Paul Patenaude a une expérience de plus de 35 années dans l'enseignement et l'animation pédagogique en mathématiques au primaire et au secondaire, au Québec et à l'étranger. Il est auteur ou co-auteur de nombreux ouvrages dédiés à l'enseignement des mathématiques dont ce lexique des termes utilisés avec les jeunes de 5 à 17 ans. Il peaufine ce lexique depuis plus de 30 ans. Prix d'excellence du GRMS (prix Claude Janvier) en 1997, prix du meilleur article de la revue l'Envol du GRMS (prix Euler) en 1992.
Il adore voyager et a déjà exploré de nombreux pays sur les cinq continents, curieux à la fois des cultures et de l'histoire. Il apprécie recevoir vos commentaires et vos questions auxquelles il s'empressera de répondre.
Graph in which you can connect, directly or indirectly, any vertex to any other vertex in the graph by a chain of edges. Example This graph is a connected graph: From any vertex in this graph, we can get to any other vertex in this graph.
Graph in which all of the vertices are either with degree greater than or equal to 2, or with degree equal to 0. Example This is a closed graph: The degree of each vertex is indicated in parentheses.
Name given to the number \(10^{100}\). The American mathematician Edward Kasner created this number to illustrate the difference between a large number and infinity. A googol is huge and much greater than the number of particles in the known universe (about \(10^{80}\)). Properties A googol is a number with 101 digits. In scientific notation, a [...]
Name given to the number \(10^\textrm{googol}\) or \(10^{10^{10}}\). The googolplex is a number defined as the number 10 raised to the power of googol. It would be physically impossible, in the decimal system, to write this number on a sheet of paper as large as we could reasonably find, because it contains more digits than [...]
Two quantities are said to be inversely proportional if their measurements evolve in opposite direction. In other words, two quantities are inversely proportional if the product of their quantities is a constant number. Example To travel a certain given distance \(d\), the speed \(v\) and the time \(t\) are two inversely proportional quantities connected in the relationship \(d [...]
Unit of measurement for mass in the International System of Units (SI). Notation The symbol for gram is “g” which signifies “gram.” In one gram, there are 1000 milligrams and we write: 1 g = 1000 mg. In one kilogram, there are 1000 grams and we write: 1 kg = 1000 g. One gram corresponds [...]
Characteristic or property of a mathematical or physical object that can be measured or calculated and that is often expressed accompanied by a unit of measurement. The measure expresses the magnitude of a measurable object in order to make this magnitude comparable to other magnitudes of the same type. The concept of magnitude is used in [...]
Term used to refer to a curve, figure, or lines in which all of the points do not belong to the same plane. Examples A spiral representing a spring is an example of a skew curve. The surface of a sphere is a skew surface. Educational Note The term "skew" is not opposed to the term [...]
In a statistical investigation, the expression of the ratio of the tally of a modality or a value of the character studied and the total number of results, in the form of a decimal number or a percentage. If the statistical character is continuous, we call the frequency f of a class the ratio of [...]
Fraction in which the numerator and the denominator have at least one common integral divisor that is different from one. Examples The fraction \(\frac{8}{18}\) is a reducible fraction, because the number 2 is an integral divisor that is common to both 8 and 18. Therefore, the simplified fraction that is equivalent to \(\frac{8}{18}\) is \(\frac{4}{9}\). The fraction [...]
Fraction in the form \(\dfrac {1}{n}\), in which \(n\) is a non-zero whole number. Properties A unit fraction is the inverse of a non-zero whole number. The fraction \(\frac{1}{1}\) is a unary fraction. Examples Fractions like \(\frac {1}{2}\), \(\frac {1}{5}\) and \(\frac {1}{10}\) are unit fractions.
Fraction in the form \(\frac {n}{n}\), in which n is a non-zero whole number. Examples Fractions like \(\frac{3}{3}\), \(\frac{5}{5}\) and \(\frac{10}{10}\) are whole fractions.
Fractions that are equal when they are simplified to their simplest expression. Fractions that represent the same rational number. Example The fractions \(\frac{12}{18}\) and \(\frac{10}{15}\) are equivalent, because they are both equivalent to the rational number \(\frac{2}{3}\).
If f is a periodic function with period P, the frequency of f is equal to the inverse of P, or \(\frac{1}{\textrm{P}}\). Educational Note In the different areas of physics, including music (sound waves), the frequency refers to the number of times that a periodic phenomenon can be reproduced by unit of time or space. [...]
Relationship between a part and the whole. The whole is divided into a certain number of equivalent parts. The numerator indicates the number of equivalent parts considered. The denominator indicates how many equivalent parts the whole was divided into. A fraction empirically represents a part of a whole expressed in the form of a ratio [...]
Fraction or fractional expression that can be expressed with a denominator that is a power of 10. Property All decimal fractions express a decimal number and all decimal numbers can be expressed as a decimal fraction. Examples The fractions \(\frac{3}{10}, \frac{17}{100}, \frac{27}{1000}\) and \(\frac{83}{1\space 000\space 000}\)are decimal fractions. The fractions \(\frac{3}{4},\frac{4}{5}, \frac{3}{20}\) and \(\frac{17}{25}\) might [...]
Fractional expression in which the numerator is greater than the denominator. This expression is outdated. This expression is most often replaced by the expression rational number depending on the context. Examples The expressions \(\frac{12}{5}, \frac{321}{100}\) and \(\frac{1000}{6}\) are improper fractions.
Fraction in which the numerator and the denominator do not have a whole common divisor that is different from 1. Fraction in which the numerator and the denominator are relatively prime, which means that they do not have a whole common divisor that is greater than 1. Examples Fractions like \(\frac{2}{3}\), \(\frac{3}{4}\), \(\frac{5}{13}\) and \(\frac{8}{19}\) [...]
A periodic fraction is equivalent to a Series in which the terms are fractions and the pattern of the terms repeats. If we exclude the case where the period is a non-zero fraction, a periodic fraction expresses a rational number that is not a decimal number. Example The fraction \(\dfrac{2}{27}\) is a periodic fraction because:\(\dfrac{2}{27}=\dfrac{74}{1000}+\dfrac{74}{1\ 000\ [...]
Mathematical statement that includes one or more variables and is true for some values attributed to these variables and false in other cases. The defined set of variables of a propositional form is called the universal set of this propositional form. An expression like 3x – 5 = 7 is a propositional form called an [...]
Form or written form of the relationship that defines a function and that highlights the parameters that transform the basic form of the function. The standard form is a parametric form of a function rule in which the parameters characterize a transformation of the function's graph. Example The standard form that defines a second-degree polynomial [...]
Simplest form or general written rule of a function that does not contain any parameters. Some authors use the term "basic function" instead. Example The basic form that defines a second-degree polynomial function is: f(x) = x² See also: Standard form of a function
Polynomial written in the form of a product of polynomials of lower degrees, when the polynomial can be factored. Factorization is possible on the set of real numbers if there is at least one real number that cancels the polynomial. Example The expression 5(x - 2)(x - 3) is the factored form of the polynomial [...]
Parametric form or written form of the defining relationship of a function that emphasizes the general form of the rule. Example The function f on the set of real numbers defined by the relation f(x) = Ax² + Bx + C is a function defined in general parametric form or general form that in this [...]
The expression of the definition of a function using a parametric equation. Example The function f on the set of real numbers defined by the relation f(x) = Ax² + Bx + C is a function defined in parametric form. This function can also be defined using the relation f(x) = a(x - h)² + k, which [...]
Form of the equation of a line in which the y-coordinate is expressed as a function of the other variables, that is y = mx + b, where m is the slope of the line and b is its y-intercept. This particular form is also called the "slope-intercept form".
Function in which the argument is an angle value, or an angle of rotation. General term used to designate one of the sine, cosine, tangent, secant, cosecant and cotangent functions that are the subject of circular trigonometry. Example The function \(f\) defined in the set of real numbers by the relation \(f\left(x\right)\) is a trigonometric [...]
A function f is periodic if there is a real positive number p so that, for all x and (x + p) of the domain of f, we have f(x + p) = f(x) or f(x – p) = f(x). Sine, cosine and tangent trigonometric functions are periodic functions. In the expression "sin (x + p)", the least value of p is the period [...]
Function defined by a relation in the form f(x) = \({a}^{x}\) where a is a strictly positive real number that is different from 1. The graph of an exponential function passes through the point (0, 1), no matter what the base of the function is. The functions defined by f(x) = \({a}^{x}\) and g(x) = [...]
A homographic function is the quotient of two first-degree polynomial functions, through an expression in the form \(f \left( x \right)=\dfrac {ax+b} {cx+d}\) with c ≠ 0. When c = 0, the function is reduced to a first degree polynomial function, represented by a line. The graphic representation of a homographic function is an equilateral [...]
If \(\left[a,b\right]\) is an interval in the domain of a function \(f\), we say that the function \(f\) is decreasing in the interval \(\left[a,b\right]\) if and only if for all elements \(x_{1}\) and \(x_{2}\) of \(\left[a,b\right]\), if \(x_{1}<x_{2}\), then \(f\left( x_{1}\right) ≥ f\left(x_{2}\right)\). Example Consider the function defined by \(f\left(x\right) = -3x+2\). If \(x_{1}=0\), then \(f\left(0\right) [...]
If [a, b] is an interval in the domain of a function f, we say that the function f is increasing in the interval [a, b] if and only if for all elements x1 and x2 of [a, b], if x1 < x2, then f(x1) ≤ f(x2). Example Consider the function defined by f(x) = [...]
Function defined from an arc of a circle or from the value of the angle at the centre corresponding to an arc of a circle. A function f is a circular function on a unit circle C if and only if f: \(\mathbb{R}\) → C : | t |→ (a, b) where | t | is the [...]
Function defined in the set of real numbers by a relation in the form f(x) = k, where k is a real number. The graph of a constant function is a horizontal line that is parallel to the x-axis. A constant function is a particular case of an affine function. Example The function defined by [...]
Relation under which each value or element in a set of departure (or domain) is associated with one and only one value or element in a set of arrival (or image), according to a rule of correspondence that describes this association. A function can be defined in extension or intension. The pairs belonging to a [...]
Set of points, connected or not, that are used to represent a given object. The term figure is a synonym for drawing, representation, diagram, image, etc. It consists of a primarily visual representation that, in the context of geometric analysis, then takes on the more restrictive signification of a set of points that has certain [...]
Each of the elements that are involved in a multiplication. Example In the multiplication 24 × 7 = 168, the numbers 24 and 7 are called factors and the number 168 is called the product. Educational Note We often confuse the terms factors and divisor. The set of the divisors of 12 is: div (12) [...]
For any positive integer n, a number that is the product of all the positive integers less than or equal to n. In set theory, the factorial n is defined as being the number of permutations of a set of n elements. We say that "n!" is a function of the set of whole numbers in itself, [...]
Expression that only contains numbers that are connected by operational symbols. Examples Here are some examples of numerical expressions: 6 + 5 3(5 + 8) + 2 \(\sqrt{7}\) – 3
Events that do not have any common elements and therefore cannot occur at the same time. We also say that these events are disjoint or mutually exclusive. Example In the experiment that consists of rolling a fair die with six faces numbered 1 to 6 and recording the result that appears on the top face, [...]
Events for which the occurrence or non-occurrence of one does not affect the probability of the occurrence of the other. An event whose occurrence does not depend on the result of another event is sometimes called a simple event. Property The probability that two independent events will occur in the same random experiment is equal [...]
Events that do not have any common elements and for which their assembly corresponds to set of all possible of a random experiment. The expression complementary events is a synonym for contrary events. Notation The complementary event of an event E is noted as E’. Therefore, we have E ∪ E’ = Ω. Example In the [...]
Events for which the occurrence or non-occurrence of one affects the probability of the occurrence of the other. Example A bag contains 10 red marbles and 20 white marbles. We draw two marbles from the bag randomly and one after the other, without putting them back after each draw. The event A: "drawing a white [...]
Singleton of the set of possible results. Each of the possible results of a random experiment is an elementary event. Example In the random experiment of rolling an honest die with six faces numbered 1 to 6, each of the 6 possible outcomes is an elementary event.
Event that corresponds to an impossible outcome of a random experiment. This event will never occur. Empty subset of the set of possible outcomes of a random experiment. Property The probability of an impossible event is 0. If the experimental probability of an event in a random experiment is close to 0, we say that [...]
Events that have results in common and that can occur at the same time. We also say that compatible events are joint or inclusive. If two events A and B are compatible, then A ∩ B ≠ ∅. Example In the random experiment that consists of rolling an honest die with six faces numbered 1 [...]
Events that have the same probability. Example In the random experiment that consists of rolling an honest die with six faces numbered 1 to 6 and recording the result that appears on the top face, the events: "A: rolling an even number" and "B: rolling an odd number" are equiprobable events, because: P(A) = \(\frac{1}{2}\) and P(B) = \(\frac{1}{2}\).
Event resulting from the union or intersection of several events. Example In an experiment that consists of rolling a regular die with six faces numbered 1 to 6 and noting the result that appears on the top face, the event “getting an even result less than 5” is a composite event that results from the [...]
Difference between the extreme values taken by a quantitative statistical attribute. The number of results does not factor into the calculation of the range. The calculation of the range is important for determining the amplitude of the statistical classes in a distribution. Examples Consider this distribution: 3, 4, 7, 9, 12, 15, 17. The range E [...]
Distance between the limits of a quartile in a statistical distribution. Notation We can note: EQ1 = |Q1 - xmin| EQ2 = |Q2 - Q1| EQ3 = |Q3 - Q2| EQ4 = |xmax - Q3| where EQ1, for example, refers to the range of the first quartile, \(x_{min}\) refers to the lowest datum in the [...]
Interval in which the upper and lower limits are the quartiles Q1 and Q3 of a statistical series. The interquartile range is a measure of dispersion of a distribution that provides information about the scattering of data around the median.
Ensemble formé de tous les résultats possibles d’une expérience aléatoire. Cet évènement va toujours se produire. Properties The probability of a certain event is 1. An event with a frequential or experimental probability that is very close to 1 can be called an event that is almost certain. Examples We drew a ball from a [...]
Subset of the set of possible outcomes of a random experiment. An event can include one or more possible outcomes. Notation As the subset of a set, an event can be noted extensionally or intentionally. In the case of a random experiment that consists of rolling an honest die with 6 faces numbered 1 to [...]
Quantity or value used when an exact value is not necessary, relevant, or possible to find, depending on the context. An approximation by estimation, or estimation, is a value that we determine to be sufficiently close to a real value that is observable but difficult to measure without more appropriate conditions and a suitable measuring [...]
Relation of equivalence between the bipoints of a geometric plane so that any bipoints (a, b) and (c, d) in a plane where a ≠ b and c ≠ d, (a, b) and (c, d) are said to be equipollent if and only if ab and cd are the opposite sides of a parallelogram.
Relationship between mathematical objects for which the representations - numerical, logical, statistical, etc. - have the same value. Examples The equations 2x + 6x - 12 = 0 and 4x = 6 are equivalent equations because they have the same solution set. A square with 10 cm sides and a rectangle that is 20 cm [...]
Sum of the products of the values of a random variable by their probability. In the case of a game of chance, a game is fair when the expected value is zero. Example When rolling a fair die with six faces numbered 1 to 6, you bet $0.50 on the 6, hoping to win 10 [...]
Sets that can be put in a one-to-one relation with one another. Example Sets A and B below are equipotent. Properties Two sets are equipotent if and only if they have the same cardinal. Two infinite sets can have different cardinals if they are not equipotent. Therefore, the set of whole numbers, which is an [...]
Transcendental equation in which the variable appears as the argument of trigonometric ratios. Example Consider the equation \(y = \sin(x) − 3\). This equation is a trigonometric equation.
Collection of distinct objects that have a common characteristic (defining property) called elements in this set. A set is extensionally defined when it is defined by the explicit list of its elements, such as: U = {6, 7, 8, 9, 10, 11, 12} or \(\mathbb{N}\) = {0, 1, 2, 3, 4, 5, 6, 7, ...}. [...]
Algebraic equation of the form Ax + By + C = 0, where A, B and C are real numbers and where A and B are not both zero. In the general form of the equation of the line Ax + By + C = 0, the parameters A, B and C are usually non-zero [...]
A scientific approach intended to discover and establish facts about a population or a sample of a population, in order to comprehensively compile information that was unknown at the start of the inquiry. When conducting a statistical inquiry, statistical tools and techniques are used to collect and analyse data and determine the results. See also: [...]
Obtaining a sum of money or value as a loan, which means it must be repaid after a certain amount of time that is determined at the time the loan is granted, with or without interest. The term to borrow is the opposite of to loan. For the lender, the borrowed amount is the amount to receive back [...]
Relationship between two quantities that have the same value. Relationship between two quantities that have the same value or between two representations of the same mathematical object. Notations The relationship of equality is denoted by the symbol "=", which is read as "is equal to". This symbol can only be used between numbers, numerical variables [...]
Each object that forms part of a given set. Examples Consider set E = {0, 2, 6, 8}. The elements of E are each of the numbers 0, 2, 6 and 8. Consider set F = {a, b, c, d}. The elements of F are each of the letters a, b, c and d. Consider [...]
Ratio or relationship between the measure of the representation of an object and the real measure of the object. Symbol The symbol generally used to compare a measure and its representation is “\(≙\)” which means “corresponds to.” Example The relationship “1 cm \(≙\) 1 km” is written on many maps, meaning that a distance of [...]
Finite subset of data that is chosen randomly as being representative of a population or a phenomenon to study. Properties Biased sample Set of individuals in a population that is supposed to represent the population, but whose selection has introduced a bias, or a source of error (positive or negative), that distorts conclusions formulated about [...]
Application of specific criteria and techniques to choose a sample of a population to study. The census is the best way to collect the most exact information about a population, because in a census, we question every individual in the population. In practice, this method is not always feasible, such as when trying to gather [...]
When counting the values of a statistical variable, the number of values that corresponds to a given characteristic or that belongs to a particular class of values. (A) If the attribute studied is discrete, the tally of a value x of this characteristic is the number of individuals for which the value of the characteristic [...]
Linear graph representing the theoretical correlation between two random variables or two quantitative statistical characteristics defined on the same population. To find the parameters of the line of regression or a good approximation of this line, we can use different methods, including: the median-median method the Mayer method the method of least squares Example
Absolute value of the difference between the values of the quartiles Q1 and Q3 in the distribution of a statistical variable. Synonym for interquartile interval or interquartile amplitude or interquartile range. The semi-interquartile deviation is half of the interquartile deviation.
Space in which an object is defined. We can talk about the domain of a study, the domain of a science, the domain of a statistical inquiry, the domain of a numerical relationship, and so on. The domain sets the range and the limits of a concept or an intention. The meanings used in mathematics [...]
Elements of information, often numerical, that serve as a point of departure for a statistical study. Statistical data can be numerical data or alphanumeric data. Raw data are data that have not yet been sorted. In a statistical distribution, an inconsistent element is a value that differs significantly from the main group of data (or [...]
In a division operation, the name given to a number to be divided. Example In the operation 12 ÷ 4 = 3, the number 12 is the dividend and the number 4 is the divisor.
In a division operation, the name given to a number that divides another. The divisor is the second term in the division operation. Examples The divisors of 15 are: div(15) = {1, 3, 5, 15}. The divisors of 18 are: div(18) = {1, 2, 3, 6, 9, 18}. The divisors of 25 are: div(25) = [...]
We sat that a number is divisible by another number when the remainder of this division is zero. Examples 12 is divisible by 4, because the remainder of 12 divided by 4 is zero. 25 is not divisible by 4, because the remainder of 25 divided by 4 is 1.
In a set of numbers, an operation under which a pair of elements (a, b) in this set, is made to correspond to a number q of this set so that this relationship is true: a ÷ b = q or a = b × q. The result of a division operation is called the quotient. The operation of division [...]
Set of experimental or theoretical values of a given statistical or random variable. Examples Experiment 1 We are randomly drawing marbles one by one from a bag that we don’t know anything about. After 100 draws, the results are: 12 white marbles, 35 black marbles, 8 red marbles and 45 blue marbles. The distribution of the [...]
The disjunction of two propositions P and Q is the proposition that is true if at least one of the two propositions P and Q is true, and false if the two propositions are false. Symbol The disjunction of the propositions P and Q is noted as "P ∨ Q" and is read as "P [...]
The exclusive disjunction of the propositions P and Q is the proposition that corresponds to (P ∨ Q) ∧ ¬(P ∧ Q), which means that it is true when one or the other of the propositions P and Q is true, but not both at the same time. Symbol The exclusive disjunction of the propositions [...]
Spread of the values of a distribution of a statistical variable around a central value. Example Deviation from the mean, variance and standard deviation are the parameters of dispersion in a statistical distribution. See also : Measure of dispersion
Attribute that distinguishes or opposes. Example A set of logic blocks used in math is formed by several objects that are distinct shapes, colours, thicknesses, and sizes. Using this material, a student can create sequences of piles of blocks that include one, two, or three differences between them, depending on the instructions. These differences concern [...]
Name given to the expression "B2 – 4AC" that is used to find the solutions of the second-degree polynomial equation Ax2 + Bx + C = 0. Symbol The symbol "Δ", read as "delta", is generally used to represent the discriminant of a second-degree polynomial equation. Property The sign of the discriminant provides information on the nature [...]
That which does not have any elements in common. Disjoint sets are sets in which the intersection is empty. Disjoint lines are lines that have no common points. Examples In a plane, distinct parallel lines are disjoint lines. In space, skew lines, are disjoint lines.
General term that refers to a schematic, visual representation of a set of data or one or more phenomena. See also : Some of the most frequently used diagrams in primary and secondary school include: Bar graphs Rod graph Broken line graph Pictograph Stem and leaf diagram Cartesian coordinate system Circle graph Carroll diagram Dispersion [...]
Diagram in which a distribution of a qualitative statistical variable is represented in the form of a rectangle extended by two segments, highlighting the maximum and minimum values of the distribution, median and quartiles Q1 and Q3. Box and whisker plots are sometimes called box plots. Example
Representation of the data of a qualitative statistical attribute based on a grid in the form of a pencil of rays with the same origin divided into units of measurement and where the frequency of the data is represented by points connected to one another by segments to form a polygon. In a web diagram, [...]
Diagram that represents a relationship of a finite set to a finite set in which each pair is represented by an arrow. Example Arrow diagram of the relationship divide in the set E of non-zero whole numbers less than 10: Etymological Note The representation of a relationship by an arrow diagram is sometimes called an [...]
Representation of one or more sets by simple closed lines in which the elements are represented by dots. The Venn diagram, like Carroll diagrams, is a graphic diagram used to represent logical relationships between sets and the elements in these sets. Properties A Venn diagram is made up of curved closed lines inside of which [...]
Diagram in which the modalities of the distribution of a qualitative statistical attribute or the values of a discrete quantitative statistical attribute are represented by drawings or images.