Author: pierremathieu

Pierre Mathieu a une expérience de 35 années dans l'enseignement des mathématiques et l'animation pédagogique. Il est auteur et co-auteur de nombreux ouvrages pour l'enseignement des mathématiques dont deux lexiques de mathématiques publiés aux Éditions du Triangle d'Or. Pierre a été personne ressource à de nombreuses occasions lors de la mise en oeuvre de nouveaux programmes d'études au Québec. Il a aussi travaillé dans le groupe PC² qui a produit du matériel d'enseignement et d'évaluation reconnu à travers tout le Québec pendant de nombreuses années.

Half-Circle Function

Function defined by a relation in the form f(x) = \(\sqrt{{r}^{2} – {x}^{2}}\) or f(x) = − \(\sqrt{{r}^{2} – {x}^{2}}\) where r is the radius of a circle centered on the origin point. Examples Here is the graph of the function defined by the relation f(x) = \(\sqrt{{4} – {x}^{2}}\) : Here is the graph of the [...]

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Half-Ellipse Function

Function defined by a relation in the form f(x) = \(\frac{b}{a}\) \(\sqrt{{a}^{2}~ – {x}^{2}}\) or f(x) = − \(\frac{b}{a}\) \(\sqrt{{a}^{2}~ – {x}^{2}}\) where a is the horizontal half-axis and b is the vertical half-axis of an ellipse centered on the origin point. Examples Here is a graph of a function defined by the f(x) = \(\dfrac{1}{3}\) \(\sqrt{{9}~ [...]

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Major Arc

The longer of two arcs linking two points one a circle. Example The major arc AB is the longer of the two arcs in the circle O between points A and B.

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Unit Rate

Rate whose consequence is equal to one. Examples The cost of one litre of gas is $1.18: \(\dfrac{$1.18}{1\space\textrm {litre}}\) A ratio of 22 clients per IT expert: \(\dfrac{22\space\textrm{clients}}{1\space\textrm {expert}}\). A speed of 100 kilometres per hour: \(\dfrac{100\space\textrm{km}}{1\space\textrm {hour}}\).

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Counting

Counting is the fact of enumerating, determining the number of objects in a given set. Example We can determine that there are 23 sheep in an enclosure by counting them.

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Hundred

Grouping of 100 ones in the decimal number system. Examples In the number 235, the digit 2 occupies the hundreds position and its place value is 200. In the number 3425, the digit 4 occupies the hundreds position and its place value is 400. The value of the digit 6 in the number 654 is [...]

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Thousands (Place Value)

Grouping of 1000 ones in the decimal number system. Description In the number 3245, the digit 3 occupies the thousands place value. In the number 14 532, the digit 4 occupies the thousands place value. The value of the digit 5 in the number 45 678 is "five thousands" or "5000". One thousands is equivalent [...]

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Rhombus-Based Pyramid

Pyramid whose base is a rhombus. If this pyramid is right, then the height passes through the point of intersection of the diagonals of the rhombus. The measure of the height is therefore the distance between the apex of the pyramid and the point of intersection of the two diagonals of the rhombus. If the [...]

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Regular Prism

Right prism whose bases are congruent regular polygons. In a regular prism, all of the lateral faces are isometric rectangles.

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Hollow Sphere

A hollow sphere is what is left of a sphere with radius \(r_{\scriptsize{2}}\), when a sphere with radius \(r_{\scriptsize{1}}\) has been removed from it, the two spheres having the same centre and \(r_{\scriptsize{1}}\) < \(r_{\scriptsize{2}}\). Formulas The volume V of a hollow ball is: \(V = \dfrac{4\pi (r_{\scriptsize{2}}^3~ - ~r_{\scriptsize{1}}^3)}{3}\)

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Right Circular Cylinder

See right cylinder.

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Right Circular Cone

See right cone.

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Fixed Line

A line that is invariant in a geometric transformation. Examples The line of symmetry is a fixed line in this symmetry. The line of reflection is a fixed line in this reflection. The lines mapping a dilation are fixed lines in this dilation.

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e (constant)

Letter that represents a constant that is the basis for the system of natural logarithms. This letter is the first letter of the word exponential. An approximate value of e is 2.718 281 … Like the number π, the number e is used in many mathematical formulas. Like the number π, the number e is [...]

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Measure of an Angle

Number that indicates how wide an angle opens. The basic unit of measure of angles is usually the degree. Angles are measured with a protractor, a measuring instrument marked in degrees. Image of a protractor:

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Measure of a Line Segment

Number that expresses a length. This number is related to the unit of measure used. Conventional units of measure such as the millimetre and the centimetre are commonly used to measure line segments. Example To measure the blue segment below, any of the 4 unconventional units of measure identified by the letters A to D can [...]

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Measure of a Surface Area

Number that expresses an area. This number is related to the unit of measure used. Formula The area of a circle is a number that expresses the measure of the surface area of the circle. The formula to calculate this measure is: A = π\({r^2}\) = π\(\frac{d^2}{4}\). The letter A represents the area of the circle. [...]

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Measure of a Space

Number that expresses a volume. This number is related to the unit of measure used. Examples This cube has an edge length of four centimetres. Its volume V is 64 cubic centimetres, or: V = 4 × 4 × 4 = 64. The unit of measure used, cubic centimetres, is a conventional unit of measure. We can [...]

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Measure of a Circle

Number, called the circumference, that expresses the length of a circle. Formula The formula to calculate the circumference of a circle is: C = 2πr = πd. The letter C represents the circumference of the circle. The letter r represents the radius of the circle. The letter d represents the diameter of the circle. The value of [...]

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Side of an Angle

Each of the rays that form an angle. Example In this figure, each of the rays AB and AC of origin A forms a side of the angle BAC.

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Side of a Polygon

Each line segment that forms the boundary of a polygon. A side of a polygon is a segment that joins two consecutive vertices of the polygon. Example

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Face of a Solid

A plane or curved surface in a solid that is bounded by edges. Examples This polyhedron has 6 faces and 12 edges. This hexagonal prism has 8 plane faces. A cone has 1 plane surface and 1 curved surface. A cylinder has 2 plane surfaces and 1 curved surface. A ball has 1 curved surface.

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Point of Intersection

Point where two elements cross. Properties Two non-parallel lines in the same plane have one point of intersection. Two parallel lines in the same plane have no point of intersection. Two circles in the same plane can have zero, one or two points of intersection. Example In a Cartesian plane, the coordinates of the point [...]

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Quantile Rank

Measure of position of an element in an ordered statistical series in one of the intervals with the same frequency determined by the quantile. See also: Quantile Quartile Percentile Decile Quintile rank

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Radius of a Torus

Line segment that joins the centre of gravity of a torus to the centre of the torus. Example Segment OC is the radius of the torus below:

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Diagonal of a Polyhedron

Line segment that joins two vertices that do not belong to the same face of a polyhedron. Example The diagonal AB joins vertices A and B of this prism. Properties Non-polyhedra do not have diagonals. A tetrahedron does not have a diagonal.

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Diagonal of a Polygon

Line segment that joins two nonconsecutive vertices of a polygon. Example Segments AC and AD are diagonals of the pentagon ABCDE. Properties A triangle does not have a diagonal. To calculate the number \(N\) of diagonals of a polygon with n sommets, vertices, you can use the formula:  \(N=\dfrac{n(n\space –\space 3)}{2}\).

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Height of a Cone

The distance between the apex of a cone and its base. This distance is always measured perpendicular to the base. Therefore, it is possible for the foot of the perpendicular not to fall on the base. Notation The symbol generally used is "h", for "height". See also: Cone

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Height of a Cylinder

The distance between the two bases of a cylinder. This distance is always measured perpendicular to the base. Therefore, it is possible for the foot of the perpendicular to fall outside the base. See also: Cylinder

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Height of a Parallelogram

The distance between two parallel sides of a parallelogram. The distance is always measured perpendicular to the base of the parallelogram. It should be noted that the foot of the perpendicular may fall on the base or on an extension of the base. In the figure below, the height is \(h\).

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Height of a Trapezoid

Distance between the two parallel sides of a trapezoid. The distance is always measured perpendicular to the bases. It is possible for the foot of the height of a trapezoid to fall outside the base. Example In the figure below, the height is \(h\).

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Height of a Prism

The distance between the two bases of a prism. The distance is always measured perpendicular to the base of the prism. Examples In the figures below, the height is \(h\).

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Height of a Pyramid

The distance between the apex and the base of a pyramid. The segment \(\textrm{AB}\) is the height \(h\) of these pyramids.

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Lines Mapping Corresponding Points in a Reflection

If s is a reflection about an axis d in the plane and P is a point in the plane, a line mapping corresponding points in the reflection s is any line that passes through a point P and its image s(P). Property The lines mapping corresponding points in a reflection form a set of lines that [...]

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Lines Mapping Corresponding Points in a Translation

If t is a translation of the plane and point P is a point in the plane, a line mapping corresponding points in the translation t is any line that passes through P and its image t(P). Property The lines mapping corresponding points in a translation form a set of parallel lines also named a pencil [...]

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Vertex of an Angular Sector

Meeting point of the boundaries of the two intersecting half-planes that form the angular sector. See also: Vertex Angular sector

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Rotational Symmetry

Property that is applied to a geometric figure by which this figure remains unchanged by a given rotation. Synonym for invariant by a rotation. Examples A square ABCD remains unchanged by a rotation of a multiple of 90° around its centre of gravity O. A sphere remains unchanged by a rotation around its centre of gravity, [...]

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Christian Goldbach (1690-1764)

Christian Goldbach was a Russian scientist. Although he was not a mathematician by profession, he was able, through his contacts with others in the field such as Bernouilli and de Moivre, and especially through his correspondence with Leonhard Euler, to develop the following famous conjecture: Every even integer greater than 2 can be expressed as [...]

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Square Root of a Real Number

In the equation n\(^{2}\) = N, the real number n whose square is equal to N. Symbols The square root of N is written as \(\sqrt[2]{N}\) and is read as "the square root of N". Examples If x\(^{2}\) = 100, then \(\sqrt[2]{100}\) = 10, since 10 × 10 = 100. If x\(^{2}\) = 0,25, then \(\sqrt[2]{0,25}\) = 0.5, since [...]

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Aryabhata (476 – 550)

Aryabhata was a famous astronomer and the first great Indian mathematician. He is known for an important treatise, translated in Europe in the 19th century, called the Aryabhatiya, written in Sanskrit (the sacred language of brahmins) in 499 about astronomy and mathematics. Unlike the geocentric doctrine of Ptolemy, which was widely read and inherited from […]

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Auguste Ferdinand Möbius (1790-1868)

Auguste Ferdinand Möbius was a mathematician and theoretical astronomer at the University of Leipzig in Germany. Starting in 1809, he studied mathematics and astronomy at the universities of Leipzig, Göttingen (where he had Carl Friedrich Gauss as a professor) and Halle. He is primarily known for his discovery of the Möbius strip, a non-orientable two-dimensional [...]

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Augustus De Morgan (1806-1871)

Augustus De Morgan was a British mathematician and logician born in India. His father passed away when he was ten years old. With his mother, who wanted him to become a priest, he lived in various cities in the southwest of England and changed schools often. His mathematical talents were discovered when he was fourteen [...]

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Gabriel Cramer (1704 – 1752)

Gabriel Cramer was a Swiss mathematician and professor of mathematics and philosophy at the University of Geneva. Cramer’s contributions to mathematics mostly focus on algebra and geometry in his unique book, a treatise on curves called Introduction à l’analyse des lignes courbes algébriques, which was published in Geneva in 1750. This treatise notably includes a [...]

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Directed Line

Line on which we have defined an order. Example The two axes in a Cartesian coordinate system are directed lines.

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Inconsistent Data

See statistical data.

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Half

Fraction of a whole that corresponds to half of an object or a set of objects. Symbolism The symbol is \(\dfrac{1}{2}\). Examples A half-circle is one half of a circle. Consider the set U = {1, 2, 3, 4, 5, 6}; half of the elements in this set are even numbers.

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Perfect Cube

A number that is the cube of a whole number. Examples 125 is a perfect cube, since 53 = 125 and 5 is a whole number. 25 is not a perfect cube, since \(\sqrt[3]{25}\) is not a whole number. Educational Note The expression "perfect cube" should not be applied to just any algebraic expression. The [...]

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Contraposition

In Boolean logic, a proposition equivalent to a conditional P → Q and defined by the conditional ¬Q →¬P. Examples In logic, the contraposition is a type of reasoning that consists of affirming the implication "if not B then not A" from the implication "if A then B". The implication "if not B then not [...]

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Arabic Numeral

Symbols that came to us from the civilizations of India through Arabic scholars and that we call symbols of the Hindu-Arabic number system. Arabic numerals use ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. When writing numbers, the digits are grouped in sets of three digits separated by a space, [...]

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Certain Digit

See significant digit.

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Conjugate Axis

 

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Transverse Axis

Synonym for the large axis in the representation of an ellipse. See also ellipse in a Cartesian plane.

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Equipollent Bipoints

The bipoints (A, B) and (C, D) are equipollent if the segments AD and BC have the same centre. If two bipoints (A, B) and (C, D) are equipollent, then ABDC is a parallelogram. To go from A to B and from C to D, the route goes in the same sense, the same direction, [...]

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Total Area of a Solid

See area of a solid.

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Lateral Area of a Solid

See area of a solid.

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Hollow Cylinder

Cylinder in which the bases are congruent annuli. Formulas The area A of a hollow cylinder is equal to the sum of the areas of its two bases (annuli), the lateral interior area \(A_1\) and the exterior area \(A_2\) : \(\begin{aligned} A&=A_b + A_1 + A_2 \\ &=2\left(πr_{\scriptsize{2}}^2-πr_{\scriptsize{1}}^2\right) + 2πr_{\scriptsize{1}}h  + 2πr_{\scriptsize{2}}h \\ &=2π\left( r_1+r_2 \right) \left(r_2-r_1+h\right) [...]

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One-To-One Correspondence

See one-to-one.

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Net of a Prism

Plane representation of all of the faces of a prism so that each face can be connected to at least one other by a common edge and all of the faces are connected to each other at least two by two. Example Here is the net of a prism:

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Non-Polyhedron

Name given to a solid that has at least one curved face (plural : non-polyhedra). Spheres, cylinders of revolution and cones of revolution are non-polyhedra. Examples Examples of non-polyhedra:

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Katherine Johnson (1918- …)

African American physicist, mathematician, and space engineer. Katherine Johnson (1918- ...) She contributed to the aeronautics and space programs at the National Aeronautics and Space Administration (NASA). Renowned for her reliability in computerized space navigation, she conducted technical work at NASA that spanned decades. In 2017, she was still living at age 99. A building [...]

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Alphametic

A cryptarithm in which the elements form words that have a meaning in the context. Examples APPLE + GRAPE = CHERRY In this alphametic, the solution is 63 374 + 90 634 = 154 008. SUN + FUN = SWIM In this alphametic, the solution is 136 + 936 = 1072

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Alphabetical Order Number

Number that is used to determine the position of a letter in alphabetical order. Examples The letter H is the eighth letter of the English alphabet; it takes the number 8. The letter A is the first letter of the English alphabet; it takes the number 1. The letter Z is the twenty-sixth letter of [...]

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Unit of Measurement of Space

The base unit to measure space is the cubic metre. The units of measurement of space are units of volume, meaning units of measurement for a three-dimensional geometric object. Properties One cubic metre is equal to 1000 cubic decimetres and we write: 1 m\(^{3}\) = 1000 dm\(^{3}\) Un mètre cube équivaut à 1 000 000 centimètres [...]

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Unit of Measurement of Surface Area

The base unit to measure surface area is the square metre. The units to measure surface are units of area, meaning units of measurement for a two-dimensional geometric object. Properties One square metre is equal to 100 square decimetres and we write: 1 m\(^{2}\) = 100 dm\(^{2}\) One square metre is equal to 10 000 square [...]

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Millenium

 

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Century

 

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Decade

A decade is equal to a duration of 10 years. See also : Unit of measurement of time

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Lustrum

 

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Year

One year is equivalent to a duration of 365 days. PROPERTIES 1 year equals 52 weeks 1 year equals 12 months 1 century equals 100 years See also : Unit of measurement of time

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Day

One day is equivalent to a duration of 24 hours. See “unit of measure of time.” Properties 1 day is equivalent to 1440 minutes 1 week is equivalent to 7 days 1 year is equivalent to 365 days

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Hour

One hour is equal to a duration of 60 minutes. Properties 1 hour is equal to 3600 seconds 1 day is equal to 24 hours See also Unit of measure of time

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Second

The second is the base unit for measuring time. Symbol The symbol used to refer to a measurement of time of seconds is s. This means that 36 s is read as: 36 seconds. The sexagesimal second is noted with the double prime symbol. This means that 36″ is read as: 36 seconds. Properties 1 minute is equal [...]

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Unit of Measurement of Time

The base unit to measure time is the second. Quantitatively, the second is defined by a number of oscillations (exactly 9 192 631 770) of a cesium atom. Measuring and counting these oscillations is carried out by electronic clocks. It is a very precise measurement. Symbols The usual symbols used for time measurement are: second [...]

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Is Less Than

See order relation.

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Is Greater Than

See order relation.

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Cardinality

See cardinal number.

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Cube

Originally, a cube was a game die. This term was later used to refer to a geometric solid, and then the third power of a number. See also cube of a number cube (geometry) perfect cube

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Number of

See denominate number.

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As Much

The terms “as much” expresses equality between two numbers or quantities. Example Marie has $10 and Jean also has $10. Therefore, they both have as much money as each other. See also comparison

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The Least

See comparison.

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The Greatest

See comparison.

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Comparison

Relationship established between two numbers, two sets, two terms or two expressions. Symbols To compare mathematical objects based on an order relation, we usually use the symbols <, >, =, ≤ and ≥. The symbol < is read as "is less than" Example : 5 < 10. The symbol > is read as "is greater [...]

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Thousand

Cardinal number representing 100 groups of 10 objects, in the decimal numeral system. See also: Thousands (Place Value)

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To Enumerate

See enumeration.

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Grouping

Action of gathering or assembling objects according to a given base. In the decimal number system, to count objects, we group them into blocks of 10. Example If we count twenty-three objects, we have 2 blocks of 10 objects and one block of three objects; therefore, we have 23 objects.

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Tens

Grouping of 10 ones in the decimal number system. Examples In the number 35, the digit 3 occupies tens position and its place value is 30. In the number 3245, the digit 4 occupies the tens position and its place value is 40. The value of the digit 6 in the number 165 is "six [...]

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To Count

See counting.

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Archimedes of Syracuse (c. 287- c. 212 BCE)

Greek mathematician, geometer, physicist and engineer. Archimedes was one of the leading scientists and the greatest mathematician of classical antiquity. In fact, he is considered to be one of the greatest of all time. Archimedes of Syracuse (c. 287- c. 212 BCE.) Historical Note Archimedes’ mathematical writings were not widely known in antiquity, but he [...]

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Cevian

Line segment joining the vertex of a triangle to its opposite side. Properties In triangles, altitudes, medians and angle bisectors are special types of cevians. The length of a cevian can be determined by using the formula: \(b^{2}m+c^{2}n = a(d^{2}+mn)\)   If the cevian is an altitude, its length is given by the formula: \(d^{2}=b^{2}-n^{2}=c^{2}-m^{2}\). [...]

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Sociable Numbers

Synonym of amicable numbers.

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Gottfried Leibniz (1646-1716)

German mathematician, philosopher, diplomat, jurist and philologist who wrote in Latin, German and French. Some authors attribute the introduction of the terms abscissa and ordinate in 1692, to Leibniz. Gottfried Leibniz (1646-1716)

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Richard Dedekind (1831-1916)

German mathematician who proposed an axiomatic definition of the set of integers as well as an explicit construction of real numbers based on rational numbers. Dedekind introduced the notation \(\mathbb{N}\) in 1888 to refer to the set of non-zero natural numbers. Today, this set is denoted by \(\mathbb{N}\)*. Dedekind also proposed the use of the symbol \(\mathbb{R}\) to [...]

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Johann Rahn (1622-1676)

Swiss mathematician. In his treatise on algebra from 1659, he introduced the symbol * as a multiplication symbol and the symbol ÷ as the division symbol (a ÷ b). Johann Rahn (1622-1676)

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Pierre Herigone (1580-1644)

French mathematician and astronomer of Basque origin. He introduced the symbol of orthogonality (⊥) and the term parallelepiped.

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John Napier (1550-1617)

Scottish mathematician, theologian, physicist and astronomer. He brought the decimal point into common use and invented logarithms. John Napier (1550-1617)

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Christian Kramp (1760-1826)

Christian Kramp, a French mathematician, was the first to use the factorial notation (n!) as we know it today, in his work Éléments d’Arithmétique Universelle in 1808. The more general concept of factorial was introduced at the same time by French mathematician Louis François Antoine Arbogast (1759-1803).

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Raphael Bombelli (1526-1572)

Italian mathematician in the 16th century. Raphael Bombelli was the son of a merchant from Bologna and became an engineer (in particular, he drained swamps). He was employed by a Roman, Alessandro Ruffini, to carry out a long project that was interrupted for a few years, which gave him time to write about algebra during [...]

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John Wallis (1616-1703)

Seventeenth century English mathematician. His work, a precursor of Newton's, also paved the way for the development of phonetics, Deaf education and speech therapy. He is credited with having introduced the infinity symbol (∞) as it is used today. He was also one of the founders of the famous Royal Society (Oxford). John Wallis (1616-1703)

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Karl Weierstrass (1815-1897)

Nineteenth century German mathematician. He introduced the use of two vertical lines to write the absolute value of mathematical expressions such as |24| or |2x + 3|, etc. Karl Weierstrass (1815-1897)

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Christopher Clavius (1538-1612)

Christophorus Clavius (Latinization of his birth name) was a German Jesuit mathematician and astronomer. He was the first to use parentheses to isolate mathematical expressions. He was commissioned by Pope Gregory XIII to write the basis for a reformed calendar. The calendar was instituted by Pope Gregory XIII in 1582 and gradually adopted throughout the [...]

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