paulpatenaude | Lexique de mathématique

Simple Chain

Chain that does not use the same edge twice. Example In this graph, the chain A-E-F-C-B-A is a simple chain. In the graph below, the chain made up of edges a, b, c, f, j and h is a chain of length 6.

Size of the neighbourhood of this vertex, or the number of edges incident to this vertex. If a vertex includes a loop, this loop counts twice because its two end points are incident to the same vertex. Examples The degree of the vertex F in this graph is 3. The degree of the vertex A [...]

Order of a Vertex in a Graph

Number of edges incident to a vertex in graph. Synonym for degree of a vertex. Example In this graph, the vertex F is of order 3 and vertex B is of order 2.

Order of a Graph

Number of vertices in a graph. Example This is a graph of order 6.

Eulerian Chain

Simple chain that passes through all of the edges in an undirected graph. Property A graph contains a Eulerian chain if the graph is connected and if it has two vertices of odd order. Example In this graph, there is no Eulerian chain because there are more than two vertices of odd order. The graph [...]

Elementary Chain

Chain that does not pass through the same vertex twice. Example In this graph, the chain A-B-C-F-D is an elementary chain.

Great Circle of a Sphere

A circle in a sphere, that has the same centre and the same radius as the sphere. A great circle of a sphere has the same diameter as the sphere. A great circle of a sphere is the boundary of the intersection of the sphere and a plane that passes through the centre of the [...]

Unit Circle

Circle in which the radius is equal to 1 unit. Synonym for trigonometric circle. Property Because the radius of a unit circle is 1, its circumference is equal to \(2π\) and its area is equal to \(π\).

Centre of a Sphere

A point at an equal distance from all the points on the sphere. Example The point O is the centre of a sphere of radius r.

Centre of a Hyperbola

Midpoint of the segment joining the foci of a hyperbola. The centre of a hyperbola is also the intersection point of its lines of symmetry and its asymptotes. Example The point O is the centre of the hyperbola.

Centre of a Regular Polygon

Meeting point of the perpendicular bisectors of the sides of a polygon. Property The centre of a regular polygon is the centre of the inscribed circle and the centre of the circumscribed circle of the polygon. Examples The centre O of various regular polygons is shown below: The point O is the centre of the [...]

Centre of a Line Segment

Point on a line segment that is halfway between the two endpoints of the segment. Synonym of midpoint of a segment.

Median of a Quadrilateral

Segment that joins the midpoints of two opposite sides of a quadrilateral. Property The medians of a quadrilateral intersect at their midpoint. Examples The segments MP and NQ are medians of the quadrilateral ABCD. The segments AB and CD are the medians of rectangle ABCD. The segment AB is the median of the trapezoid CDEF [...]

Centre of Symmetry of a Geometric Figure

Point in relation to which all the points in the figure are symmetric in pairs. Therefore, the centre of symmetry of a figure is the centre of a point symmetry that overlays the figure onto itself. Examples Each endpoint of a segment whose endpoints are on the edge of the polygon and that passes through [...]

Centre of an Arc of a Circle

Point at an equal distance from all the points on an arc of a circle. The centre of an arc of a circle is not a point on the arc, just as the centre of a circle is not a point on the circle. Example In this figure, point O is the centre of the [...]

Percentile Rank

Measure of the position of a datum in an ordered statistical series divided into 100 equal parts. When we say that a student’s percentile rank is 85, this means that: 1% of the students in her group have the same percentile rank as her, 84% of students in her group belong to intervals that are [...]

Multiplication Table

Table of numbers, entered twice, representing all of the products of the whole numbers from 0 to 10 or from 0 to 12. Example Educational Note Memorizing the multiplication table in the decimal numeral system allows you to multiply any two numbers using only the knowledge of the products of two numbers from 0 to [...]

Divisibility Rule

Rule that helps to determine if a number is divisible by another number without needing to carry out the division. The properties of a number that allow us to determine if this number is divisible by another number are properties that characterize this divisibility. Because these are also conditions of divisibility, we also talk about [...]

Vertex of an Angle

The point where two sides of an angle meet. Example Angle AOB has vertex O.

Amplitude of a Sinusoidal Function

In a sinusoidal function defined in its parametric form, which is \(f(x)=a·\sin (b(x-h))+k\), the amplitude A of the function is provided by the absolute value of the parameter a : A = |a|. Examples In this graph, the function defined by f(x) = 2 sin(x) has an amplitude of 2.

Base of a Geometric Solid

Specific face of a solid identified for the purpose of providing a measure or a definition. Certain solids do not have a base in the strict sense of the word. This is the case with spheres, most Platonic solids, bipyramids, etc. Examples In this illustration, the polygon on which each triangular lateral face rests is [...]

Surface Area of a Solid

The sum of the areas of all the faces of a solid. The following distinction is made between the lateral area and the total area of a solid: Lateral area of a solid: The sum of the areas of the lateral faces of some solids. Total area of a solid: The sum of the areas [...]

Area of a Polygon

The measurement of a two-dimensional region enclosed by a polygonal chain. Formulas The following formulas can be used to find the area A of some common polygons: Area of a square A = s\(^2\) s: length of one side Area of a rhombus A = \(\frac {d\space ×\space D}{2}\) d: length of the shorter diagonal D: [...]

Area of a Circle

The measurement of the surface enclosed by a circle. Formulas A = \(πr^{2}\) or A = \(π\frac{d^{2}}{4}\) Example Consider a circle with a radius of 10 cm. \(A = πr^{2} = π × 10^{2} = 100π\). \(A ≈ 314.16\)

Order of a Bipyramid

Number of sides of the polygonal base of the two pyramids that form a bipyramid. Examples A regular octahedron is a bipyramid of order 4. A hexagonal bipyramid is a bipyramid of order 6:

Addition Of Irrational Numbers

Addition under which any pair \((a, b)\) of irrational numbers is made to correspond to an irrational number \((a + b)\) called the sum of a and b. Because an irrational number is expressed by an unlimited, non-repeating decimal sequence, we cannot express the sum of two irrational numbers in the form of a repeating [...]

Degree of a Polynomial

The degree of a polynomial is the highest degree of any of its monomials. Examples The degree of the monomial 10a7b3c2 is 12, since: 7 + 3 + 2 = 12. The degree of the polynomial 4x² − 7x + 15 is 2. The degree of the polynomial 6x3y2 − x2y4 + 2xy + 3 is 6.

Order of a Polynomial

Synonym for degree of a polynomial..

Coefficient of Determination

Measure of the quality of the prediction of a linear regression. This coefficient varies between 0 and 1, between a low predictive power and a high predictive power. The coefficient of determination (R², the square of the coefficient of the linear correlation r) is an indicator that makes it possible to judge the quality of a [...]

Coefficient of Linear Correlation

Numerical value that characterizes the link – sense and importance – that exists between two random variables or two statistical variables. This value can be positive, negative, or zero. It should be noted that the value of the coefficient is zero when the two variables in question are independent. Examples This scatter plot illustrates a [...]

Addition of Rational Numbers

An operation in which any pair \((a, b)\) of rational numbers is made to correspond to a rational number \((a + b)\) called the sum of a and b. We can always express the sum of two rational number in the form of a repeating decimal sequence, which is not the case for irrational numbers. [...]

Square Brackets

Symbol used to denote intervals. Square brackets are always used in pairs. Sometimes, square brackets are used as a symbol to group operations, like this: 8 + 5[4 + 3(8 – 4) – 2 (9 – 8) + 6] – 12 = 96 Notation The square bracket symbol is "[ ]". Examples The interval [3, [...]

Parentheses

Grouping symbols frequently used to isolate operations. Parentheses are always used in pairs. In a sequence of operations, we solve the operations in parentheses first. Parentheses help to set a certain order in which to solve the operations. We also use parentheses to group together two elements used to locate a point on a cartesian [...]

Braces

Symbol that denotes a grouping of elements in a set. Braces are always used in pairs. Braces can be used as a symbol to group operations. Ex. : {8 + 3(5 – 2) – 5} Notation The brace symbol is { }. Examples P = {0, 1, 2, 3, 4, 5, 6, 7} Q = [...]

Median of a Triangle

The segment in a triangle that joins a vertex to the midpoint of the opposite side. Example Segment AM is the median from vertex A in triangle ABC.

Centre of Gravity of a Triangle

Meeting point of the medians of a triangle. Example Point G is the centre of gravity of triangle ABC. Educational Note The expression "centre of gravity" is a synonym of "balance point" or centroid. The centre of gravity of a segment is its midpoint. The centre of gravity of a quadrilateral is the meeting point [...]

Rounding

A process that consists of providing a close and simpler value of a number that is known or to be calculated that is less precise but easier to use. The normal method of rounding a number to a certain position consists of keeping the last digit in this position unchanged if it is followed by [...]

Edge of a Geometric Solid

Line segment where two surfaces of a geometric solid meet. In a solid, an edge is a segment at the intersection of two plane or curved surfaces. Property Euler's formula can be applied in polyhedra that can be represented using a connected graph. In this case, the sum of the number of vertices (V) and [...]

Vertex in a Graph

Each of the points that belongs to a graph. Strictly speaking, a vertex in a graph is always the endpoint of an edge. However, in a broader sense, a vertex of a graph cannot be connected to any of the other vertices. Examples In this graph, the vertices are the points identified by the letters [...]

Edge of a Graph

Line that joins two consecutive vertices (distinct or not) in a undirected graph. Some authors sometimes use the term branch to refer to an edge in a graph.

Radius of a circle

A line segment that joins the centre of a circle to any point on the circle. The term radius of a circle often refers to the length of a segment that joins the centre of the circle to a point on the circle. Example In this circle with centre O and points A and [...]

Tangent of an Angle

Ratio between the sine and the cosine of an angle. Consider a right triangle with a hypotenuse that measures 1 unit, or a trigonometric circle in which r = 1. In this right triangle, we have the relations: \(\sin \left( \theta \right) = y\) and \(\cos \left( \theta \right) = x\). Therefore, \(\tan \left( \theta \right) = [...]

Probability Tree

Directed and valued graph responding to these rules: The sum of the values (probabilities) associated with the edges coming from the same vertex gives 1, The probability of a path is the product of the probabilities associated with the edges that compose it, The value of an edge that goes from vertex S1 to vertex [...]

Factor Tree

A visual representation of the breakdown of a number into a product of its prime factors. Example The prime factorization of 60 is 22 × 3 × 5. The terminal branches reveal the breakdown into prime factors of the number 60, which is: 60 = 22 × 3 × 5.

Order of Magnitude of a Number

Number that characterizes an approximate value of a measure or of a more precise number. The order of magnitude of a number can be the power of 10 closest to the number or a specific place value chosen to represent it. Examples The order of magnitude of the height of a person is the foot, [...]

Slant Height of a Regular Pyramid

Distance a from the vertex O of the pyramid to any of the edges of its base. Example This distance is the length of the perpendicular segment drawn from the vertex of the regular pyramid to any of the sides of its base. Formula The formula for the measure of the apothem of a regular polygon [...]

Apothem of a Regular Polygon

Distance from the centre of the polygon to any side of the polygon. The word "apothem" also refers to the line segment drawn perpendicularly from the centre of the polygon to one of its sides. Formula The general formula to find the length \(a\) of the apothem of a regular polygon with \(n\) sides that have a [...]

Slant Height of a Right Circular Cone

Distance a from vertex A of a cone (generally called its apex) to its directrix or to any point on the circle of the base. The directrix is the perimeter of the base of a cone. In the case of a right circular cone, the directrix is a circle. A right circular cone is also called [...]

Finite Sequence

Finite subset of elements in a sequence. Example If we consider the sequence of even numbers, then the sequence of even numbers under 100 is a finite sequence.

Decimal Sequence

Name given to the decimal expansion of a real number. Example The decimal expansion of the rational number \(\frac{1}{7}\) is :\(\frac{1}{7}\) = 0.142 857 142 857 ... Therefore, we can write: \(\frac{1}{7}\) = \(\overline {142\space857}\) This is a periodic decimal sequence, because the sequence of digits 142 857 repeats infinitely. Educational Note We can express a [...]

Arithmetic Sequence

Sequence of numbers in which the difference r between two consecutive terms is constant. Synonym for arithmetic progression. The constant difference between two consecutive terms is called the difference of the arithmetic sequence. Example Consider the number sequence S = {1, 3, 5, 7, 9, 11, …} Notice that there is a constant difference of [...]

Polar Coordinates of a Point

In a polar coordinate system, the coordinates of an ordered pair (r, θ) associated with the position of a point P, where r is the distance from the origin to point P and θ is the angle of rotation. The origin O of the coordinate system is called the pole. The distance r from the [...]

Cartesian Coordinates of a Point

In a Cartesian diagram of degree n, the n-tuple of numbers used to determine the position of this point with respect to each axis. The values on each of the axes are determined with parallel projections of the point onto the axes. Examples In the diagram below, the Cartesian coordinates of point A are (6, [...]

Sexagesimal Number System

A number system with 60 as its base. The number system with 60 as its base is currently used only in a few specific fields to measure time, angles and geographic coordinates. Examples The units of measure for time are hours, minutes and seconds, in the following relationships: 1 hour is equal to 60 minutes [...]

Circular System

Name given to the system for measuring angles in which the base unit is the radian. This system of measurement is called the circular system because of the association between the definition of the radian and the circle. In trigonometry, we most often use measures of angles expressed in radians. Properties In this system of [...]

Amplitude of a Periodic Function

Half of the distance between the maximum and the minimum of a periodic function. If the function has several local maxima and minima, the amplitude is half of the distance between the greatest maximum and the least minimum. Example In this graph of the function defined by f(x) = cos(x), we can see that the amplitude [...]

Amplitude of a Statistical Class

When we divide data about a population studied in statistics into classes of the same range, this range is called the class interval. Example In this bar graph, we can observe that the total data has been divided into classes for which the interval is 10 percentage points.

Amplitude of an Interval

Distance between the bounds of a given interval.

Positional Numeral System

Numeral system in which a digit represents different values depending on the position it occupies in a number. Examples In the decimal numeral system, the digit 8 occupies the tens position in the number 285; this means that it represents 8 tens, or 80 ones. In the number 1825, the digit 8 occupies the hundreds [...]

Numeral System

System including symbols and rules of use for these symbols allowing us to write and refer to various numbers. Property A numeral system normally includes writing rules that concern the value of symbols depending on their position in the writing. If the rules apply to the position of the digits (symbols) in the written form [...]

Decimal Number System

A positional numeral system that groups objects by ten and uses the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. We say that the decimal number system is a base 10 numeral system. Reading numbers in the decimal system uses the written form of digits like one, two, three, etc, but also ten, twenty, thirty, [...]

Binary Numeral System

Positional numeral system that groups objects by two and only uses the digits 0 and 1. The binary numeral system is a base 2 system. However, in the binary system, the digit 2 does not exist. The binary numeral system is also a positional numeral system like the one that we can use in the [...]

Equivalent Propositions

Propositions that have the same truth value. Synonym for identity. Symbol The equivalence of the two propositions P and Q is noted as: P \(\iff\) Q and is read as: "P is equivalent to Q" or "P if and only if Q". Examples For the real values of x, the following propositions are equivalent: 4x + [...]

Program

In mathematics and computer science, the set of instructions or restrictions established to describe the execution of a finite sequence of operations. Examples An algorithm describing the sequence of actions to execute to solve the Euclidean division of two whole numbers is a program. The sequence of geometric constructions to carry out to determine the [...]

Universe

Synonym for universal set or universal. See also: Universe of events Universe of possible outcomes

Universe of Events

Set of the parts of the universe of possible outcomes of a random experiment Each part of the universe of possible outcomes is called an event in a random experiment. Example In the experiment that consists of rolling a regular die with six faces and noting the result that appears on the top face, the [...]

Universe of Possible Outcomes

Set of all of the possible outcomes of a random experiment. Notation This set is noted as: Ω. Example In the experiment that consists of rolling an honest die with six faces numbered 1 to 6 and noting the result that appears on the top face, the universe of possible outcomes is the set {1, [...]

Ones

Quantity that is used to count by one in a numeral system. The ones is a generalization of the digit 1 in all numeral systems. Property The ones is the only element in a set of numbers that is invertible. Examples In the number 542, the number 2 occupies the ones place value. There are 763 [...]

Unit of Measurement

Finite quantity that is used as the base for measurements of other quantities of the same type. The seven base units of the International System of Units are: Quantity Units Symbol Description Length Metre m Length defined by taking the fixed numerical value of the speed of light in vacuum c to be 299 792 458 when [...]

Approximation by Truncation

The process of replacing a number with an approximate value found by removing all of the digits from the fractional part that is located to the right of a given position. The retained part is called the truncation. All of the digits in a truncation are exact digits. Example Imagine a baseball game in a [...]

Lattice

Cartesian network made up of criss-crossing lines where objects are represented by network nodes. See also: Lattice of divisors of a number

Transitivity

Property of a transitive relation.

Theorem

In a theory, a demonstrable proposition that results from other propositions that are already demonstrated or accepted without proof, such as axioms or postulates. Etymological Note The word theorem comes from the Greek théorêma (θεωρημα), which means "object of study" or "object of contemplation". The words theory and theatre have the same root.

Rate

Name given to certain ratios that generally include different kinds of quantities. A rate can be expressed in decimal notation, fractional notation, or as a percentage. A rate can express how one variable changes based on another one, as is the case for speed, where the distance covered depends on how much time has passed: [...]

Term

Each element in a sequence, ratio, addition, subtraction, polynomial, proportion or fraction. Examples In the sequence 1, 3, 5, 7, … , each number is a term of the sequence. In the fraction \(\frac{5}{8}\), the numbers 5 and 8 are the terms of the fraction. In the ratio \(\frac{12}{5}\), the numbers 12 and 5 are the terms [...]

Tautology

Logical proposition that is always true, regardless of the truth values of its components. Synonym for proposition that is always true. Example The proposition composed P → (Q → P) is a tautology. Here is its truth table: P Q Q → P P → (Q → P) T T F F T F T F T T [...]

Population Size

Number of elements or individuals in the population that is the subject of a statistical study. Example At a school, there are 220 students: 120 girls and 100 boys. For a survey, we decided to question one-quarter of the girls and one-quarter of the boys. Therefore, we surveyed 30 girls and 25 boys, chosen at [...]

Table

Series of data or information arranged in lines or columns in a clear, ordered fashion to facilitate consultation, and which sometimes includes images. See also : Real frequency table Correlation table Frequency table

Real Frequency Table

Table that presents the actual frequencies of a statistical attribute and in which these frequencies are compared to each of the modalities, values or statistical classes taken by this characteristic. Example Here is a table that presents the distribution of students at a primary school: Ages of students at a primary school Ages 6 7 [...]

Graph Size

Number of edges or arcs in a graph. See also : Graph Order of a graph Edge of a graph Arc of a graph

Sample Size

Number of elements in a sample of a population being studied. Example At a school, there are 220 students: 120 girls and 100 boys. For a survey, we decided to question one-quarter of the girls and one-quarter of the boys. Therefore, we surveyed 30 girls and 25 boys, chosen at random. The population size is [...]

Matrix Size

Synonym for dimension of a matrix.

Correlation Table

Table that presents the values of a distribution with two statistical attributes. Example Results of grade 6 students on two different exams out of 50 points Students Grade 1 Grade 2 Alice 42 38 Axel 38 46 Beatrice 49 50 Cindy 48 45 Denis 41 32 Dominique 45 30 Louise 36 29 Lucy 31 44 [...]

Number Table

Table in which the numbers are arranged methodically in rows and columns. Examples Table of squares 0 1 2 3 4 5 6 7 8 9 0 1 4 9 16 25 36 49 64 81 Table of the first 20 prime numbers 2 3 5 7 11 13 17 19 23 29 31 37 [...]

Pythagorean Table

Table of n rows and n columns in which the boxes are located by pairs of elements in a set with n elements that present the results of an operation. Example Pythagorean table of the addition of integers modulo 4. + 0 1 2 3 0 0 1 2 3 1 1 2 3 0 [...]

System of Equations

A finite set of equations that can be verified simultaneously. A system of equations in two variables can be represented in a Cartesian plane using the graph of each equation. The points of intersection of these graphs are the solutions to the system. Example Consider the following system of equations: \(\left\{\begin{matrix} y=2x\\ y=4x^2-1 \end{matrix}\right.\) The [...]

Sequence

Set of elements (numbers or figures) indexed by whole numbers called the ranks of these elements in the sequence. Example The sequence: 3, 7, 11, 15, 19, 23, … is a sequence of numbers (numerical sequence) Above, these figures illustrate the first few elements in the sequence of triangular numbers. See also : Numerical sequence [...]

Proportional Sequences

Two sequences where the second sequence can be found by multiplying each term in the first sequence by the same number are proportional sequences. This number is called the coefficient of proportionality. The relationship of a term in the first sequence and the term in the same rank in the second sequence is called the [...]

Descriptive Statistics

Branch of applied mathematics based on the observation of experimental events from which we try to establish hypotheses that will allow us to predict events in analogous situations.

Digital Sum

Sum of the values associated with the digits in a whole number. Example The digital sum DS of the number 4067 is 17. DS = 4 + 0 + 6 + 7 = 17 See also : Divisibility characteristic Digital root

Subtraction

Operation under which each pair of numbers, called the terms of the subtraction, is made to correspond to a new number called the difference of these terms. We can define a subtraction operation in different sets such as number sets, relationship sets, geometric figure sets, etc. The inverse operation of subtraction is addition. Symbol The [...]

Sub-Graph of a Graph

A graph G = (L, S) given as such, a sub-graph of G is the graph G1 = (L1, S1) formed by a subset S1 of vertices of G and a sub-set L1 of edges of L. Example Consider the graph G below, defined by G = (L, S), where L = {a, b, c, [...]

Subset

Given a set E, a subset of E is the set in which all of the elements belong to set E. Synonym for part of a set. A proper or strict subset of a set E is a subset of E that is not equal to E. A large subset of a set E is [...]

Survey

Statistical operation conducted with a representative sample of a population in order to study one or more of its characteristics. Example We surveyed 39 Grade 6 students to find out which of the following six colours was their favourite: white, orange, yellow, red, black, green Here is the data collected: Study population: Grade 6 students [...]

Sum

The result of the addition of two quantities. Examples In the operation 16 + 24 = 40, 40 is the sum of 16 and 24. In the operation 3.8 + 6.5 = 10.3, 10.3 is the sum of 3.8 and 6.5. In the operation 2x + 5x = 7x, the term 7x is the sum [...]

Solution to an Equation

Each of the values that can be substituted for the variables in an equation to find a true equality. The set of all these values is called the solution set of the equation. Example Consider the equation "2x + y = 12" in which the variables x and y are defined in the set E [...]

Solution to a System of Equations

The set of all values that can be substituted for the variables in each equation of a system and that satisfy all the equations of the system simultaneously. Example Consider the system of equations y = 2x and y = 4x² − 1 : The solutions to this system are the ordered pairs of coordinates (\(\frac{1 [...]

singleton

Set of cardinality one (1). Set that only contains one single element. Examples If A = {0}, then set A is a singleton. The set of points that form the intersection of two intersecting lines is a singleton. The GCD of two relatively prime numbers is {1}.

Statistical Series

Set of values of a statistical characteristic X compiled in an inquiry regarding a sample of a given population. If the inquiry focuses on two statistical characteristics X and Y, the statistical series will then be formed by ordered pairs of values (x1, y1), (x2, y2), … A statistical series is defined with a population [...]