DDe Moivre’s Theorem

A formula useful for finding powers and roots of complex numbers.

 

 

 Decagon

A polygon with ten sides.

 

Decagon Regular Decagon

Mathwords for ICP Program

 

Deciles

The 10th and 90th percentiles of a set of data.

Decreasing Function

A function with a graph that moves downward as it is followed from left to right. For example, any line with a negative slope is decreasing.

Note: If a function is differentiable, then it is decreasing at all points where its derivative is negative.

 

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.

Formally, a definite integral is the limit of a Riemann sum as the norm of the partition

approaches zero. That is, .

 

 

 Integral Rules

For the following, a, b, c, and C are constants; for definite integrals, these represent real number constants. The rules only apply when the integrals exist.

 

Indefinite integrals (These rules all apply to definite integrals as well)

Mathwords for ICP Program

1.

2.

3.

4.

5. Integration by parts:

 

Definite integrals

1.

2.

3. If f(u) ≤ g(u) for all a ≤ u ≤ b, then

4. If f(u) ≤ M for all a ≤ u ≤ b, then

5. If m ≤ f(u) for all a ≤ u ≤ b, then

6. If a ≤ b, then

 

Mathwords for ICP Program

Degenerate

An example of a definition that stretches the definition to an absurd degree.

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.

 

 

Degenerate Conic Sections

Plane figures that can be obtained by the intersection of a double cone with a plane passing through the apex. These include a point, a line, and intersecting lines. Like other conic sections, all degenerate conic sections have equations of the form Ax2 + Bxy + Cy2 + Dx + Ey + F = 0.

 

 

Degree of a Polynomial

The highest degree of any term in the polynomial.

 

Mathwords for ICP Program

Degree of a Term

For a term with one variable, the degree is the variable's exponent. With more than one variable, the degree is the sum of the exponents of the variables.

 

 

Del Operator

The symbol , which stands for the "vector" or .

 

Dependent Variable

A variable that depends on one or more other variables. For equations such as y = 3x – 2, the dependent variable is y. The value of y depends on the value chosen for x. Usually the dependent variable is isolated on one side of an equation. Formally, a dependent variable is a variable in an expression, equation, or function that has its value determined by the choice of value(s) of other variable(s).

Mathwords for ICP Program

 

 

Derivative

A function which gives the slope of a curve; that is, the slope of the line tangent to a function. The derivative of a function f at a point x is commonly written f '(x). For example, if f(x) = x3 then f '(x) = 3x2. The slope of the tangent line when x = 5 is f '(x) = 3·52 = 75.

 

Derivative of a Power Series

The derivative of a function defined by a power series can be found by differentiating the series term-by-term.

 

Mathwords for ICP Program

 

Derivative Rules

A list of common derivative rules is given below.

 

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.

Mathwords for ICP Program

 

Determinant

A single number obtained from a matrix that reveals a variety of the matrix's properties. Determinants of small matrices are written and evaluated as shown below. Determinants may also be found using expansion by cofactors.

Note: Although a determinant looks like an absolute value it is not. The determinant of a matrix may be negative or positive.

Diagonal Matrix

A square matrix which has zeros everywhere other than the main diagonal. Entries on the main diagonal may be any number, including 0.

 

Determinant

A single number obtained from a matrix that reveals a variety of the matrix's properties. Determinants of small matrices are written and evaluated as shown below. Determinants may also be found using expansion by cofactors.

Note: Although a determinant looks like an absolute value it is not. The determinant of a matrix may be negative or positive.

Mathwords for ICP Program

 

Diagonal of a Polygon

A line segment connecting non-adjacent vertices of a polygon. Note: An n -gon has diagonals.

 

 

Diameter of a Circle or Sphere

A line segment between two points on the circle or sphere which passes through the center. The word diameter is also also refers to the length of this line segment.

 

 

Diametrically Opposed

Mathwords for ICP Program

Two points directly opposite each other on a circle or sphere. More formally, two points are diametrically opposed if they are on opposite ends of a diameter.

 

 

 

Difference

The result of subtracting two numbers or expressions. For example, the difference between 7 and 12 is 12 – 7, which equals 5.

Sum/Difference Identities

Trig identities which show how to find the sine, cosine, or tangent of the sum or difference of two given angles.

 

Sum/Difference Identities

Difference Quotient

Mathwords for ICP Program

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.

 

 

Differentiable

A curve that is smooth and contains no discontinuities or cusps. Formally, a curve is differentiable at all values of the domain variable(s) for which the derivative exists.

Differential

An tiny or infinitesimal change in the value of a variable. Differentials are commonly written in the form dx or dy.

 

Differential Equation

An equation showing a relationship between a function and its derivative(s). For example,

is a differential equation with solutions y = Ce–x.

Differentiation

The process of finding a derivative.

 

Digit

Mathwords for ICP Program

Any of the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 used to write numbers. For example, the digits in the number 361 are 3, 6, and 1.

 

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).

Note: Some high school textbooks erroneously use the word dilation to refer to all transformations in which the figure changes size, whether the figure becomes larger or smaller. Unfortunately the English language has no word that refers collectively to both stretching and shrinking.

Pronunciation: Dilation (die-LAY-shun) has only three syllables, not four.

 

Dilation of a Geometric Figure

A transformation in which all distances are lengthened by a common factor. This is done by stretching all points away from some fixed point P.

 

 

 

StretchDilation of a Graph

Mathwords for ICP Program

A transformation in which all distances on the coordinate plane are lengthened by multiplying either all x-coordinates (horizontal dilation) or all y-coordinates (vertical dilation) by a common factor greater than 1.

Note: When the common factor is less than 1 the transformation is called a compression.

 

 

Dimensions

On the most basic level, this term refers to the measurements describing the size of an object. For example, length and width are the dimensions of a rectangle.

In a more advanced sense, the number of dimensions a set, region, object, or space possesses indicates how many mutually perpendicular directions of movement are possible. For example, a line is one dimensional, a plane is two dimensional, the space in which we live is three dimensional, and a point is zero dimensional.

 

Dimensions of a Matrix

The number of rows and columns of a matrix, written in the form rows×columns. The matrix below has 2 rows and 3 columns, so its dimensions are 2×3. This is read aloud, "two by three."

Note: One way to remember that Rows come first and Columns come second is by thinking of RC Cola®.

 

 

Mathwords for ICP Program

Direct ProportionDirect VariationDirectly Proportional

A relationship between two variables in which one is a constant multiple of the other. In particular, when one variable changes the other changes in proportion to the first.

If b is directly proportional to a, the equation is of the form b = ka (where k is a constant).

 

Equation: y = 4x

Variable y is directly proportional to x.

Doubling x causes y to double. Tripling x causes y to triple.

x y

1 4

2 8

3 12

 

 

Directrices of an Ellipse

Two parallel lines on the outside of an ellipse perpendicular to the major axis. Directrices can be used to define an ellipse. Formally, an ellipse is the locus of points such that the ratio of the distance to the nearer focus to the distance to the nearer directrix equals a constant that is less than one. This constant is the eccentricity.

 

 

 

Mathwords for ICP Program

Directrices of a Hyperbola

Two parallel lines which are perpendicular to the major axis of a hyperbola. The directrices are between the two parts of a hyperbola and can be used to define it as follows: A hyperbola is the locus of points such that the ratio of the distance to the nearer focus to the distance to the nearer directrix equals a constant that is greater than one. This constant is the eccentricity.

 

 

 

Directrix of a Parabola

A line perpendicular to the axis of symmetry used in the definition of a parabola. A parabola is defined as follows: For a given point, called the focus, and a given line not through the focus, called the directrix, a parabola is the locus of points such that the distance to the focus equals the distance to the directrix.

 

 

Mathwords for ICP Program

Example:  

This is a graph of the parabola with all its major features labeled: axis of symmetry, focus, vertex, and directrix.

 

Discontinuity

A point at which the graph of a relation or function is not connected. Discontinuities can be classified as either removable or essential. There are several kinds of essential discontinuities, one of which is the step discontinuity.

 

Discontinuous Function

A function with a graph that is not connected.

 

Mathwords for ICP Program

 

 

Discrete

A set with elements that are disconnected. The set of integers is discrete. The set of real numbers is not discrete; it is continuous.

Formally, a set of numbers is discrete if each number in the set is contained in a neighborhood that contains no other elements of the set.

 

Discriminant of a Quadratic

The number D = b2 – 4ac determined from the coefficients of the equation ax2 + bx + c = 0. The discriminant reveals what type of roots the equation has.

Note: b2 – 4ac comes from the quadratic formula.

 

 

Mutually ExclusiveDisjoint Events

Mathwords for ICP Program

Events that have no outcomes in common.

 

Disjoint SetsNon-Overlapping Sets

Two or more sets which have no elements in common. For example, the sets A = {a,b,c} and B = {d,e,f} are disjoint.

Disjunction

A statement which connects two other statements using the word or.

For example, "A polygon with four sides can be called a quadrilateral or a quadrangle" contains the disjunction "quadrilateral or quadrangle".

Disk

The union of a circle and its interior.

 

 

Disk Method

A technique for finding the volume of a solid of revolution. This method is a specific case of volume by parallel cross-sections.

 

Mathwords for ICP Program

 

Distance Formula

The formula is the distance between points (x1, y1) and (x2, y2).

 

Distance from a Point to a Line

The length of the shortest segment from a given point to a given line. A formula is given below.

 

 

 

Distinct

Different. Not identical.

DistributeExpand

Mathwords for ICP Program

To multiply out the parts of an expression. Distributing is the opposite of factoring.

Example 1: 3x(x + 8) = 3x·x + 3x·8                = 3x2 + 24x

Example 2: (x + 2)(x + 5) = x(x + 5) + 2(x + 5)                       = x·x + x·5 + 2·x + 2·5                       = x2 + 7x + 10

Example 3: (x – 3)4 = (x – 3)(x – 3)(x – 3)(x – 3)             = (x·x – x·3 – 3·x + 3·3)(x·x – x·3 – 3·x + 3·3)             = (x2 – 6x + 9)(x2 – 6x + 9)             = x2(x2 – 6x + 9) – 6x(x2 – 6x + 9) + 9(x2 – 6x + 9)             = x2·x2 – x2·6x + x2·9 – 6x·x2 + 6x·6x – 6x·9 + 9·x2 – 9·6x + 9·9             = x4 – 6x3 + 9x2 – 6x3 + 36x2 – 54x + 9x2 – 54x + 81             = x4 – 12x3 + 54x2 – 108x + 81

 

Distributing Rules

Algebra rules for distributing expressions. See factoring rules as well.

 

A. Multiplication

1. addition:  a(b + c) = ab + ac  and  (b + c)a = ba + ca

2. subtraction:  a(b – c) = ab – ac  and  (b – c)a = ba – ca

3. FOIL:  (a + b)(c + d) = ac + ad + bc + bd

Careful!!

a(bc) ≠ ab·ac

 

B. Division

Mathwords for ICP Program

1. addition:  

2. subtraction:  

Careful!!

 

C. Exponents   (a ≥ 0, b ≥ 0)

1. multiplication:  (ab)x = axbx

2. division:        (b ≠ 0)

Careful!!

(a + b)x ≠ ax + bx

(a – b)x ≠ ax – bx

 

D. Roots   (x ≥ 0, y ≥ 0)

1. multiplication:  

2. division:        ( y ≠ 0)

Careful!!

 

Mathwords for ICP Program

E. Logarithms   (x > 0, y > 0, a > 0, a ≠ 1)

1. multiplication:  loga (xy) = loga x + loga y

2. division:  

3. powers:  loga (xp) = p loga x

Careful!!

loga (x + y) ≠ loga x + loga y

loga (x – y) ≠ loga x – loga y

 

F. Trig

1.

2.

3.

 

Diverge

To fail to approach a finite limit. There are divergent limits, divergent series, divergent sequences, and divergent improper integrals.

Divergent Sequence

A sequence that does not converge. For example, the sequence 1, 2, 3, 4, 5, 6, 7, ... diverges since its limit is infinity (∞). The limit of a convergent sequence must be a real number.

 

Divergent Series

Mathwords for ICP Program

A series that does not converge. For example, the series 1 + 2 + 3 + 4 + 5 + ··· diverges. Its sequence of partial sums 1, 1 + 2, 1 + 2 + 3 , 1 + 2 + 3 + 4 , 1 + 2 + 3 + 4 + 5, ... diverges.

Dodecagon

A polygon with 12 sides.

 

Dodecagon Regular Dodecagon

 

Domain

The set of values of the independent variable(s) for which a function or relation is defined. Typically, this is the set of x-values that give rise to real y-values.

Note: Usually domain means domain of definition, but sometimes domain refers to a restricted domain.

 

Mathwords for ICP Program

DodecahedronRegular Dodecahedron

A polyhedron with 12 faces. A regular dodecahedron has faces that are all regular pentagons.

Note: It is one of the five platonic solids.

Regular Dodecahedron

a = length of an edge

Volume =

Surface Area =

Rotate me if your browser is

Java-enabled.

 

Domain of DefinitionNatural Domain

Alternate terms for domain used to make it clear that the domain being referred to is not a restricted domain.

 

Dot Product

In two dimensions, (ai + bj)•(ci + dj) = ac + bd. In three dimensions, (ai + bj + ck)•(di + ej + fk) = ad + be + cf. In either case, u • v = |u| |v| cos θ, where θ is the angle between the vectors.

 

Double Cone

A geometric figure made up of two right circular cones placed apex to apex as shown below. Typically a double cone is considered to extend infinitely far in both directions, especially when working with conic sections and degenerate conic sections.

Note: The graph of the equation z2 = x2 + y2 is a standard way to represent a double cone. That is the equation for the image below.

 

Mathwords for ICP Program

Double Cone

Double Angle IdentitiesDouble Number Identities

Trig identities that show how to find the sine, cosine, or tangent of twice a given angle.

 

Double Angle Identities

Doubling Time

For a substance growing exponentially, the time it takes for the amount of the substance to double.

 

Mathwords for ICP Program

 

Double Root

A root of a polynomial equation with multiplicity 2. Also refers to a zero of a polynomial function with multiplicity 2.

Mathwords for ICP Program