Arithmetical operations :  addition, congruence calculation, division, factorization, multiplication, power computation, root extraction, and subtraction.

Arithmetic Series :  A series in which the difference between any two consecutive terms is a constant.

Base :  The number of digits in a number system. The same word is used in the context of logarithms.

Cartesian Coordinates :  The usual coordinate system, originally described by Descartes, in which points are specified as distances to a set of perpendicular axes. Also called rectangular coordinates.

Decimal Expansion :  The usual "base 10" representation of a real number.

Factorial :  The product of the first n positive integers, denoted n!.

Fraction:  A rational number expressed in the form a/b, where a is known as the numerator and b as the denominator.

Function Graph :  The set of points showing the values taken by a function. This type of plot is called simply a "graph" in common parlance, but is distinct from a collection of points and lines that mathematicians refer to when they speak of a "graph."

Geometric Series :  A series in which the ratio of any two consecutive terms is always the same.

Greatest Common Divisor :  For two or more integers, the largest integer dividing all of them. Integer: One of the numbers ..., -2, -1, 0, 1, 2, ....

Intersection :  (1) For two sets A and B, the set of elements common to A and B. (2) For two or more geometric objects, the set of points that are common to both of them.

Interval :  A connected piece of the real number line. An interval can be open or closed at either end.

Irrational Number :  A number that cannot be written as a fraction. Irrational numbers have decimal expansions that neither terminate nor become periodic.

Least Common Multiple :  For two or more integers, the smallest number that is a multiple of all of them.

Polynomial :  A mathematical expression involving a sum of powers in one or more variables multiplied by coefficients.

Power :  An exponent to which a given quantity is raised.

Prime Factorization :  The factorization of a number into its constituent primes. Also called prime decomposition.

Prime Number :  A positive integer that has exactly one positive integer divisor other than 1 (i.e., no factors other than 1 and itself). Prime numbers are often simply called primes.

Pythagorean Theorem :  An equation relating the lengths of the sides of a right triangle. Given two sides, the length of the third can be determined.

Rational Number :  A real number that can be written as a quotient of two integers.

Real Number :  The set of all rational and irrational numbers.

Sequence : :  A (possibly infinite) ordered list of numbers.

Series :  An often infinite sum of terms specified by some rule.

Set :  A finite or infinite collection of objects in which order has no significance and multiplicity is generally also ignored.

Square Number :  An integer that is the square (i.e., second power) of another integer.

Square Root :  A square root of x is a number r such that r*r = x.

Aptitude arithmetic:  Average, problems on numbers and ages, surds and indices, percentage, profit and loss, ratio and proportion, time and work, time and distance, time and speed, alligation or mixture, word problems, simple interest, compound interest, rate.

Abstract Algebra :  The set of advanced topics in algebra that deal with abstract algebraic structures rather than the usual number systems.

Boolean Algebra :  An algebra where the multiplication and addition also satisfy the properties of the AND and OR operations from logic.

Isomorphism :  A map between mathematical objects such as groups, rings, or fields that is one-to-one, onto, and preserves the properties of the object.

Abelian Group :  A group for which the binary operation is commutative.

Cyclic Group :  An abstract group generated by a single element. Cyclic groups are always Abelian

Dihedral Group :  The symmetry group for a regular polygon.

Finite Group :  A group with a finite number of elements.

Group :  A set of elements and a binary operation that together satisfy the four fundamental properties of closure, associativity, the identity property, and the inverse property.

Group Action :  The association of each of a group's elements with a permutation of the elements of a set. The group is said to act on the set. This is done in a manner consistent with the group multiplication.

Group Representation :  A group action on a vector space.

Group Theory :  The study of abstract groups.

Normal Subgroup :  A subgroup that is fixed under conjugation by any element.

Simple Group :  A group whose only normal subgroups are the trivial subgroup of order one and the improper subgroup consisting of the entire original group.

Subgroup :  A subset of a group that is also a group.

Symmetric Group :  The group of all permutations of a given set.

Symmetry Group :  A group of symmetry-preserving operations, i.e., rotations, reflections, and inversions.

Rings and Fields Algebra :  (1) A subject taught in grade school and high school which includes the solution of polynomial equations in one or more variables, and basic properties of functions and graphs. It is sometimes referred to as "arithmetic." (2) Abstract Algebra. (3) A vector space that also possesses a vector multiplication.

Algebraic Number :  A number that is the root of some polynomial with integer coefficients. Algebraic numbers can be real or complex and need not be rational.

Field :  A ring in which every nonzero element has a multiplicative inverse. The real numbers and the complex numbers are both fields.

Finite Field :  A field with a finite number of elements. The number of elements in a finite field is always a power of a prime.

Gaussian Integer :  A complex number a+bi, where a and b are integers and i is the imaginary unit.

Ideal :  A subset of a ring that is closed under addition and multiplication by any element of the ring.

Module :  A generalization of a vector space in which the scalars form a ring rather than a field.

Quaternion :  A member of a four-dimensional noncommutative division algebra (i.e., a ring in which every nonzero element has a multiplicative inverse, but multiplication is not necessarily commutative) over the real numbers.

Ring :  An Abelian group together with a rule for multiplying its elements.

Geometric Mean :  A method of averaging n numbers by multiplying them together and taking the nth root (rather than the usual "arithmetic" method of adding and dividing).

Quadratic Equation :  A polynomial equation in a single variable containing the square of the variable, but no higher powers.

Subset :  A portion of a set. Subsets can also consist of the entire set or be empty.

Union :  For two sets A and B, the set of all elements appearing either in A or in B, including any duplicates.

Bernoulli Number :  One of a sequence of signed rational numbers that can be defined using a certain simple generating function. They are very important in number theory and analysis and commonly arise in series expansions of trigonometric functions.

Calculus of Variations :  A generalization of calculus that seeks to find the path, curve, surface, etc., for which a given function has a stationary value (which, in physical problems, is usually a minimum or maximum).

Cantor Set :  An example of an uncountable set of measure zero.

Convolution :  An integral that expresses the amount of overlap of one function g as it is shifted over another function f.

Delta Function :  A generalized function that has the property that the convolution integral of it with any function equals the value of that function at zero.

Fourier Series :  An expansion of a periodic function in terms of an infinite sum of sines and cosines.

Gamma Function :  An extension of the factorial to real and complex arguments.

Lebesgue Measure :  An extension of the classical notions of length and area to more complicated sets.

Functional Analysis :  A branch of mathematics concerned with infinite-dimensional vector spaces and mappings between them.

Hilbert Space :  A vector space that has a complete inner product. Hilbert spaces are important in the study of infinite-dimensional vector spaces.

Eigenvalue :  One of a set of special scalars associated with a linear system of equations that describe that system's fundamental modes. An eigenvector is associated with each eigenvalue.

Eigenvector :  One of a special set of vectors associated with a linear system of equations. An eigenvalue is associated with each eigenvector.

Euclidean Space :  The space of all n-tuples of real numbers. It is the generalization of the two dimensional plane and three dimensional space.

Inner Product :  (1) In a vector space, a way to multiply vectors together, with the result of this multiplication being a scalar. (2) A synonym for dot product.

Linear Algebra :  The study of linear systems of equations and their transformation properties.

Linear Transformation :  A function from one vector space to another. If bases are chosen for the vector spaces, a linear transformation can be given by a matrix.

Matrix :  A concise and useful way of uniquely representing and working with linear transformations. In particular, for every linear transformation, there exists exactly one corresponding matrix, and every matrix corresponds to a unique linear transformation. The matrix is an extremely important concept in linear algebra.

Matrix Inverse :  Given a matrix M, the inverse is a new matrix M-1 that when multiplied by M, gives the identity matrix.

Matrix Multiplication :  The process of multiplying two matrices (each of which represents a linear transformation), which forms a new matrix corresponding to the matrix representation of the two transformations' composition.

Norm :  A quantity that describes the length, size, or extent of a mathematical object.

Vector Space :  A set that is closed under finite vector addition and scalar multiplication. The basic example is n-dimensional Euclidean space.

Chain Rule :  A formula for the derivative of the composition of two functions in terms of their derivatives.

Continuous Function :  A function with no jumps, gaps, or undefined points.

Critical Point :  A point of a function's graph where the derivative is either zero or undefined.

Definite Integral :  An integral with upper and lower limits.

Derivative :  The infinitesimal rate of change in a function with respect to one of its parameters. The derivative is one of the key concepts in calculus.

Discontinuity :  A point at which a function jumps suddenly in value, blows up, or is undefined. The opposite of continuity.

Extreme Value Theorem :  The theorem that a continuous function on a closed interval has both a maximum and minimum value.

First Derivative Test :  A method for determining the maximum and minimum values of a function using its first derivative.

Fundamental Theorems of Calculus :  Deep results which express definite integrals of continuous functions in terms of antiderivatives.

Implicit Differentiation :  The procedure of differentiating an implicit equation (one which has not been explicitly solved for one of the variables) with respect to the desired variable, treating other variables as unspecified functions of it.

Indefinite Integral :  An integral without upper and lower limits.

Inflection Point :  A point on a curve at which the concavity changes.

Integral :  A mathematical object that can be interpreted as an area or a generalization of area. Integrals and derivatives are the fundamental objects of calculus.

Intermediate Value Theorem :  The theorem that if f is continuous on a closed interval [a, b], and c is any number between f(a) and f(b) inclusive, then there is at least one number x in [a, b] such that f(x)=c.

Limit :  The value a function approaches as the variable approaches some point. If the function is not continuous, the limit could be different from the value of the function at that point.

Maximum :  The largest value of a set, function, etc.

Mean-Value Theorem :  The theorem that if f(x) is differentiable on the open interval (a, b) and continuous on the closed interval [a, b], there is at least one point c in (a, b) such that (b - a) f'(c) = f(b) - f(a).

Minimum :  The smallest value of a set, function, etc.

Newton's Method :  An iterative method for numerically finding a root of a function.

Riemann Sum :  An estimate, using rectangles, of the area under a curve. An definite integral is defined as a limit of Riemann sums.

Second Derivative Test :  A method for determining a function's maxima, minima, and points of inflection by using its first and second derivatives

Convergent Series :  A series for which partial sums become arbitrarily close to some fixed number.

Exponential Growth :  The increase in a quantity according to an exponential function.

Harmonic Series :  The sum of the reciprocals of the positive integers. This series diverges.

Maclaurin Series :  A Taylor series expansion of a function around zero.

Power Series :  A sum of powers of a variable. A power series is essentially an infinite polynomial.

Radius of Convergence :  Half the width of the interval inside which a power series converges absolutely.

Ratio Test :  A test for determining whether a series converges. It is used for calculating the radius of convergence for power series.

Surface of Revolution :  A surface generated by rotating a two-dimensional curve about an axis.

Taylor Series :  The power series of a function around a given point.

Differential Equation :  An equation that involves the derivatives of a function as well as the function itself.

Euler Forward Method :  An numerical method for solving ordinary differential equations

Fourier Transform :  A generalization of complex Fourier series that expresses a function in terms of frequency components. Fourier transforms arise quite commonly not only in mathematics, but also in optics, signal processing, and many other areas of science and engineering.

Laplace Transform :  An integral transform perhaps second only to the Fourier transform in its utility in solving physical problems. The Laplace transform is particularly useful in solving linear ordinary differential equations such as those arising in the analysis of electronic circuits.

Ordinary Differential Equation :  An equality involving a function and its derivatives.

Partial Differential Equation :  An equation involving a function and its partial derivatives.

Second-Order Ordinary Differential Equation :  An ordinary differential equation that contains derivatives of second order but of no higher orders.

Separation of Variables :  A method of solving differential equations.

Slope Field :  A particular visualization of a linear, first-order differential equation in which the derivative at a given point is represented by a line segment of the corresponding slope.

Differential Geometry :  A field of mathematics that studies properties such as distance and curvature on surfaces and manifolds.

Differential k-Form :  A tensor of rank k that is antisymmetric under exchange of any pair of indices.

Gaussian Curvature :  One measure of the amount of "bending" a surface undergoes at a given point. The Gaussian curvature is independent of the coordinate system used to describe it.

Mean Curvature :  The amount of "bending" of a surface at given point defined as the average of the two so-called "principal curvatures."

Tangent Bundle :  Given a manifold, a new manifold which consists of the tangent spaces for each point pasted together in a continuous fashion.

Tensor :  A generalization of scalars, vectors, and matrices to an arbitrary number of indices.

Binary :  The "base 2" method of counting, in which only the digits 0 and 1 are used.

Logic :  The formal mathematical study of the methods, structure, and validity of mathematical deduction and proof.

CombinatoricsBinomial Coefficient :  The number of ways of picking k unordered outcomes from n possibilities, also known as a combination or combinatorial number.

Binomial Theorem :  A formula describing how to expand powers of a binomial (x+a)n using binomial coefficients.

Combinatorics :  The branch of mathematics studying the enumeration, combination, and permutation of sets of elements and the mathematical relations that characterize these properties.

Fibonacci Number :  A member of the Fibonacci sequence. The Fibonacci sequence is generated by beginning with 1, 1, 2, 3 and continuing so that subsequent terms are the sum of the two previous numbers.

Generating Function :  For a sequence, a power series whose coefficients are the members of that sequence.

Magic Square :  A square array of positive integers such that the sum of any row, column, or main diagonal equals that of any other.

Pascal's Triangle :  A triangular array of binomial coefficients that can visually illustrate several of their properties.

Permutation :  A rearrangement of the elements in an ordered list S into a one-to-one correspondence with S itself. Combinatorics studies the number of possible ways of doing this under various conditions.

Recurrence Relation :  A mathematical relationship expressing the members of a sequence as some combination of their predecessors.

Graph TheoryChromatic Number :  The smallest number of colors necessary to color the vertices of a graph or the regions of a surface such that no two adjacent vertices or regions are the same color.

Complete Graph :  A graph in which every pair of vertices is connected by an edge.

Connected Graph :  A graph for which there is a path between any pair of vertices.

Cycle Graph :  A graph containing a single cycle which passes through all its vertices.

Directed Graph :  A graph in which each edge is specified as going in a particular direction.

Graph :  A collection of points together with lines that connect some subset of the points.

Graph Cycle :  Any of a graph's edge-set subsets that forms a path, the first node of which is also the last.

Tree :  A graph that contains no cycles.

Geometry :  The branch of mathematics that studies figures, objects, and their relationships to each other. This contrasts with algebra, which studies numerical quantities and attempts to solve equations.

Similar :  A property of two figures whose corresponding angles are all equal and whose distances are all increased by the same ratio

High-Dimensional Solid :  A generalization of a solid such as a cube or a sphere to more than three dimensions.

Hypercube :  The generalization of a cube to more than three dimensions.

Hyperplane :  The generalization of a plane to more than two dimensions.

Hypersphere :  he generalization of a sphere to more than three dimensions.

Polytope :  A generalization of a polyhedron to more than three dimensions.

Plane GeometryAcute Angle :  An angle with measure less than ninety degrees.

Altitude :  A line segment from a vertex of a triangle which meets the opposite side at a right angle.

Angle :  The amount of rotation about the point of intersection of two lines or line segments that is required to bring one into correspondence with the other.

Area :  The amount of material that would be needed to "cover" a surface completely.

Circle :  The set of points in a plane that are equidistant from a given center point.

Complementary Angle :  A pair of angles whose measures add up to ninety degrees.

Geometric Construction :  A construction of a geometric figure using only straightedge and compass. Such constructions were studied by the ancient Greeks.

Golden Ratio :  Generally represented as phi. Given a rectangle having sides in the ratio 1:phi, partitioning the original rectangle into a square and new rectangle results in the new rectangle having sides with the ratio 1:phi. phi is approximately equal to 1.618.

Golden Rectangle :  A rectangle in which the ratio of the sides is equal to the golden ratio. Such rectangles have many visual properties and are widely used in art and architecture.

Hypotenuse :  The longest side of a right triangle (i.e., the side opposite the right angle).

Supplementary Angle :  For a given angle, the angle that when added to it totals 180 degrees.

Triangle Inequality :  The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

PolygonsEquilateral Triangle :  A triangle in which all three sides are of equal length. In such a triangle, the angles are all equal as well.

Parallelogram :  A quadrilateral with opposite sides parallel and therefore opposite angles equal.

Polygon :  A two-dimensional figure that consists of a collection of line segments, joined at their ends.

Quadrilateral :  A four-sided polygon.

Convex Hull :  For a set of points S, the intersection of all convex sets containing S.

Cross Section :  The plane figure obtained by a solid's intersection with a plane

Cube :  A Platonic solid consisting of six equal square faces that meet each other at right angles. It has eight vertices and twelve edges.

Cylinder :  A solid of circular cross section in which the centers of the circles all lie on a single line.

Dodecahedron :  A Platonic solid consisting of twelve pentagonal faces, 30 edges, and 20 vertices.

Icosahedron :  (1) A 20 sided polyhedron. (2) The Platonic solid consisting of twenty equilateral triangles.

Octahedron :  A Platonic solid consisting of 8 triangular faces, 8 edges, and 6 vertices.

Platonic Solid :  A convex solid composed of identical regular polygons. There are exactly five Platonic solids.

Polyhedron :  A three-dimensional solid that consists of a collection of polygons, joined at their edges.

Prism :  A polyhedron with two congruent polygonal faces and with all remaining faces parallelograms.

Pyramid :  A polyhedron with one face (known as the "base") a polygon and all the other faces triangles meeting at a common polygon vertex (known as the "apex").

Solid Geometry :  That portion of geometry dealing with solids, as opposed to plane geometry.

Sphere :  The set of all points in three-dimensional space that are located at a fixed distance from a given point.

Surface :  A two-dimensional piece of three-dimensional space.

Surface Area :  The area of a surface which lies in three-dimensional space, or the total area of all surfaces that bound a solid.

Tetrahedron :  A Platonic solid consisting of four equilateral triangles.

Volume :  The amount of space occupied by a closed three-dimensional object.

Congruence :  An equation in modular arithmetic, i.e., one in which only the remainders relative to some base, known as the "modulus," are significant.

Continued Fraction :  A real number as a nested fraction, useful in number theory.

Convergent :  (1) A partial sum of a continued fraction. (2) Tending towards some definite finite value.

>Diophantine Equation :  An equation for which only integer solutions are allowed.

Divisor Function :  (1) The number of distinct divisors of a given integer. (2) The sum of some fixed power of the divisors.

Elliptic Curve :  A curve defined by an irreducible cubic polynomial in two variables.

Euclidean Algorithm :  An algorithm for finding the greatest common divisor of two numbers.

Euler-Mascheroni Constant :  The limit of the difference between the nth partial sum of the harmonic series and the natural logarithm of n, and is approximately equal to 0.577.

Fermat's Last Theorem :  A famous discovery by Fermat that took 350 years to prove, that any number that is a power greater than two cannot be the sum of two like powers.

Number Theory :  A field of mathematics sometimes called "higher arithmetic" consisting of the study of the properties of integers. Primes and prime factorization are especially important concepts in number theory.

Partition :  A way of writing a whole number as a sum of positive integers in which the order of the addends is not significant.

Perfect Number :  A positive integer that equals the sum of its divisors.

Prime Counting Function :  Given x, the prime counting function returns the number of primes less than or equal to x.

Riemann Zeta Function :  A special function of mathematics and physics that is intimately related to deep results surrounding the prime number theorem.

Squarefree :  A positive integer is squarefree if it is not divisible by any perfect square greater than one.

Totient Function :  The number of positive integers less than or equal to a given number that are relatively prime to it.

Transcendental Number :  A number that is not the root of any polynomial with integer coefficients. The opposite of algebraic number.

Curve :  A continuous map from a one-dimensional space to an n-dimensional space. Loosely speaking, the word "curve" is often used to mean the function graph of a two- or three-dimensional curve.

Determinant :  1) A function that assigns a scalar to a square matrix (or, equivalently, its linear transformation). (2) The value of this function for a particular matrix. The matrix has an inverse if and only if its determinant is non-zero.

Parametric Equations :  A set of equations that together express a set of quantities as explicit functions of a number of independent variables, which are known as parameters.

Plane :  A two-dimensional surface defined by linear equations.

Plane Curve :  A curve that lies in a single plane. A plane curve may be closed or open.

Polar Coordinates :  A two-dimensional coordinate system in which points in two dimensions are given by an angle and a distance from the origin.

Rational Function :  A function that can be written as the quotient of two polynomials.

Scalar :  A value (such as a measurement) that has only magnitude but not direction. This contrasts with a vector, which has direction as well as magnitude.

Spherical Coordinates :  A coordinate system in which points in three-dimensional space are given by two angles and a distance from the origin.

Tangent Line :  A line that touches but does not cross a curve at a given point.

Translation :  A transformation consisting of a constant shift with no rotation or stretching.

Complex Conjugate :  The result of changing the sign of the imaginary part of a complex number.

Complex Number :  A number consisting of a real part and an imaginary part. A complex number is an element of the complex plane.

Complex Plane :  The set of all complex numbers. Just as all real numbers can be imagined as sitting on a line, all complex numbers can be thought of as points in a plane.

Conic Section :  The nondegenerate curves generated by the intersections of a plane with one or two nappes of a cone. A conic section can also be realized as the zero set of a quadratic equation in two variables.

Ellipse :  A conic section with eccentricity less than one. It resembles a squashed circle.

Hyperbola :  A conic section with eccentricity greater than one. A hyperbola consists of two separate branches.

Locus :  The set of all points (usually forming a curve or surface) satisfying some condition. For example, the locus of points in a plane that are equidistant from a given point is a circle.

Parabola :  A conic section with eccentricity equal to one. Parabolas appear as the graphs of quadratic equations and the trajectories of projectiles.

Exponents and Logarithmse :  The base of the natural logarithm, approximately equal to 2.718. After pi, e is the most important constant in mathematics.

Exponential Function :  The function consisting of the base of the natural logarithm e taken to the power of the variable.

Logarithm :  The power to which a number (called the base) is raised to produce a given number; e.g. The logarithm of 100 to the base 10 is 2.

Natural Logarithm :  The logarithm having base e.

FunctionsDomain :  (1) The set of values for which a function is defined. (2) In topology, a connected, open set.

Function :  A relation that uniquely associates members of one set with members of another set. The term "function" is sometimes implicitly understood to mean continuous function, linear function, or function into the complex numbers.

Inverse Function :  For a function f, the function f-1 for which f(f-1(x)) = x for any x.

Range :  (1) The set of all values that a function can take. (2) The difference between the minimum and the maximum values of a data set.

Cross Product :  A product of two vectors that results in a vector perpendicular to both.

Dot Product :  A product of two vectors, which results in a scalar.

Normal Vector :  A vector perpendicular to a surface.

Vector :  (1) A mathematical entity that has both magnitude (which can be zero) and direction. (2) An element of a vector space.

Cosine :  In a right triangle, the ratio of the length of a given angle's adjacent side to the length of the hypotenuse. The cosine function is one of the basic functions encountered in trigonometry (the others being the sine and tangent).

Double-Angle Formulas :  A set of formulas that express trigonometric functions of twice an angle in terms of the trigonometric functions of the original angle.

Half-Angle Formulas :  A set of formulas that express trigonometric functions of half an angle in terms of the trigonometric functions of the original angle.

Law of Cosines :  A formula relating the side lengths and angles of a general triangle. It is a generalization of the Pythagorean theorem.

Law of Sines :  For general triangles, a formula relating the ratios of sines of angles to the opposite side lengths.

Sine :  In a right triangle, the ratio of the length of a given angle's opposite side to the length of the hypotenuse. The sine function is one of the basic functions encountered in trigonometry (the others being the cosine and tangent).

Tangent :  In a right triangle, the ratio of the length of a given angle's opposite side to the length of the angle's adjacent side. The tangent function is one of the basic functions encountered in trigonometry (the others being the cosine and sine).

Trigonometric Addition Formulas :  A set of formulas that express trigonometric functions of sums of angles in terms of the trigonometric functions of the original angles.

Trigonometry :  The study of angles and of the angular relationships of planar and three-dimensional figures.

Unit Circle :  A circle of radius one.