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In arithmetic, the result of the division of two integers usually cannot be expressed with an integer quotient, unless a remainder —an amount "left over"— is also accepted. The remainder for natural numbers If a and d are natural numbers, with d non-zero, it can be proved that there exist unique integers q and r , such that a = qd + r and 0 = r < d . The number q is called the quotient , while r is called the remainder . Examples When dividing 13 by 10, 1 is the quotient and 3 is the remainder,

Trigonometric functions

In mathematics, the trigonometric functions (also called circular functions ) are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle. Trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications. The most familiar trigonometric functions are the sine, cosine, and tangent. In the context of the standard unit circle with radius 1, where a triangle is formed by a ray ori

Point (geometry)

In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they have do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics dealing with set theory, an element is often referred to as a point . A point could also be defined as a sphere which has a diameter of zero. [edit] Points in Euclidean geometry A finite set of points

Invertible matrix

In linear algebra an n -by- n (square) matrix A is called invertible or nonsingular or nondegenerate, if there exists an n -by- n matrix B such that where I n denotes the n -by- n identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix B is uniquely determined by A and is called the inverse of A , denoted by A -1 . It follows from the theory of matrices that if for square matrices A and B , then also [ 1 ] Non-square matrices ( m -by-


In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and -23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits continue in some way; or, the real numbers may be thought of as points on an infinitely long number line. These descriptions of the real numbers, while intuitively


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


In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. Specifically, if c times b equals a , written: where b is not zero, then a divided by b equals c , written: For instance, since . In the above expression, a is called the dividend , b the divisor and c the quotient . Conceptually, division describes two distinct but related settings. Partitioning involves taking a set of size a and forming b groups that are equal in s


In mathematics, an average , or central tendency [ 1 ] of a data set is a measure of the "middle" or "expected" value of the data set. There are many different descriptive statistics that can be chosen as a measurement of the central tendency of the data items. An average is a single value that is meant to typify a list of values. If all the numbers in the list are the same, then this number should be used. If the numbers are not all the same, an easy way to get a representative value from a li

Elementary algebra

Elementary algebra is a fundamental and relatively basic form of algebra taught to students who are presumed to have little or no formal knowledge of mathematics beyond arithmetic . While in arithmetic only numbers and their arithmetical operations (such as +, -, ×, ÷) occur, in algebra one also uses symbols (such as x and y , or a and b ) to denote numbers. These are called variables . This is useful because: It allows the generalization of arithmetical equations (and inequalities ) to be stat

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

Calculus-Parametric Equations

Parametric Equations Think of a curve being traced out over time, sometimes doubling back on itself or crossing itself. Such a curve cannot be described by a function y = f(x). Instead, we will describe our position along the curve at time t by x = x(t) y = y(t) . Then x and y are related to each other through their dependence on the parameter t. Example Suppose we trace out a curve according to x = t 2 - 4t y = 3t . t x y 0 0 0 1 -3 3 2 -4 6 3 -3 9 4 0 12 where t ³ 0. Drag the point along the

Calculus - Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors We review here the basics of computing eigenvalues and eigenvectors. Eigenvalues and eigenvectors play a prominent role in the study of ordinary differential equations and in many applications in the physical sciences. Expect to see them come up in a variety of contexts! Definitions Let A be an n ×n matrix. The number l is an eigenvalue of A if there exists a non-zero vector v such that Av = l v . In this case, vector v is called an eigenvector of A corresponding to

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


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

Digital geometry

Digital geometry deals with discrete sets (usually discrete point sets) considered to be digitized models or images of objects of the 2D or 3D Euclidean space. Simply put, digitizing is replacing an object by a discrete set of its points. The images we see on the TV screen, the raster display of a computer, or in newspapers are in fact digital images. Its main application areas are computer graphics and image analysis. Main aspects of study are: Constructing digitized representations of objects


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

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

Great circle

A great circle of a sphere is a circle that runs along the surface of that sphere so as to cut it into two equal halves, as distinct from a small circle. The great circle therefore has both the same circumference and the same center as the sphere. It is the largest circle that can be drawn on a given sphere. Great circles serve as the analogue of "straight lines" in spherical geometry. See also spherical trigonometry and geodesic. The great circle, also known as the Riemannian circle, is the pa


In mathematics, modular arithmetic (sometimes called clock arithmetic ) is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus . The Swiss mathematician Leonhard Euler pioneered the modern approach to congruence in about 1750, when he explicitly introduced the idea of congruence modulo a number N . Modular arithmetic was further advanced by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae , published in 1801. Time-keeping on

Babylonian numerals

Babylonian numerals were written in cuneiform, using a wedge-tipped reed stylus to make a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record. The Babylonians, who were famous for their astronomical observations and calculations (aided by their invention of the abacus), used a sexagesimal (base-60) positionalnumeral system inherited from the Sumerian and also Akkadian civilizations. Neither of the predecessors was a positional system (having a con

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