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Arithmetical operations

Addition (addends, sum). Subtraction (minuend, subtrahend, difference). Multiplication (multiplicand, multiplier, product, factors). Division (dividend, divisor, quotient, dividing integers, fraction, divisible numbers, remainder, division without remainder, division with remainder). Raising to a power (power, base of a power, index or exponent of a power, value of a power). Extraction of a root (root, radicand, index or degree of a root, value of a root, square root, cube root). Mutually inver

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

Finite model theory

In mathematical logic, finite model theory is a subfield of model theory that focuses on properties of logical languages, such as first-order logic, over finite structures, such as finite groups, graphs, databases, and most abstract machines. It focuses in particular on connections between logical languages and computation, and is closely associated with discrete mathematics, complexity theory, and database theory. Many important results of first-order logic and classical model theory fail when

Calculus-Taylor's Theorem

Taylor's Theorem Suppose we're working with a function f(x) that is continuous and has (n+1) continuous derivatives on an interval about x = 0. We can approximate f near 0 by a polynomial P n (x) of degree n: For n = 0, the best constant approximation near 0 is P 0 (x) = f(0) which matches f at 0. For n = 1, the best linear approximation near 0 is P 1 (x) = f(0)+f ¢ (0)x. Note that P 1 matches f at 0 and P 1 ¢ matches f ¢ at 0. For n = 2, the best quadratic approximation near 0 is P 2 (

Modus ponens

In classical logic, modus ponendo ponens ( Latin for the way that affirms by affirming ; [ 1 ] often abbreviated to MP or modus ponens ) is a valid , simple argument form sometimes referred to as affirming the antecedent or the law of detachment . It is closely related to another valid form of argument, modus tollens . Modus ponens is a very common rule of inference, and takes the following form: If P , then Q . P . Therefore, Q Formal notation The modus ponens rule may be written in sequent no

Primary mathematics:Adding numbers

Adding two single digit numbers [edit] When teaching young students how to add you will need to use physical items to help them understand how addition works, use items that are similar to avoid confusing younger children. It is very important that they understand the number system before trying to add or subtract. The following is an example of how to show a child how to add. I have two blocks, Mary has two blocks, if Mary gives me two blocks how many blocks do I have? First show that you do i

division by zero

In mathematics, division by zero is a term used if the divisor (denominator) is zero. Such a division can be formally expressed as a / 0 where a is the dividend (numerator). Whether this expression can be assigned a well-defined value depends upon the mathematical setting. In ordinary (real number) arithmetic, the expression has no meaning, as there is no number which, multiplied by 0, gives a ( a ?0). In computer programming, an attempt to divide by zero may, depending on the programming langu

GRAPH

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

Binary arithmetic

Binary arithmetic Arithmetic in binary is much like arithmetic in other numeral systems. Addition, subtraction, multiplication, and division can be performed on binary numerals. Addition The circuit diagram for a binary half adder, which adds two bits together, producing sum and carry bits. The simplest arithmetic operation in binary is addition. Adding two single-digit binary numbers is relatively simple, using a form of carrying: 0 + 0 ? 0 0 + 1 ? 1 1 + 0 ? 1 1 + 1 ? 10, carry 1 (since 1 + 1

Paradox

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

AVERAGE

In mathematics, an average , 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. These include means, the median and the mode. Other statistical measures such as the standard deviation and the range are called measures of spread and describe how spread out the data is. An average is a single value that is meant to typif

REAL NUMBERS

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

MATRIX

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

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

HYPERBOLA

In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror images of each other and resembling two infinite bows. The hyperbola is one of the four kinds of conic section, formed by the intersection of a plane and a cone. The other conic sections are the parabola,

Combining Probability

Decision theory

Decision theory in philosophy, mathematics and statistics is concerned with identifying the values, uncertainties and other issues relevant in a given decision, its rationality , and the resulting optimal decision. It is very closely related to the field of game theory. Normative and descriptive decision theory Most of decision theory is normative or prescriptive , i.e. , it is concerned with identifying the best decision to take, assuming an ideal decision maker who is fully informed, able to

CONSERVATIVE VECTOR FIELD

In vector calculus a conservative vector field is a vector field which is the gradient of a scalar potential. There are two closely related concepts: path independence and irrotational vector fields. Every conservative vector field has zero curl (and is thus irrotational), and every conservative vector field has the path independence property. In fact, these three properties are equivalent in many 'real-world' applications. Definition A vector field is said to be conservative if there e

RATIONAL NUMBER

In mathematics, a rational number is any number that can be expressed as the quotient a / b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number. The set of all rational numbers is usually denoted by a boldface Q (or blackboard bold , Unicode U+211a ℚ ), which stands for quotient. The decimal expansion of a rational number always either terminates after finitely many digits or begins to repeat the same finite sequence of

Decimal Superbase

Many numeral systems with base 10 use a superimposed larger base of 100, 1000, 10000 or 1000000. It is a power of 10 and might be called a superbase or superradix of the numeral system. The superbase is mainly used in the spoken/written language but also apparent when writing large numbers with digits by grouping of digits, as a mental aid of measuring the number. Mathematical description The spoken numeral system uses the ten arithmetic numerals zero, one, two , three, four, five, six, seven,







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