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Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria (e.g. "two counties over"). In mathematics, a distance function or metric is a generalization of the concept of physical distance. A metric is a function that behaves according to a specific set of rules, and provides a concrete way of describing what it means for elements of some space to be "close to" or "far awa

Primary mathematics:Numbers

Teaching Number This page is for teachers or home-schoolers. It is about teaching the basic concepts and conventions of simple number. Developing a sound concept of number Children typically learn about numbers at a very young age by learning the sequence of words, "one, two, three, four, five" etc. Usually, in chanting this in conjunction with pointing at a set of toys, or mounting a flight of steps for example. Typically, 'mistakes' are made. Toys or steps are missed or counted twice,

Greatest common divisor

In mathematics, the greatest common divisor (gcd) , also known as the greatest common factor (gcf) , or highest common factor (hcf) , of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder. This notion can be extended to polynomials, see greatest common divisor of two polynomials. Overview The greatest common divisor of a and b is written as gcd( a , b ), or sometimes simply as ( a , b ). For example, gcd(12, 18) = 6, gcd(-4, 14) = 2. Two

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


Volume Many three-dimensional solids can be generated by revolving a curve about the x-axis or y-axis. For example, if we revolve the semi-circle given by f(x) = ______ Ö r 2 -x 2 about the x-axis, we obtain a sphere of radius r. We can derive the familiar formula for the volume of this sphere. Finding the Volume of a Sphere Consider a cross-section of the sphere as shown. This cross-section is a circle with radius f(x) and area p [f(x)] 2 . Informally speaking, if we "slice" the sphere vertica

Brahmagupta interpolation formula

In trigonometry, the Brahmagupta interpolation formula is a special case of the Newton-Stirling interpolation formula to the second-order, which Brahmagupta used in 665 to interpolate new values of the sine function from other values already tabulated. The formula gives an estimate for the value of a function f at a value a + xh of its argument (with h > 0 and -1 = x = 1) when its value is already known at a - h, a and a + h . The formula for the estimate is: where ? is the first-order forward-


Combinatorics is a branch of pure mathematics concerning the study of discrete (and usually finite ) objects. It is related to many other areas of mathematics , such as algebra , probability theory , ergodic theory and geometry , as well as to applied subjects in computer science and statistical physics . Aspects of combinatorics include "counting" the objects satisfying certain criteria ( enumerative combinatorics ), deciding when the criteria can be met, and constructing and analyzing objects

Additive inverse

In mathematics, the additive inverse , or opposite , of a number n is the number that, when added to n , yields zero. The additive inverse of F is denoted - F . For example, the additive inverse of 7 is -7, because 7 + (-7) = 0, and the additive inverse of -0.3 is 0.3, because -0.3 + 0.3 = 0. In other words, the additive inverse of a number is the number's negative. For example, the additive inverse of 8 is -8, the additive inverse of 10002 is -10002 and the additive inverse of x² is -x². T


This article is about addition in mathematics. For addition reaction in chemistry, see addition reaction. Addition is the mathematical operation of combining or adding two numbers to obtain an equal simple amount or total. Addition also provides a model for related processes such as joining two collections of objects into one collection. Repeated addition of the number one is the most basic form of counting. Performing addition is one of the simplest numerical tasks, accessible to infants as yo


A composite number is a positive integer which has a positive divisor other than one or itself. In other words, if n > 0 is an integer and there are integers 1 < a , b < n such that n = a × b then n is composite. By definition, every integer greater than one is either a prime number or a composite number. The number one is a unit - it is neither prime nor composite. For example, the integer 14 is a composite number because it can be factored as 2 × 7. Likewise, the integers 2 and 3 are not comp

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

Negative and non-negative numbers

Being negative or non-negative is a property of a number which is real, or a member of a subset of real numbers such as rational and integer numbers. A negative number is one that is less than zero, such as - , -1.414, -1. A positive number (e.g., positive real number, positive rational number, positive integer) is one that is greater than zero, such as , 1.414, 1. Zero itself is neither positive nor negative . The non-negative numbers are the numbers that are not negative (they are positive or

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

Algebra-Composition of Functions: Composing Functions at Points

Composition of Functions: Composing Functions at Points Suppose you are given two functions, f (x) = 2x + 3 and g(x) = –x2 + 5. Composition means that you can plug g(x) into f (x). This is written as "( f o g)(x)", which is read as "f-compose-g of x". And "( f o g)(x)" means " f (g(x))". That is, you plug something in for x, then you plug that value into g, simplify, and then plug the result into f. This is just like what I did above, except that I will be evaluating functions to find values, r


A paradox is a statement or group of statements that leads to a contradiction or a situation which defies intuition. The term is also used for an apparent contradiction that actually expresses a non-dual truth ( cf. kōan, Catuskoti ). Typically, the statements in question do not really imply the contradiction, the puzzling result is not really a contradiction, or the premises themselves are not all really true or cannot all be true together. The word paradox is often used interchangeably w


Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. Most contemporary authors simply write algebra instead of abstract algebra. The term abstract algebra now refers to the study of all algebraic structures, as distinct from the elementary algebra ordinarily taught to children, which teaches the correct rules for manipulating formulas and algebraic expressions involving real and complex number

Right-angled triangle

Right-angled triangle definitions A right triangle always includes a 90° (p/2 radians) angle, here labeled C. Angles A and B may vary. Trigonometric functions specify the relationships among side lengths and interior angles of a right triangle. (Top): Trigonometric function sin ? for selected angles ? , p - ? , p + ? , and 2 p - ? in the four quadrants. (Bottom) Graph of sine function versus angle. Angles from the top panel are identified. The notion that there should be some standard correspon


Sexagesimal ( base 60 ) is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, it was passed down to the ancientBabylonians, and it is still used — in a modified form — for measuring time, angles, and the geographic coordinates that are angles. The number 60, a highly composite number, has twelve factors, namely { 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 } of which two, three, and five are prime numbers. With so many factors, many fraction


A complex number can be visually represented as a pair of numbers forming a vector on a diagram called an Argand diagram In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit , denoted i , which satisfies: [ 1 ] Every complex number can be written in the form a + bi , where a and b are real numbers called the real part and the imaginary part of the complex number, respectively. Complex numbers are a field, and thus have addition, subtra


In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices , and the links that connect some pairs of vertices are called edges . Typically, a graph is depicted in diagrammatic form as a set of dots for the vertices, joined by lines or curves for the edges. The edges may be directed (asymmetric) or undirected (symmetric). For example, if

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