Absolute Value of a Complex NumberModulus of a Complex NumberThe distance between a complex number and the origin on the complex plane

Affine Transformation A transformation which involves any combination of translations, reflections, stretches, shrinks, or rotations. Note: Collinearity and concurrency are invariant under affine transformations.Aleph Null cardinality of a countably infinite set.Alternating Series Remainder A quantity that measures how accurately the nth partial sum of an alternating series estimates the sum of the series.Annulus/Washer The region between two concentric circles which have different radii.Antipodal Points Two points directly opposite each other on a sphere. That is, two points on opposite ends of a sphere's diameter. Note: For a sphere, antipodal means the same thing as diametrically opposed.Argand Plane /Complex Plane The coordinate plane used to graph complex numbers. The x-axis is called the real axis and the y-axis is called the imaginary axis. The complex number x + yi is graphed as the point (x, y).Argument of a Complex Number/Polar Angle of a Complex Number The angle describing the direction of a complex number on the complex plane. The argument is measured in radians as an angle in standard position. For a complex number in polar form r(cos θ + isin θ) the argument is θ.Argument of a Function The variable, term or expression on which a function operates. For example, the argument of is x, the argument of sin(2A) is 2A, and the argument of e x – 5 is x – 5. The argument of f(x) is x.Argument of a Vector The angle describing the direction of a vector. The argument is measured as an angle in standard position.ASA Congruence Angle-side-angle congruence. When two triangles have corresponding angles and sides that are congruent as

Asymptote A line or curve that the graph of a relation approaches more and more closely the further the graph is followed.Augmented Matrix A matrix form of a linear system of equations obtained from the coefficient matrix as shown below. It is created by adding an additional column for the constants on the right of the equal signs. The new column is set apart by a vertical line.Average Value of a Function The average height of the graph of a function. For y = f(x) over the domain [a, b], the formula for average value is

given below.

Back-Substitution The process of solving a linear system of equations that has been transformed into row-echelon form or reduced row-echelon form. The last equation is solved first, then the next-to-last, etcBearing Two similar ways of indicating direction

Bernoulli Trials An experiment in which a single action, such as flipping a coin, is repeated identically over and over. The possible results of the action are classified as "success" or "failure". The binomial probability formula is used to find probabilities for Bernoulli trials.Note: With Bernoulli trials, the repeated actions must all be independent.Boundary Value ProblemBVP A differential equation or partial differential equation accompanied by conditions for the value of the function but with no conditions for the value of any derivatives.Note: Boundary value problem is often abbreviated BVP.Bounded Set of Geometric Points A figure or a set of points in a plane or in space that can be enclosed in a finite rectangle or box.Box-and-Whisker Plot A visual display of the five number summary. A simplified boxplot taught to beginners. It does not show outliers. The whiskers extending all the way to the minimum and maximum values regardless of how far out they may be.

Boxplot/Modified Boxplot A data display that shows the five-number summary. The whiskers, stretching outward from the first quartile and third quartile as shown below, are no longer than 1.5 times the interquartile range (IQR). Outliers beyond that are marked separately.Note: Beginners are sometimes taught to draw box-and-whisker plots, which do not show outliers. Modified boxplot is a name sometimes used for boxplots to distinguish them from box-and-whisker plots.

Brachistochrone A cycloid hanging downwards. Also a tautochrone, the bead will take the same amount of time to reach the bottom no matter how high or low the release point.

Cardinal Numbers The numbers 1, 2, 3, . . . as well as some types of infinity. Cardinal numbers are used to describe the number of elements in either finite or infinite sets.Cardioid A curve that is somewhat heart shaped. A cardioid can be drawn by tracing the path of a point on a circle as the circle rolls around a fixed circle of the same radius. The equation is usually written in polar coordinates.Note: A cardioid is a special case of the limaçon family of curves.Cardioid: r = a ± a cos θ (horizontal) or r = a ± a sin θ (vertical)

Cavalieri’s Principle A method, with formula given below, of finding the volume of any solid for which cross-sections by parallel planes have equal areas. This includes, but is not limited to, cylinders and prisms.Formula:Volume = Bh, where B is the area of a cross-section and h is the height of the solid.Ceiling Function/Least Integer Function A step function of x which is the least integer greater than or equal to x

Ceva's Theorem A theorem relating the way three concurrent cevians of a triangle divide the triangle's three sides.

Cevian A line segment, ray, or line that extends from a vertex of a triangle to the opposite side (which may be extended). Medians, altitudes, and angle bisectors are all examples

of cevians.Coincident Identical, one superimposed on the other. That is, two or more geometric figures that share all points. For example, two coincident lines would look like one line since one is on top of the other.Collinear Lying on the same line.Conjugate Pair Theorem An assertion about the complex zeros of any polynomial which has real numbers as coefficients.Consistent System of Equations system of equations that has at least one solution.

Absolute Convergence/Absolutely Convergent Describes a series that converges when all terms are replaced by their absolute values. To see if a series converges absolutely, replace any subtraction in the series with addition. If the new series converges, then the original series converges absolutely.Conditional Convergence Describes a series that converges but does not converge absolutely. That is, a convergent series that will become a divergent series if all negative terms are made positive. Convergent Sequence A sequence with a limit that is a real number. For example, the sequence 2.1, 2.01, 2.001, 2.0001, . . . has limit 2, so the sequence converges to 2. On the other hand, the sequence 1, 2, 3, 4, 5, 6, . . . has a limit of infinity (∞). This is not a real number, so the sequence does not converge. It is a divergent sequence.Cusp A sharp point on a curve. Note: Cusps are points at which functions and relations are not differentiable.

De Moivre’s Theorem A formula useful for finding powers and roots of complex numbers.Deciles The 10th and 90th percentiles of a set of data.Definite Integral An integral which is evaluated over an interval. A definite

integral is written . Definite integrals are used to find the area between the graph of a function and the x-axis. There are many other applications.Degenerate A degenerate triangle is the "triangle" formed by three collinear points. It doesn’t look like a triangle, it looks like

a line segment.A parabola may be thought of as a degenerate ellipse with one vertex at an infinitely distant point.Degenerate examples can be used to test the general applicability of formulas or concepts. Many of the formulas developed for triangles (such as area formulas) apply to degenerate triangles as well.

Deleted Neighborhood The proper name for a set such as {x: 0 < |x – a| < δ}. Deleted neighborhoods are encountered in the study of limits. It is the set of all numbers less than δ units away from a, omitting the number a itself.

Descartes' Rule of Signs A method for determining the maximum number of positive zeros for a polynomial. This maximum is the number of sign changes in the polynomial when written as shown below.Difference Quotient For a function f, the formula . This formula computes the slope of the secant line through two points on the graph of f. These are the points with x-coordinates x and x + h. The difference quotient is used in the definition the derivative.Dihedral Angle An angle formed by intersecting planes.Dilation A transformation in which a figure grows larger. Dilations may be with respect to a point (dilation of a geometric figure) or with respect to the axis of a graph (dilation of a graph).Diverge To fail to approach a finite limit.Equivalence Relation Any relation that satisfies the reflexive, symmetric, and transitive properties. For example, modular equivalence is an equivalence relation. So is cardinality of a set.Explicit Function A function in which the dependent variable can be written explicitly in terms of the independent variable.For example, the following are explicit functions: y = x2 – 3, , and y = log2 x.Extreme Value Theorem/Min/Max Theorem A theorem which guarantees the existence of an absolute max and an absolute min for any continuous function over a closed interval.

Extreme Values of a Polynomial The graph of a polynomial of degree n has at most n – 1

extreme values (minima and/or maxima). The total number of extreme values could be n – 1 or n – 3 or n – 5 etc.Extremum An extreme value of a function. In other words, the minima and maxima of a function. Extrema may be either relative (local) or absolute (global).Note: The first derivative test and the second derivative test are common methods used to find extrema.Fibonacci Sequence The sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, . . . for which the next term is found by adding the previous two terms. This sequence is encountered in many settings, from population models to botany.Note: The sequence of ratios of consecutive terms has the Golden Mean as its limit.Reflection/Flip A transformation in which a geometric figure is reflected across a line, creating a mirror image. That line is called the axis of reflection.Fractal A figure that is self-similar. That is, no matter how far you zoom in on the figure, the portion you look at is similar to the original figure. The Koch edge, below, is a fractal.Note: The word fractal is often used loosely to describe figures that do not quite meet this definition.

Fundamental Theorem of Algebra The theorem that establishes that, using complex numbers, all polynomials can be factored. A generalization of the theorem asserts that any polynomial of degree n has exactly n zeros, counting multiplicity.Fundamental Theorem of Algebra:A polynomial p(x) = anxn + an–1xn–1 + ··· + a2x2 + a1x + a0 with degree n at least 1 and with coefficients that may be real or complex must have a factor of the form x – r, where r may be real or complex.

Fundamental Theorem of Arithmetic The assertion that prime factorizations are unique. That is, if you have found a prime factorization for a positive integer then you have found the only such factorization. There is no different factorization lurking out there somewhere.Fundamental Theorem of Calculus The theorem that establishes the connection between derivatives, antiderivatives, and definite integrals. The fundamental theorem of calculus is typically given in two parts.

Gaussian Integer A complex number of the form a + bi for which both a and b are integers. For example, 2 + 3i, 8 – 7i, –5, and 12i are all Gaussian integers.Glide Reflection The transformation that is a combination of a reflection and a translation. 

Golden MeanGolden Ratio The number  , or about 1.61803. The Golden Mean arises in many settings, particularly in connection with the Fibonacci sequence. Note: The reciprocal of the Golden Mean is about 0.61803, so the Golden Mean equals its reciprocal plus one. It is also a root of x2 – x – 1 = 0.Note: The Greek letter phi, φ, is often used as a symbol for the Golden Mean. Occasionally the Greek letter tau, τ, is used as well.

Golden Rectangle Golden Rectangle

A rectangle which has its ratio of length to width equal to the Golden Mean. This is supposedly the rectangle which is most pleasing to the eye.

 Golden Spiral A spiral that can be drawn in a golden rectangle as shown below. The figure forming the structure for the spiral is made up entirely of squares and golden rectangles. 

Googol The number 10100. This number can be written as a 1 followed by 100 zeros.Googolplex The number 10googol, or 1 followed by a googol number of zeros. This is reputed to be the largest number with a name.Note: This can also be written 10(10^100). Third QuartileHigh QuartileHigher QuartileQ3

For a set of data, a number for which 75% of the data is less than that number. The third quartile is the same as the median of the part of the data which is greater than the median. Same as 75th percentile. 

Removable DiscontinuityHole A hole in a graph. That is, a discontinuity that can be "repaired" by filling in a single point. In other words, a removable discontinuity is a point at which a graph is not connected but can be made connected by filling in a single point.Formally, a removable discontinuity is one at which the limit of the function exists but does not equal the value of the function at that point; this may be because the function does not exist at that point. 

Inconsistent System of Equations Note: Attempts to solve inconsistent systems typically result in impossible statements such as 0 = 3. Indefinite Integral The family of functions that have a given function as a common derivative. The indefiniteintegral of f(x) is

written∫ f(x) dx. 

Independent Variable

  Induction A method for proving a proposition that is valid for infinitely many different values of avariable. For example, it can be used to prove the formula 1 + 2 + 3 + 4 + . . . + n =  . Infinitesimal A hypothetical number that is larger than zero but smaller than any positive real number. Although the existence of such numbers makes no sense in the real number system, many worthwhile results can be obtained by overlooking this obstacle.Note: Sometimes numbers that aren't really infinitesimals are called infinitesimals anyway. The word infinitesimal is occasionally used for tiny positive real numbers that are nearly equal to zero. Inflection Point A point at which a curve changes from concave up to concave down, or vice-versa.Note: If a function has a second derivative, the value of the second derivative is either 0 or undefined at each of that function's inflection points.

Intermediate Value TheoremIVT

A theorem verifying that the graph of a continuous function is connected. 

Interval of Convergence For a power series in one variable, the set of values of the variable for which the seriesconverges. The interval of convergence may be as small as a single point or as large as the set of all real numbers. 

Invariant A word describing of a property which can not be changed by a given transformation.

 Invertible MatrixNonsingular Matrix

A square matrix which has an inverse. A matrix is nonsingular if and only if its determinantdoes not equal zero.Step DiscontinuityJump Discontinuity Step Discontinuity

Jump Discontinuity

A discontinuity for which the graph steps or jumps from one connected piece of the graph to another. Formally, it is a discontinuity for which the limits from the left and right both exist but are not equal to each other.

Lemma Lemma

A helping theorem. A lemma is proven true, just like a theorem, but is not interesting or important enough to be a theorem. It is of interest only because it is a stepping stone towards the proof of a theorem.

Lemniscate A curve usually expressed in polar coordinates that resembles a

figure eight.

Limaçon A famliy of related curves usually expressed in polar coordinates. The cardioid is a special kind of limaçon. Limaçon:   r = b + a cos θ (horizontal, pictured below)  or  r = b + a sin θ (vertical)

Note: If a = b the curve is a cardioid.Maclaurin Series The power series in x for a function f(x). 

Expected ValueMean of a Random Variable A quantity equal to the average result of an experiment after a large number of trials. For example, if a fair 6-sided die is rolled, the expected value of the number rolled is 3.5. This is a correct interpretation even though it is impossible to roll a 3.5 on a 6-sided die. This sort of thing often occurs with expected values. 

Mean Value Theorem A major theorem of calculus that relates values of a function to a value of its derivative. Essentially the theorem states that for a "nice" function, there is a tangent line parallel to anysecant line. 

Mean Value Theorem for Integrals

A variation of the mean value theorem which guarantees that a continuous function has at least one point where the function equals the average value of the function. 

Menelaus’s TheoremTheorem of Menelaus

A theorem relating the way two cevians of a triangle divide each other and two of the triangle'ssides. 

 

Extreme Value TheoremMin/Max Theorem

A theorem which guarantees the existence of an absolute max and an absolute min for anycontinuous function over a closed interval. 

Möbius StripMobius StripMoebius Strip

A one-sided surface pictured below. A model of a Möbius strip model can be made by taking a strip of paper and taping the two ends together with a half-turn in the middle. Note: In addition to having only one "face", a Möbius strip also only has one "edge". 

 Modular Arithmetic Regular addition, subtraction, and multiplication, but with the answer given modulo   n . 

Modular Equivalence Two integers are equivalent mod n if they leave the same remainder when divided by n.For example, 5 and 17 are equivalent mod 4 since they both have remainder 1 when divided by 4. We write5 ≡ 17 (mod 4). 

Modulo nModular Numbers The value of an integer modulo n is equal to the remainder left when the number is divided byn. Modulo n is usually written mod n. Modus Ponens A logical argument of this form: 

  

Modus Tolens A logical argument of the form shown below. This is essentially the argument employed inproof by contradiction. 

Multivariable CalculusMultivariable AnalysisVector CalculusThe use of calculus (limits, derivatives, and integrals) with two or more independent variables, or two or more dependent variables. This can be thought of as the calculus of three dimensional figures .Common elements of multivariable calculus include parametric equations, vectors, partial derivatives, multiple integrals, line integrals, and surface integrals. Most of multivariable calculus is beyond the scope of this website. 

Negatively Associated Data A relationship in paired data in which one variable's values tend to increase when the other decreases, and vice-versa. In a scatterplot, negatively associated data tend to follow a pattern from the upper left to the lower right. Negatively associated data have a negative correlation coefficient. 

Neighborhood A neighborhood of a number a is any open interval containing a. One common notation for a neighborhood of a is {x: |x – a| < δ}. Using interval notation this would be (a – δ, a + δ). Newton's Method An iterative process using derivatives that can often (but not always) be used to find zeros of adifferentiable function. The basic idea is to start with an approximate guess for the zero, then use the formula below to turn that guess into a better approximation. This process is repeated until, after only a few steps, the approximation is extremely close to the actual value of the zero.Non-Euclidean Geometry Any system of geometry in which the parallel postulate does not hold. Two commonly studied non-Euclidean geometries are hyperbolic

geometry and elliptic geometry. Elliptic geometry is also known as Riemannian geometry.Nontrivial A solution or example that is not trivial. Often, solutions or examples involving the numberzero are considered trivial. Nonzero solutions or examples are considered nontrivial.Mesh of a PartitionNorm of a Partition The width of the largest sub-interval in a partition.

Taylor Polynomialnth Degree Taylor Polynomial

An approximation of a function using terms from the function's Taylor series. An nth degree Taylor polynomial uses all the Taylor series terms up to and including the term using the nth derivative. 

 One-to-One Function A function for which every element of the range of the function corresponds to exactly one element of the domain. One-to-one is often written 1-1.Note: y = f(x) is a function if it passes the vertical line test. It is a 1-1 function if it passes both the vertical line test and the horizontal line test. Another way of testing whether a function is 1-1 is given below. 

Overdetermined System of Equations  A linear system of equations in which there are more equations than there are variables. For example, a system with three equations and only two unknowns is overdetermined. Note that an overdetermined system might be either consistent or inconsistent, depending on the equations.Pappus’s TheoremTheorem of Pappus

A method for finding the volume of a solid of revolution. The volume equals the product of the area of the region being rotated times the distance traveled by the centroid of the region in one rotationParametrize To write in terms of parametric equations. 

Example:The line x + y = 2 can be parametrized as x = 1 + t, y = 1 –t.Piecewise Continuous Function A function made up of a finite number of continuous pieces. Piecewise continuous functions may not have vertical asymptotes. In fact, the only possible types of discontinuities for a piecewise continuous function are removable and step discontinuities. 

Pinching TheoremSandwich TheoremSqueeze Theorem

A theorem which allows the computation of the limit of an expression by trapping the expression between two other expressions which have limits that are easier to compute. Power Series A series which represents a function as a polynomial that goes on forever and has no highestpower of x. 

 Power Series Convergence A theorem that states the three alternatives for the way a power series may converge. 

  Quartiles The collective term for the first quartile and third quartile of a set of data. That is, the 25th and 75th percentiles. Quintic Polynomial A polynomial of degree 5.Examples: x5 – x3 + x, y5 + y4 + y3 + y2 + y + 1, and

42a3b2. Quintiles The 20th and 80th percentiles of a set of data. Quintuple Multiple of FiveRadius of Convergence The distance between the center of a power series' interval of convergence and its endpoints. If the series only converges at a single point, the radius of convergence is 0. If the series converges over all real numbers, the radius of convergence is ∞. 

 Rational Root TheoremRational Zero Theorem A theorem that provides a complete list of possible rational roots of the polynomial equationanxn + an–1xn–1 + ··· + a2x2 + a1x + a0 = 0 where all coefficients are integers.This list consists of all possible numbers of the form c/d, where c and d are integers. c must divide evenly into the constant term a0. d must divide evenly into the leading coefficient an. 

 Recursive Formula For a sequence a1, a2, a3, . . . , an, . . . a recursive formula is a formula that requires thecomputation of all previous terms in order to find the value of an .Note: Recursion is an example of an iterative procedure. 

Remainder of a Series The difference between the nth partial sum and the sum of a series. 

 Elliptic GeometryRiemannian Geometry A non-Euclidean geometry in which there are no parallel lines. This geometry is usually thought of as taking place on the surface of a sphere. The "lines" are great circles, and the "points" are pairs of diametrically opposed points. As a result, all "lines" intersect.

Rolle's Theorem A theorem of calculus that ensures the existence of a critical point between any two points on a "nice" function that have the same y-value. 

 Rose Curve A smooth curve with leaves arranged symmetrically about a common center.Note: The examples below all have polar equations using cosine. Sine may be used as well. 

 

Row-Echelon Form of a MatrixEchelon Form of a Matrix A matrix form used when solving linear systems of equations. 

 Pinching TheoremSandwich TheoremSqueeze TheoremA theorem which allows the computation of the limit of an expression by trapping the expression between two other expressions which have limits that are easier to compute. 

Scatterplot A graph of paired data in which the data values are plotted as (x, y) points. 

Simpson's Rule A method for approximating a definite integral   using parabolic approximations of f. The parabolas are drawn as shown below.Skew Lines Lines in three dimensional space that do not intersect and are not parallel. 

 Extraneous SolutionSpurious Solution A solution of a simplified version of an equation that does not satisfy the original equation. Watch out for extraneous solutions when solving equations with a variable in the denominatorof a rational expression, with a variable in the argument of a logarithm, or a

variable as theradicand in an nth root when n is an even number. Surd An irrational number that can be expressed as a radical, such as   or  . Surface of Revolution A surface that is obtained by rotating a plane curve in space about an axis coplanar to the curve.Tautochrone A cycloid hanging downwards.Taylor Series The power series in x – a for a function f . Note: If a = 0 the series is called a Maclaurin series. 

 

Taylor Series Remainder A quantity that measures how accurately a Taylor polynomial estimates the sum of a Taylor series. 

Tessellate To cover a plane with identically shaped pieces which do not overlap or leave blank spaces. The pieces do not have to be oriented identically. A tessellation may use tiles of one, two, three, or any finite number of shapes. 

 Trichotomy The property of real numbers which guarantees that for any two real

numbers a and b, exactly one of the following must be true: a < b, a = b, or a > b.

Trivial

A solution or example that is ridiculously simple and of little interest. Often, solutions or examples involving the number 0 are considered trivial. Nonzero solutions or examples are considered nontrivial.

For example, the equation x + 5y = 0 has the trivial solution x = 0, y = 0. Nontrivial solutions include x = 5, y = –1 and x = –2, y = 0.4.

Vinculum The horizontal line drawn as part of a fraction or radical, such as   or  .Note: The vinculum serves the same function as parentheses, so we do not have to write  or  .