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Approximation
Related concepts and fundamentals: * Agnosticism
Agnosticism
* Epistemology
Epistemology
* Presupposition * Probability
Probability
* v * t * e An APPROXIMATION is anything that is similar but not exactly equal to something else. CONTENTS * 1 Etymology and usage * 2 Mathematics * 3 Science * 4 Unicode
Unicode
* 5 LaTeX Symbols * 6 See also * 7 References * 8 External links ETYMOLOGY AND USAGEThe word approximation is derived from Latin
Latin
approximatus, from proximus meaning very near and the prefix ap- (ad- before p) meaning to. Words like approximate, approximately and approximation are used especially in technical or scientific contexts. In everyday English, words such as roughly or around are used with a similar meaning
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Functional Analysis
FUNCTIONAL ANALYSIS is a branch of mathematical analysis , the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product , norm , topology , etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous , unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations . The usage of the word functional goes back to the calculus of variations , implying a function whose argument is a function and the name was first used in Hadamard 's 1910 book on that subject
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Real Number
In mathematics , a REAL NUMBER is a value that represents a quantity along a line . The adjective real in this context was introduced in the 17th century by René Descartes
René Descartes
, who distinguished between real and imaginary roots of polynomials . The real numbers include all the rational numbers , such as the integer −5 and the fraction 4/3, and all the irrational numbers , such as √2 (1.41421356..., the square root of 2 , an irrational algebraic number ). Included within the irrationals are the transcendental numbers , such as π (3.14159265...). Real numbers can be thought of as points on an infinitely long line called the number line or real line , where the points corresponding to integers are equally spaced. Any real number can be determined by a possibly infinite decimal representation , such as that of 8.632, where each consecutive digit is measured in units one tenth the size of the previous one
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Rational Number
In mathematics , a RATIONAL NUMBER is any number that can be expressed as the quotient or fraction p/q of two integers , a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number. The set of all rational numbers, often referred to as "THE RATIONALS", the FIELD OF RATIONALS or the FIELD OF RATIONAL NUMBERS is usually denoted by a boldface Q (or blackboard bold Q {displaystyle mathbb {Q} } , Unicode ℚ); it was thus denoted in 1895 by Giuseppe Peano
Giuseppe Peano
after quoziente, Italian for "quotient ". The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number. These statements hold true not just for base 10 , but also for any other integer base (e.g. binary , hexadecimal )
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Irrational Number
In mathematics , the IRRATIONAL NUMBERS are all the real numbers which are not rational numbers , the latter being the numbers constructed from ratios (or fractions ) of integers . When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being incommensurable , meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself. Among irrational numbers are the ratio π of a circle's circumference to its diameter, Euler's number e , the golden ratio φ , and the square root of two ; in fact all square roots of natural numbers , other than of perfect squares , are irrational. It can be shown that irrational numbers, when expressed in a positional numeral system (e.g
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Information
INFORMATION is that which informs. In other words, it is the answer to a question of some kind. It is thus related to data and knowledge , as data represents values attributed to parameters, and knowledge signifies understanding of real things or abstract concepts. As it regards data, the information's existence is not necessarily coupled to an observer (it exists beyond an event horizon , for example), while in the case of knowledge, the information requires a cognitive observer . At its most fundamental level, information is any propagation of cause and effect within a system. Information
Information
is conveyed either as the content of a message or through direct or indirect observation of anything.That which is perceived can be construed as a message in its own right, and in that sense, information is always conveyed as the content of a message
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Lenition
In linguistics , LENITION is a kind of sound change that alters consonants , making them more sonorous (vowel-like). The word lenition itself means "softening" or "weakening" (from Latin
Latin
lenis = weak). Lenition can happen both synchronically (i.e. within a language at a particular point in time) and diachronically (i.e. as a language changes over time ). Lenition can involve such changes as making a consonant more sonorous, causing a consonant to lose its place of articulation (a phenomenon called debuccalization , which turns a consonant into a glottal consonant like or ), or even causing a consonant to disappear entirely. An example of synchronic lenition in American English
American English
is found in flapping in some dialects: the /t/ of a word like wait becomes the more sonorous in the related form waiting
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Number
A NUMBER is a mathematical object used to count , measure , and label . The original examples are the natural numbers 1 , 2 , 3 , 4 and so forth. A notational symbol that represents a number is called a numeral . In addition to their use in counting and measuring, numerals are often used for labels (as with telephone numbers ), for ordering (as with serial numbers ), and for codes (as with ISBNs ). In common usage, number may refer to a symbol, a word , or a mathematical abstraction . In mathematics , the notion of number has been extended over the centuries to include 0 , negative numbers , rational numbers such as 1/2 and −2/3, real numbers such as √2 and π , and complex numbers , which extend the real numbers by adding a square root of −1 . Calculations with numbers are done with arithmetical operations , the most familiar being addition , subtraction , multiplication , division , and exponentiation . Their study or usage is called arithmetic
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Function (mathematics)
In mathematics , a FUNCTION is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x2. The output of a function f corresponding to an input x is denoted by f(x) (read "f of x"). In this example, if the input is −3, then the output is 9, and we may write f(−3) = 9. Likewise, if the input is 3, then the output is also 9, and we may write f(3) = 9. (The same output may be produced by more than one input, but each input gives only one output.) The input variable(s) are sometimes referred to as the argument(s) of the function. Functions of various kinds are "the central objects of investigation" in most fields of modern mathematics . There are many ways to describe or represent a function. Some functions may be defined by a formula or algorithm that tells how to compute the output for a given input
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Shape
A SHAPE is the form of an object or its external boundary, outline, or external surface , as opposed to other properties such as color, texture, or material composition. Psychologists have theorized that humans mentally break down images into simple geometric shapes called geons . Examples of geons include cones and spheres. CONTENTS * 1 Classification of simple shapes * 2 Shape
Shape
in geometry * 2.1 Equivalence of shapes * 2.2 Congruence and similarity * 2.3 Homeomorphism * 3 Shape
Shape
analysis * 4 Similarity classes * 5 See also * 6 References * 7 External links CLASSIFICATION OF SIMPLE SHAPES Main article: Lists of shapes A variety of polygonal shapes. Some simple shapes can be put into broad categories. For instance, polygons are classified according to their number of edges as triangles , quadrilaterals , pentagons , etc
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Asymptotic Analysis
In mathematical analysis , ASYMPTOTIC ANALYSIS is a method of describing limiting behavior. As an illustration, suppose that we are interested in the properties of a function f(n) as n becomes very large. If f(n) = n2 + 3n, then as n becomes very large, the term 3n becomes insignificant compared to n2. The function f(n) is said to be "asymptotically equivalent to n2, as n → ∞". This is often written symbolically as f(n) ~ n2, which is read as "f(n) is asymptotic to n2". CONTENTS * 1 Definition * 2 Properties * 3 Asymptotic expansion
Asymptotic expansion
* 4 Applications * 5 Method of dominant balance * 6 See also * 7 Notes * 8 References * 9 External links DEFINITIONFormally, given functions f and g of a variable x, we define a binary relation f g ( as x ) {displaystyle fsim gquad ({text{as }}xto infty )} if and only if (de Bruijn, 1981, §1.4) lim x f ( x ) g ( x ) = 1
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Scientific Experiment
An EXPERIMENT is a procedure carried out to support, refute, or validate a hypothesis . Experiments provide insight into cause-and-effect by demonstrating what outcome occurs when a particular factor is manipulated. Experiments vary greatly in goal and scale, but always rely on repeatable procedure and logical analysis of the results. There also exists natural experimental studies . A child may carry out basic experiments to understand gravity, while teams of scientists may take years of systematic investigation to advance their understanding of a phenomenon. Experiments and other types of hands-on activities are very important to student learning in the science classroom. Experiments can raise test scores and help a student become more engaged and interested in the material they are learning, especially when used over time. Experiments can vary from personal and informal natural comparisons (e.g. tasting a range of chocolates to find a favorite), to highly controlled (e.g
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Gravity
GRAVITY, or GRAVITATION, is a natural phenomenon by which all things with mass are brought toward (or gravitate toward) one another, including planets , stars and galaxies . Since energy and mass are equivalent , all forms of energy , including light , also cause gravitation and are under the influence of it. On Earth
Earth
, gravity gives weight to physical objects and causes the ocean tides . The gravitational attraction of the original gaseous matter present in the Universe
Universe
caused it to begin coalescing, forming stars – and the stars to group together into galaxies – so gravity is responsible for many of the large scale structures in the Universe. Gravity
Gravity
has an infinite range, although its effects become increasingly weaker on farther objects
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Iteration
ITERATION is the act of repeating a process, either to generate an unbounded sequence of outcomes, or with the aim of approaching a desired goal, target or result. Each repetition of the process is also called an "iteration", and the results of one iteration are used as the starting point for the next iteration. In the context of mathematics or computer science , iteration (along with the related technique of recursion ) is a standard building block of algorithms . CONTENTS * 1 Mathematics * 2 Computing * 3 Education * 4 Relationship with recursion * 5 Other terminology * 6 See also * 7 References MATHEMATICSITERATION in mathematics may refer to the process of iterating a function i.e. applying a function repeatedly, using the output from one iteration as the input to the next
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