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Primary mathematics:Fractions

In the early primary grades, fractions are generally thought of as "a part" of a whole, and are used to express numbers less that one. Yet, fractions encompass much more mathematical territory than this. They can of course be larger than one. Rational numbers are by definition fractions.. in essence the same as division. Fractions are also used to express ratios. By the end of the primary grades, students should be making the connections between all of these possible meanings of for instance, t



Coordinates are numbers which describe the location of points in a plane or in space. For example, the height above sea level is a coordinate which is useful for describing points near the surface of the earth. A coordinate system , in a plane or in space, is a systematic method of assigning a pair or a triple of numbers to each point in the plane or in space (respectively) which describe its position uniquely. For example, the triple consisting of latitude , longitude and altitude (height abov

Calculus-Partial Differentiation

Partial Differentiation Suppose you want to forecast the weather this weekend in Los Angeles. You construct a formula for the temperature as a function of several environmental variables, each of which is not entirely predictable. Now you would like to see how your weather forecast would change as one particular environmental factor changes, holding all the other factors constant. To do this investigation, you would use the concept of a partial derivative... Let the temperature T depend on vari

Roman numerals

Roman numerals Roman numerals stem from the numeral system of ancient Rome . They are based on certain letters of the alphabet which are combined to signify the sum (or, in some cases, the difference) of their values. The first ten Roman numerals are: I, II, III, IV, V, VI, VII, VIII, IX, and X. The Roman numeral system is decimal but not directly positional and does not include a zero . It is a cousin of the Etruscan numerals , and the letters derive from earlier non-alphabetical symbols; over


A polygon is a closed figure made by joining line segments, where each line segment intersects exactly two others. Examples: The following are examples of polygons: The figure below is not a polygon, since it is not a closed figure: The figure below is not a polygon, since it is not made of line segments: The figure below is not a polygon, since its sides do not intersect in exactly two places each: Regular Polygon A regular polygon is a polygon whose sides are all the same length, and whose an

Coordinate System

In mathematics and its applications, a coordinate system is a system for assigning an n -tuple of numbers or scalars to each point in an n -dimensional space. This concept is part of the theory of manifolds. [1] "Scalars" in many cases means real numbers, but, depending on context, can mean complex numbers or elements of some other commutative ring. For complicated spaces, it is often not possible to provide one consistent coordinate system for the entire space. In this case, a collection of co


Counting is the action of finding the number of elements of a finite set of objects. The traditional way of counting consists of continually increasing a (mental or spoken) counter by a unit for every element of the set, in some order, while marking (or displacing) those elements to avoid visiting the same element more than once, until no unmarked elements are left; if the counter was set to one after the first object, the value after visiting the final object gives the desired number of elemen

Barcan formula

In quantified modal logic, the Barcan formula and the converse Barcan formula (more accurately, schemata rather than formulae) (i) syntactically state principles or interchange between quantifiers and modalities; (ii) semantically state a relation between domains of possible worlds. The formulae were introduced as axioms by Ruth Barcan Marcus, in the first extensions of modal propositional logic to include quantification. [ 1 ] Related formulas include the Buridan formula , and the converse Bur

calculus-Change of Basis

Change of Basis Let V be a vector space and let S = { v 1 , v 2 , ¼ , v n } be a set of vectors in V. Recall that S forms a basis for V if the following two conditions hold: S is linearly independent . S spans V. If S = { v 1 , v 2 , ¼ , v n } is a basis for V, then every vector v Î V can be expressed uniquely as a linear combination of v 1 , v 2 , ¼ , v n : v = c 1 v 1 + c 2 v 2 + ¼ + c n v n . Think of é ê ê ê ê ê ë c 1 c 2 : c n ù ú ú ú ú ú û as the coordinates of v relative to the basis S.


The integers (from the Latin integer, which means with untouched integrity, whole, entire) are the set of numbers consisting of the natural numbers including 0 (0, 1, 2, 3, ...) and their negatives (0, -1, -2, -3, ...). They are numbers that can be written without a fractional or decimal component, and fall within the set {... -2, -1, 0, 1, 2, ...}. For example, 65, 7, and -756 are integers; 1.6 and 1½ are not integers. In other terms, integers are the numbers one can count with items such as a

Apollonian circles

Apollonian circles are two families of circles such that every circle in the first family intersects every circle in the second family orthogonally, and vice versa. These circles form the basis for bipolar coordinates. They were discovered by Apollonius of Perga, a renowned Greek geometer. Definition The Apollonian circles are defined in two different ways by a line segment denoted CD . Each circle in the first family (the blue circles in the figure) is associated with a positive real number r

Parabolic coordinates

Parabolic coordinates are a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal parabolas. A three-dimensional version of parabolic coordinates is obtained by rotating the two-dimensional system about the symmetry axis of the parabolas. Parabolic coordinates have found many applications, e.g., the treatment of the Stark effect and the potential theory of the edges. Two-dimensional parabolic coordinates Two-dimensional parabolic coordinates (s,t) are defined b


A ratio is an expression that compares quantities relative to each other. The most common examples involve two quantities, but any number of quantities can be compared. Ratios are represented mathematically by separating each quantity with a colon – for example, the ratio 2:3, which is read as the ratio "two to three". The quantities separated by colons are sometimes called terms . Examples The quantities being compared in a ratio might be physical quantities such as speed or temperature, or ma


In mathematics, a matrix (plural matrices , or less commonly matrixes ) is a rectangular array of numbers, such as An item in a matrix is called an entry or an element. The example has entries 1, 9, 13, 20, 55, and 4. Entries are often denoted by a variable with two subscripts, as shown on the right. Matrices of the same size can be added and subtracted entrywise and matrices of compatible sizes can be multiplied. These operations have many of the properties of ordinary arithmetic, except that


The integers (from the Latin integer , literally "untouched", hence "whole": the word entire comes from the same origin, but via French [ 1 ] ) are formed by thenatural numbers including 0 (0, 1, 2, 3, ...) together with the negatives of the non-zero natural numbers (-1, -2, -3, ...). Viewed as a subset of the real numbers, they are numbers that can be written without a fractional or decimal component, and fall within the set {... -2, -1, 0, 1, 2, ...}. For example, 65, 7, and -759 are integers


In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339.... The real numbers include rational numbers, such as 42 and -23/129, and irrational numbers, such as p and the square root of 2, and can be represented as points along an infinitely long number line. A more rigorous definition of the real numbers was one of the most important developments of 19th century mathematics. Popular definitions in use t


Fraction is part of a whole. The following illustrate the concept;The figure has 4 parts and only 1 part of the figure is shaded. Now, suppose you go to Domino's pizza, you may order a medium pizza. If your pizza has 8 slices and you did not eat the whole pizza, this means that you only ate parts of the whole pizza. Let's say you don't have a big appetite and eat only 2 out of those 8 slices. The following figure illustrate the situation we write 2/8 and we call 2 the numerator and

Calculus-Lines, Planes, and Vectors

Lines, Planes, and Vectors In this tutorial, we will use vector methods to represent lines and planes in 3-space. Displacement Vector The displacement vector v with initial point (x 1 ,y 1 ,z 1 ) and terminal point (x 2 ,y 2 ,z 2 ) is v = (x 2 -x 1 ,y 2 -y 1 ,z 2 -z 1 ) Why? That is, if vector v were positioned with its initial point at the origin, then its terminal point would be at (x 2 -x 1 ,y 2 -y 1 ,z 2 -z 1 ). Example The vector v with initial point (-1,4,5) and final point (4,-3,2) is v


In mathematics, a multiset (or bag ) is a generalization of a set. While each member of a set has only one membership, a member of a multiset can have more than one membership (meaning that there may be multiple instances of a member in a multiset, not that a single member instance may appear simultaneously in several multisets). The term "multiset" was coined by Nicolaas Govert de Bruijn in the 1970s. [ 1 ] The use of multisets in mathematics predates the name "multiset" by nearly 90 years. Ri

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