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Home >> Educational Materials >> Mathematics## BINARY OPERATION

In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division. More precisely, a binary operation on a set S is a binary relation that maps elements of the Cartesian product S × S to S : If f is not a function, but is instead a partial function, it is called a partial operation . For instance, division of real numbers is a parti## Circumference

The circumference is the distance around a closed curve. Circumference is a special perimeter. Circumference of a circle The circumference of a circle is the length around it. The circumference of a circle can be calculated from its diameter using the formula: Or, substituting the radius for the diameter: where r is the radius and d is the diameter of the circle, and the Greek letter p is defined as the ratio of the circumference of the circle to its diameter. The numerical value of p is 3.141## 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## Quotient

In mathematics, a quotient is the result of a division. For example, when dividing 6 by 3, the quotient is 2, while 6 is called the dividend, and 3 the divisor. The quotient can also be expressed as the number of times the divisor divides into the dividend. A quotient can also mean just the integer part of the result of dividing two integers. For example, the quotient of 13 ÷ 5 would be 2 while the remainder would be 3. For more, see the division algorithm. In more abstract branches of mathemat## Partial-Fraction Decomposition

Partial-Fraction Decomposition: General Techniques Previously, you have added and simplified rational expressions, such as: Partial-fraction decomposition is the process of starting with the simplified answer and taking it back apart, of "decomposing" the final expression into its initial polynomial fractions. To decompose a fraction, you first factor the denominator. Let's work backwards from the example above. The denominator is x 2 + x , which factors as x ( x + 1) . Then you write the f## 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## 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## Vector fields in cylindrical and spherical coordinates

Vector fields Vectors are defined in cylindrical coordinates by (?,f,z), where ? is the length of the vector projected onto the X-Y-plane, f is the angle of the projected vector with the positive X-axis (0 = f < 2p), z is the regular z-coordinate. (?,f,z) is given in cartesian coordinates by: or inversely by: Any vector field can be written in terms of the unit vectors as: The cylindrical unit vectors are related to the cartesian unit vectors by: Note: the matrix is an orthogonal matrix, that i## Least common multiple

In arithmetic and number theory, the least common multiple or lowest common multiple ( LCM ) or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple of both of a and of b . Since it is a multiple, it can be divided by a and b without a remainder. If either a or b is 0, so that there is no such positive integer, then LCM( a , b ) is defined to be zero. The definition is sometimes generalized for more than two integers: The lowest common multiple of## Divisibility criteria

Divisibility of numbers by 2, 4, 8, 3, 9, 6, 5, 25, 10, 100, 1000, 11. Divisibility by 2 . A number is divisible by 2, if its last digit is 0 or is divisible by 2. Numbers, which are divisible by 2 are called even numbers. Otherwise, numbers are called odd numbers. Divisibility by 4 . A number is divisible by 4, if its two last digits are zeros or they make a two-digit number, which is divisible by 4. Divisibility by 8 . A number is divisible by 8, if its three last digits are zeros or they mak## ANGLE

In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. [ 1 ] The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept out when one ray is rotated about the vertex to coincide with the other (see "Measuring angles", below). Where there is no possibility of confusion, the term "angle" is used interchangeably for both the geometric configuration## Combinatorics

Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size ( enumerative combinatorics ), deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria (as in combinatorial designs and matroid theory ), finding "largest", "smallest", or "optimal" objects ( extremal combinatorics and combinatorial optimization ), and studying co## Axiom

In traditional logic , an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident , or subject to necessary decision . Therefore, its truth is taken for granted, and serves as a starting point for deducing and inferring other (theory dependent) truths. In mathematics , the term axiom is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms" . In both senses, an axiom is any mathematical statement that ser## Analytic combinatorics

Analytic combinatorics is a branch of combinatorics that describes combinatorial classes using generating functions, with formal power series that often correspond to analytic functions. Given a generating function, analytic combinatorics attempts to describe the asymptotic behavior of a counting sequence using algebraic techniques. This often involves analysis of the singularities of the associated analytic function. Two types of generating functions are commonly used — ordinary and exponentia## Isomorphism

In abstract algebra, an isomorphism (Greek: ἴs?? isos "equal", and µ??fή morphe "shape") is a bijective map f such that both f and its inverse f -1 are homomorphisms, i.e., structure-preserving mappings. In the more general setting of category theory, an isomorphism is a morphism f : X ? Y in a category for which there exists an "inverse" f -1 : Y ? X , with the property that both f -1 f = id X and f f -1 = id Y . Informally, an isomorphism is a kind of mapping between objects that s## ORDER THEORY

Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of ordering, providing a framework for saying when one thing is "less than" another. This article gives a detailed introduction to the field and includes some of the most basic definitions. For a quick lookup of order-theoretic terms, there is also an order theory glossary. A list of order topics collects the various articles in the vicinity of order theory. Background and mo## Calculus - Convergence Tests for Infinite Series

Convergence Tests for Infinite Series In this tutorial, we review some of the most common tests for the convergence of an infinite series ¥ å k = 0 a k = a 0 + a 1 + a 2 + ¼ The proofs or these tests are interesting, so we urge you to look them up in your calculus text. Let s 0 = a 0 s 1 = a 1 : s n = n å k = 0 a k : If the sequence { s n } of partial sums converges to a limit L, then the series is said to converge to the sum L and we write ¥ å k = 0 a k = L. For j ³ 0, ¥ å k = 0 a k converges## Algebra

Algebra is a branch of mathematics concerning the study of structure, relation and quantity. The name is derived from the treatise written by the Persian [1] mathematician, astronomer, astrologer and geographer, Muhammad bin Mūsā al-Khwārizmī titled (in Arabic الكتاب الجبر والمقابلة ) Al-Kitab al-Jabr wa-l-Muqabala (meaning " The Compendious Book on Calcula## Theory of computation

The theory of computation is the branch of computer science that deals with whether and how efficiently problems can be solved on a model of computation, using an algorithm. The field is divided into two major branches: computability theory and complexity theory, but both branches deal with formal models of computation. In order to perform a rigorous study of computation, computer scientists work with a mathematical abstraction of computers called a model of computation. There are several model## Radian and degree measures of angles

Degree and radian measures of angles. Relation of a circle radius and a circumference length. Table of degree and radian measures for some most used angles. A degree measure. Here a unit of measurement is a degree (its designation is ° or deg ) – a turn of a ray by the 1 / 360 part of the one complete revolution. So, the complete revolution of a ray is equal to 360 deg. One degree is divided into 60 minutes (a designation is ‘ or min ); one minute – correspondingly into 60 seconds (a designatio$.ajax({
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