambda I)\mathbf{v} = \mathbf{0}$ for each eigenvector.

A matrix is diagonalizable when it has a full set of independent eigenvectors — guaranteed when all eigenvalues are distinct. The spectral theorem says every real symmetric matrix is orthogonally diagonalizable with real eigenvalues.

4 · Abstract Algebra & Number Theory

The rest of the ~25% algebra area. Group theory, rings and fields, and elementary number theory. Questions here reward knowing definitions and theorem statements precisely.^[1]

Group Theory

A group is a set with an associative operation, an identity, and inverses. Abelian means the operation also commutes (the integers under addition). The order of an element is the smallest $n$ with $a^n = e$. By Lagrange’s theorem, the order of any subgroup — and of any element — divides the order of the group.

Rings & Fields

A ring has addition (an abelian group) and an associative, distributive multiplication. An integral domain is a commutative ring with unity and no zero divisors. A field goes further: every nonzero element has a multiplicative inverse — so $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$ are fields, but $\mathbb{Z}$ is only an integral domain.

Number Theory

Modular arithmetic and the classic theorems show up in fast computation questions. The Euclidean algorithm finds $\gcd$ by repeated division. Fermat’s little theorem reduces big powers: if $p$ is prime and $p \nmid a$, then $a^{p-1} \equiv 1 \pmod{p}$.

5 · Real Analysis, Topology & Complex Variables

Part of the ~25% additional-topics area. The rigorous side of calculus (analysis), the abstract study of open and closed sets (topology), and functions of a complex variable.^[1]

Real Analysis

Analysis makes the limit precise. The epsilon–delta definition says $\lim_{x\to a} f(x) = L$ when, for every $\varepsilon > 0$, some $\delta > 0$ forces $|f(x) - L| < \varepsilon$ whenever $0 < |x - a| < \delta$. Key results: the Bolzano–Weierstrass theorem (every bounded sequence has a convergent subsequence) and that $\mathbb{R}$ is complete (Cauchy sequences converge).

Point-Set Topology

A set is open if every point has a neighborhood inside it, and closed if its complement is open. A compact set is one where every open cover has a finite subcover; in $\mathbb{R}^n$, the Heine–Borel theorem makes that the same as closed and bounded. The continuous image of a compact set is compact, and of a connected set is connected.

Complex Variables

A complex number $z = a + bi$ has modulus $|z| = \sqrt{a^2 + b^2}$ and polar form $re^{i\theta}$ (Euler’s formula). A function is analytic where it satisfies the Cauchy–Riemann equations, and Cauchy’s integral theorem says $\oint_C f(z)\,dz = 0$ for an analytic $f$ on and inside a simple closed contour.

6 · Probability, Statistics & Discrete Math

The rest of the ~25% additional-topics area. Counting, probability, the common distributions, and discrete-math staples. These are some of the most computational — and most learnable — questions on the test.^[1]

Counting & Combinatorics

Permutations count ordered arrangements, $P(n,k) = \frac{n!}{(n-k)!}$; combinations count unordered choices, $\binom{n}{k} = \frac{n!}{k!(n-k)!}$. For arrangements with repeated items, divide by the factorials of the repeats.

Probability

For equally likely outcomes, $P(E) = \frac{\text{favorable}}{\text{total}}$. Know the core rules:

Statistics & Distributions

The expected value $E[X] = \sum x_i p_i$ is the long-run average; the variance $\operatorname{Var}(X) = E[X^2] - (E[X])^2$ measures spread. Expectation is linear even for dependent variables. Know the named distributions:

How to Use This Study Guide

A study guide is a map, not the whole territory — use it alongside the official ETS GRE Mathematics Test Practice Book, which has a full-length real form with an answer key.^[2] Because calculus is half the exam, spend the most time in Modules 1 and 2; they have the highest return. The test is fast — about 2.5 minutes per question — so practice for speed and breadth, not just accuracy, and remember that scoring is rights-only, so never leave a question blank.

GRE Math Glossary

Quick definitions for the theorems and terms you’ll see most across the GRE Mathematics Subject Test:

Cauchy–Riemann equations

The conditions $\partial u/\partial x = \partial v/\partial y$ and $\partial u/\partial y = -\partial v/\partial x$ that an analytic complex function $f = u + iv$ must satisfy.

Chain rule

The rule for differentiating a composition: if $y = f(g(x))$, then $\frac{dy}{dx} = f'(g(x))\,g'(x)$.

Combination

An unordered selection of $k$ from $n$ items: $\binom{n}{k} = \frac{n!}{k!\,(n-k)!}$.

Compact set

A set in which every open cover has a finite subcover; in $\mathbb{R}^n$ (Heine–Borel) this is the same as being closed and bounded.

Determinant

A scalar attached to a square matrix; nonzero exactly when the matrix is invertible, and equal to the product of the eigenvalues.

Eigenvalue

A scalar $\lambda$ with $A\mathbf{v} = \lambda \mathbf{v}$ for some nonzero vector $\mathbf{v}$; found from $\det(A - \lambda I) = 0$.

Eigenvector

A nonzero vector whose direction is preserved (up to scaling) by a matrix: $A\mathbf{v} = \lambda \mathbf{v}$.

Euler's formula

$e^{i\theta} = \cos\theta + i\sin\theta$; the polar form of a complex number is $re^{i\theta}$.

Expected value

The long-run average of a random variable, $E[X] = \sum x_i p_i$ (or $\int x f(x)\,dx$).

Fermat's little theorem

If $p$ is prime and $p \nmid a$, then $a^{p-1} \equiv 1 \pmod{p}$.

Field

A commutative ring with unity in which every nonzero element has a multiplicative inverse — for example $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$.

Fundamental Theorem of Calculus

The theorem linking differentiation and integration: the derivative of $\int_a^x f(t)\,dt$ is $f(x)$, and $\int_a^b f(x)\,dx = F(b) - F(a)$ for any antiderivative $F$.

Gradient

The vector $\nabla f$ of partial derivatives of a scalar function; it points in the direction of steepest increase.

GRE Mathematics Subject Test

An ETS exam for prospective math (and quantitative-field) graduate students. About 66 multiple-choice questions over 2 hours 50 minutes, reported on a 200–990 scaled score. Calculus is about 50%, algebra 25%, and additional topics 25%.

Green's theorem

A theorem converting a line integral around a positively oriented simple closed curve into a double integral of $\partial Q/\partial x - \partial P/\partial y$ over the region it bounds.

Group

A set with an associative operation, an identity element, and inverses for each element. Abelian if the operation also commutes.

L'Hôpital's rule

A method for an indeterminate limit ($0/0$ or $\infty/\infty$): replace it with the limit of the ratio of derivatives, $\lim f/g = \lim f'/g'$.

Lagrange's theorem

In a finite group, the order of any subgroup divides the order of the group.

Normal subgroup

A subgroup $N$ with $gN = Ng$ for all $g$; exactly the condition needed to form the quotient group $G/N$.

p-series

A series $\sum 1/n^p$. It converges exactly when $p > 1$; at $p = 1$ it is the divergent harmonic series.

Permutation

An ordered arrangement of $k$ from $n$ items: $P(n,k) = \frac{n!}{(n-k)!}$.

Rank

The dimension of a matrix's column space — the number of pivots in row echelon form.

Rank–nullity theorem

For a linear map $T : V \to W$, $\operatorname{rank}(T) + \operatorname{nullity}(T) = \dim V$.

Ratio test

A convergence test: with $L = \lim |a_{n+1}/a_n|$, the series converges if $L < 1$, diverges if $L > 1$, and is inconclusive if $L = 1$.

Rights-only scoring

A scoring method that counts only correct answers, with no penalty for wrong ones — so you should answer every question, guessing when unsure.

Scaled score

The reported GRE Subject Test score, from 200 to 990 in 10-point increments, derived from the raw number-correct (rights-only) score. Each scaled score also carries a percentile rank.

Taylor series

An expansion of a function as $\sum \frac{f^{(n)}(a)}{n!}(x-a)^n$; a Maclaurin series is the case $a = 0$.

Variance

A measure of spread, $\operatorname{Var}(X) = E[X^2] - (E[X])^2$; the standard deviation is its square root.

Vector space

A set closed under vector addition and scalar multiplication that obeys the eight axioms (associativity, commutativity, a zero vector, inverses, and the distributive/identity laws).

Free GRE Math Study Materials & Resources

Everything you need to prepare for the GRE Mathematics Subject Test is free here — no paywall, no credit card. This guide is the foundation; pair it with the rest of our free GRE Math study materials for active recall and timed practice:

• GRE Math Practice Test — exam-style questions across calculus, algebra, and additional topics, with worked explanations.
• GRE Math Flashcards — 200 active-recall cards covering the high-yield theorems, definitions, and formulas.

GRE Math Study Guide FAQ

References

1. 1.ETS. “GRE Subject Tests — Mathematics Test Content and Structure.” ETS. ↑
2. 2.ETS. “GRE Mathematics Test Practice Book.” ETS. ↑
3. 3.ETS. “About the GRE Subject Tests.” ETS. ↑
4. 4.ETS. “GRE Subject Tests — Scores.” ETS. ↑

Sources for the concept answers

Every answer in the GRE Math concept questions above is drawn from an official ETS source:

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