query: the difference of the absolute value of two numbers

pos: Absolute difference Absolute difference The absolute difference of two real numbers "x", "y" is given by |"x" − "y"|, the absolute value of their difference. It describes the distance on the real line between the points corresponding to "x" and "y". It is a special case of the L distance for all 1 ≤ "p" ≤ ∞ and is the standard metric used for both the set of rational numbers Q and their completion, the set of real numbers R. As with any metric, the metric properties hold: By contrast, simple subtraction is not non-negative or commutative, but it does obey the

pos: Relative change and difference maximum absolute value of the two numbers. if at least one of the values does not equal zero. This approach is especially useful when comparing floating point values in programming languages for equality with a certain tolerance. Another application is in the computation of approximation errors when the relative error of a measurement is required. Another way to define the relative difference of two numbers is to take their absolute difference divided by some functional value of the two numbers, for example, the absolute value of their arithmetic mean: This approach is often used when the two numbers reflect a

pos: Absolute value inequality for complex numbers" below.) Some additional useful properties are given below. These are either immediate consequences of the definition or implied by the four fundamental properties above. Two other useful properties concerning inequalities are: These relations may be used to solve inequalities involving absolute values. For example: The absolute value, as "distance from zero", is used to define the absolute difference between arbitrary real numbers, the standard metric on the real numbers. Since the complex numbers are not ordered, the definition given at the top for the real absolute value cannot be directly applied to complex numbers. However the

pos: Relative change and difference reference value, the absolute difference can be considered in place of the actual change. The absolute difference between two values is not always a good way to compare the numbers. For instance, the absolute difference of 1 between 6 and 5 is more significant than the same absolute difference between 100,000,001 and 100,000,000. We can adjust the comparison to take into account the "size" of the quantities involved, by defining, for positive values of "x": The relative change is not defined if the reference value ("x") is zero. For values greater than the reference value, the relative change should be

pos: Absolute difference Absolute difference The absolute difference of two real numbers "x", "y" is given by |"x" − "y"|, the absolute value of their difference. It describes the distance on the real line between the points corresponding to "x" and "y". It is a special case of the L distance for all 1 ≤ "p" ≤ ∞ and is the standard metric used for both the set of rational numbers Q and their completion, the set of real numbers R. As with any metric, the metric properties hold: By contrast, simple subtraction is not non-negative or commutative, but it does obey the

pos: Relative change and difference In general we can say that the absolute difference is being scaled by some function of the values "x" and "y", say . As with relative change, the relative difference is undefined if is zero. Several common choices for the function would be: Measures of relative difference are unitless numbers expressed as a fraction. Corresponding values of percent difference would be obtained by multiplying these values by 100 (and appending the % sign to indicate that the value is a percentage). One way to define the relative difference of two numbers is to take their absolute difference divided by the

pos: Mean absolute difference Mean absolute difference The mean absolute difference (univariate) is a measure of statistical dispersion equal to the average absolute difference of two independent values drawn from a probability distribution. A related statistic is the relative mean absolute difference, which is the mean absolute difference divided by the arithmetic mean, and equal to twice the Gini coefficient. The mean absolute difference is also known as the absolute mean difference (not to be confused with the absolute value of the mean signed difference) and the Gini mean difference (GMD). The mean absolute difference is sometimes denoted by Δ or as MD. The

pos: Mean absolute difference for all "p" between 0 and 1. Mean absolute difference The mean absolute difference (univariate) is a measure of statistical dispersion equal to the average absolute difference of two independent values drawn from a probability distribution. A related statistic is the relative mean absolute difference, which is the mean absolute difference divided by the arithmetic mean, and equal to twice the Gini coefficient. The mean absolute difference is also known as the absolute mean difference (not to be confused with the absolute value of the mean signed difference) and the Gini mean difference (GMD). The mean absolute difference is sometimes

pos: Mean absolute difference mean absolute difference is defined as the "average" or "mean", formally the expected value, of the absolute difference of two random variables "X" and "Y" independently and identically distributed with the same (unknown) distribution henceforth called "Q". Specifically, in the discrete case, In the continuous case, When the probability distribution has a finite and nonzero arithmetic mean, the relative mean absolute difference, sometimes denoted by Δ or RMD, is defined by The relative mean absolute difference quantifies the mean absolute difference in comparison to the size of the mean and is a dimensionless quantity. The relative mean absolute difference is

pos: Absolute difference second and fourth properties above, since if and only if "x" = "y", and The absolute difference is used to define other quantities including the relative difference, the L norm used in taxicab geometry, and graceful labelings in graph theory. When it is desirable to avoid the absolute value function – for example because it is expensive to compute, or because its derivative is not continuous – it can sometimes be eliminated by the identity This follows since |"x" − "y"| = ("x" − "y") and squaring is monotonic on the nonnegative reals. Absolute difference The absolute difference of two

pos: Relative change and difference , can be called their "actual difference". When "y" is a "reference value" (a theoretical/actual/correct/accepted/optimal/starting, etc. value; the value that "x" is being compared to) then Δ is called their "actual change". When there is no reference value, the sign of Δ has little meaning in the comparison of the two values since it doesn't matter which of the two values is written first, so one often works with , the absolute difference instead of Δ, in these situations. Even when there is a reference value, if it doesn't matter whether the compared value is larger or smaller than the

pos: Relative change and difference relationship of the value with respect to the reference value (that is, larger or smaller) does not matter in a particular application, the absolute difference may be used in place of the actual change in the above formula to produce a value for the relative change which is always non-negative. Defining relative difference is not as easy as defining relative change since there is no "correct" value to scale the absolute difference with. As a result, there are many options for how to define relative difference and which one is used depends on what the comparison is being used for.

pos: Scale-invariant feature transform individual values. R is minimum when the eigenvalues are equal to each other. Therefore, the higher the absolute difference between the two eigenvalues, which is equivalent to a higher absolute difference between the two principal curvatures of D, the higher the value of R. It follows that, for some threshold eigenvalue ratio formula_41, if R for a candidate keypoint is larger than formula_42, that keypoint is poorly localized and hence rejected. The new approach uses formula_43. This processing step for suppressing responses at edges is a transfer of a corresponding approach in the Harris operator for corner detection. The difference

pos: Nearest neighbor value interpolation image interpolation artefacts which reduce the quality of image, and as a result, several future developments have emerged, drawing on various aspects of the pixel value corresponding to the smallest absolute difference. Nearest neighbor value interpolation In mathematics applied to computer graphics, nearest neighbor value interpolation is an advanced method of image interpolation. This method uses the pixel value corresponding to the smallest absolute difference when a set of four known value pixels has no mode. Proposed by Olivier Rukundo in 2012 in his PhD dissertation, the first work presented at the fourth International Workshop on Advanced Computational Intelligence, was

pos: Nearest neighbor value interpolation Nearest neighbor value interpolation In mathematics applied to computer graphics, nearest neighbor value interpolation is an advanced method of image interpolation. This method uses the pixel value corresponding to the smallest absolute difference when a set of four known value pixels has no mode. Proposed by Olivier Rukundo in 2012 in his PhD dissertation, the first work presented at the fourth International Workshop on Advanced Computational Intelligence, was based only on the pixel value corresponding to the smallest absolute difference to achieve high resolution and visually pleasant image. This approach was since upgraded to deal with a wider class of

pos: Symmetric mean absolute percentage error Symmetric mean absolute percentage error Symmetric mean absolute percentage error (SMAPE or sMAPE) is an accuracy measure based on percentage (or relative) errors. It is usually defined as follows: where "A" is the actual value and "F" is the forecast value. The absolute difference between "A" and "F" is divided by half the sum of absolute values of the actual value "A" and the forecast value "F". The value of this calculation is summed for every fitted point "t" and divided again by the number of fitted points "n". The earliest reference to similar formula appears to be Armstrong (1985,

pos: Law of large numbers approaches 1/2, almost surely the absolute difference in the number of heads and tails will become large as the number of flips becomes large. That is, the probability that the absolute difference is a small number, approaches zero as the number of flips becomes large. Also, almost surely the ratio of the absolute difference to the number of flips will approach zero. Intuitively, expected absolute difference grows, but at a slower rate than the number of flips, as the number of flips grows. The Italian mathematician Gerolamo Cardano (1501–1576) stated without proof that the accuracies of empirical statistics tend to

pos: Symmetric mean absolute percentage error / "A") This measure is easier to analyse statistically, and has valuable symmetry and unbiasedness properties. When used in constructing forecasting models the resulting prediction corresponds to the geometric mean (Tofallis, 2015). Symmetric mean absolute percentage error Symmetric mean absolute percentage error (SMAPE or sMAPE) is an accuracy measure based on percentage (or relative) errors. It is usually defined as follows: where "A" is the actual value and "F" is the forecast value. The absolute difference between "A" and "F" is divided by half the sum of absolute values of the actual value "A" and the forecast value "F". The

pos: Mean absolute error the Y values. MAE has a clear interpretation as the average absolute difference between "y" and "x". Many researchers want to know this average difference because its interpretation is clear, but researchers frequently compute and misinterpret the Root Mean Squared Error (RMSE), which is not the average absolute error. As the name suggests, the mean absolute error is an average of the absolute errors formula_4, where formula_5 is the prediction and formula_6 the true value. Note that alternative formulations may include relative frequencies as weight factors. The mean absolute error uses the same scale as the data being measured. This

pos: Deviation (statistics) observed value and an "estimate" of the true value (such an estimate may be a sample mean) is a residual. These concepts are applicable for data at the interval and ratio levels of measurement. In statistics, the absolute deviation of an element of a data set is the absolute difference between that element and a given point. Typically the deviation is reckoned from the central value, being construed as some type of average, most often the median or sometimes the mean of the data set. where For an unbiased estimator, the average of the signed deviations across the entire set

pos: Relative change and difference $110,000, the percentage change of its value can be expressed as It can then be said that the worth of the house went up by 10%. More generally, if "V" represents the old value and "V" the new one, Some calculators directly support this via a or function. When the variable in question is a percentage itself, it is better to talk about its change by using percentage points, to avoid confusion between relative difference and absolute difference. If a bank were to raise the interest rate on a savings account from 3% to 4%, the statement that "the interest

neg: Prime number formula_160 of any rational number formula_107 is then defined as formula_162. Multiplying an integer by its formula_20-adic absolute value cancels out the factors of formula_20 in its factorization, leaving only the other primes. Just as the distance between two real numbers can be measured by the absolute value of their distance, the distance between two rational numbers can be measured by their formula_20-adic distance, the formula_20-adic absolute value of their difference. For this definition of distance, two numbers are close together (they have a small distance) when their difference is divisible by a high power of formula_20. In the same

neg: Absolute value line, for real numbers, or in the complex plane, for complex numbers, and more generally, the absolute value of the difference of two real or complex numbers is the distance between them. The standard Euclidean distance between two points and in Euclidean -space is defined as: This can be seen as a generalisation, since for formula_68 and formula_69 real, i.e. in a 1-space, according to the alternative definition of the absolute value, and for formula_71 and formula_72 complex numbers, i.e. in a 2-space, The above shows that the "absolute value"-distance, for real and complex numbers, agrees with the standard Euclidean

neg: Absolute value the absolute value of a real number is that number's distance from zero along the real number line, and more generally the absolute value of the difference of two real numbers is the distance between them. Indeed, the notion of an abstract distance function in mathematics can be seen to be a generalisation of the absolute value of the difference (see "Distance" below). Since the square root symbol represents the unique "positive" square root (when applied to a positive number), it follows that is equivalent to the definition above, and may be used as an alternative definition of the absolute

neg: Absolute value distance, which they inherit as a result of considering them as one and two-dimensional Euclidean spaces, respectively. The properties of the absolute value of the difference of two real or complex numbers: non-negativity, identity of indiscernibles, symmetry and the triangle inequality given above, can be seen to motivate the more general notion of a distance function as follows: A real valued function on a set is called a metric (or a "distance function") on , if it satisfies the following four axioms: The definition of absolute value given for real numbers above can be extended to any ordered ring. That

neg: Prime number places (extensions to complete fields in which the given field is a dense set, also called completions). The extension from the rational numbers to the real numbers, for instance, is a place in which the distance between numbers is the usual absolute value of their difference. The corresponding mapping to an additive group would be the logarithm of the absolute value, although this does not meet all the requirements of a valuation. According to Ostrowski's theorem, up to a natural notion of equivalence, the real numbers and formula_20-adic numbers, with their orders and absolute values, are the only valuations, absolute

neg: Number line second is less than the first). The distance between them is the magnitude of their difference—that is, it measures the first number minus the second one, or equivalently the absolute value of the second number minus the first one. Taking this difference is the process of subtraction. Thus, for example, the length of a line segment between 0 and some other number represents the magnitude of the latter number. Two numbers can be added by "picking up" the length from 0 to one of the numbers, and putting it down again with the end that was 0 placed on top

neg: Ostrowski's theorem Ostrowski's theorem In number theory, Ostrowski's theorem, due to Alexander Ostrowski (1916), states that every non-trivial absolute value on the rational numbers Q is equivalent to either the usual real absolute value or a -adic absolute value. Raising an absolute value to a power less than 1 always results in another absolute value. Two absolute values formula_1 and formula_2 on a field "K" are defined to be equivalent if there exists a real number such that The trivial absolute value on any field "K" is defined to be The real absolute value on the rationals Q is the standard absolute

neg: (ε, δ)-definition of limit "function "f" approaches the limit "L" near "a" (symbolically, formula_19) if we can make "f(x)" as close as we like to "L" by requiring that "x" be sufficiently close to, but unequal to, "a"." When we say that two things are close (such as "f(x)" and "L" or "x" and "a") we mean that the distance between them is small. When "f(x)", "L", "x", and "a" are real numbers, the distance between two numbers is the absolute value of the difference of the two. Thus, when we say "f(x)" is close to "L" we mean formula_20 is small. When we

neg: Ostrowski's theorem as Ostrowski's theorem. Ostrowski's theorem In number theory, Ostrowski's theorem, due to Alexander Ostrowski (1916), states that every non-trivial absolute value on the rational numbers Q is equivalent to either the usual real absolute value or a -adic absolute value. Raising an absolute value to a power less than 1 always results in another absolute value. Two absolute values formula_1 and formula_2 on a field "K" are defined to be equivalent if there exists a real number such that The trivial absolute value on any field "K" is defined to be The real absolute value on the rationals Q is

neg: Absolute value Absolute value In mathematics, the absolute value or modulus of a real number is the non-negative value of without regard to its sign. Namely, for a positive , for a negative (in which case is positive), and . For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its distance from zero. Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is also defined for the complex numbers, the

neg: Absolute value (algebra) an absolute value to a power less than 1 results in another absolute value, but raising to a power greater than 1 does not necessarily result in an absolute value. (For instance, squaring the usual absolute value on the real numbers yields a function which is not an absolute value because it would violate the rule |"x" + "y"| ≤ |"x"| + |"y"|.) Absolute values up to equivalence, or in other words, an equivalence class of absolute values, is called a place. Ostrowski's theorem states that the nontrivial places of the rational numbers Q are the ordinary absolute value and

neg: Linear difference equation the absolute value of the characteristic root is less than 1. If the absolute value equals 1, the term will stay constant as "t" grows if the root is +1 but will fluctuate between two values if the root is –1. If the absolute value of the root is greater than 1 the term will become larger and larger over time. A pair of terms with complex conjugate characteristic roots will converge to 0 with dampening fluctuations if the absolute value of the modulus "M" of the roots is less than 1; if the modulus equals 1 then constant amplitude

neg: Absolute value value of real numbers. The absolute value has the following four fundamental properties ("a", "b" are real numbers), that are used for generalization of this notion to other domains: Non-negativity, positive definiteness, and multiplicativity are readily apparent from the definition. To see that subadditivity holds, first note that one of the two alternatives of taking as either or guarantees that formula_6 Now, since formula_7 and formula_8, it follows that, whichever is the value of , one has formula_9 for all real formula_10. Consequently, formula_11, as desired. (For a generalization of this argument to complex numbers, see "Proof of the triangle

neg: Algebraic number theory the integers. In addition to the usual absolute value function |·| : Q → R, there are p-adic absolute value functions |·| : Q → R, defined for each prime number "p", which measure divisibility by "p". Ostrowski's theorem states that these are all possible absolute value functions on Q (up to equivalence). Therefore, absolute values are a common language to describe both the real embedding of Q and the prime numbers. A place of an algebraic number field is an equivalence class of absolute value functions on "K". There are two types of places. There is a formula_15-adic absolute

neg: Rational number set Then defines a metric on Q. The metric space (Q,"d") is not complete, and its completion is the "p"-adic number field Q. Ostrowski's theorem states that any non-trivial absolute value on the rational numbers Q is equivalent to either the usual real absolute value or a "p"-adic absolute value. Rational number In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . Since may be equal to 1, every integer is a rational number. The set of all rational numbers, often referred to

neg: Archimedean property to as the ultrametric triangle inequality, respectively. A field or normed space satisfying the ultrametric triangle inequality is called non-Archimedean. The concept of a non-Archimedean normed linear space was introduced by A. F. Monna. The field of the rational numbers can be assigned one of a number of absolute value functions, including the trivial function formula_12 when formula_13, the more usual formula_14, and the "p"-adic absolute value functions. By Ostrowski's theorem, every non-trivial absolute value on the rational numbers is equivalent to either the usual absolute value or some -adic absolute value. The rational field is not complete with respect

neg: Mean absolute error the identity line. The Mean Absolute Error is given by: formula_1 It is possible to express MAE as the sum of two components: Quantity Disagreement and Allocation Disagreement. Quantity Disagreement is the absolute value of the Mean Error. Allocation Disagreement is MAE minus Quantity Disagreement. The Mean Error is given by: formula_2 It is also possible to identify the types of difference by looking at an formula_3 plot. Allocation difference exists if and only if points reside on both sides of the identity line. Quantity difference exists when the average of the X values does not equal the average of

neg: P-adic number P-adic number In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of "closeness" or absolute value. In particular, -adic numbers are considered to be close when their difference is divisible by a high power of : the higher the power, the closer they are. This property enables -adic numbers to encode congruence information in a way that turns out to have

neg: Complete metric space always possible to "fill all the holes", leading to the "completion" of a given space, as explained below. The space Q of rational numbers, with the standard metric given by the absolute value of the difference, is not complete. Consider for instance the sequence defined by formula_2 and formula_3. This is a Cauchy sequence of rational numbers, but it does not converge towards any rational limit: If the sequence did have a limit "x", then by solving formula_4 necessarily "x" = 2, yet no rational number has this property. However, considered as a sequence of real numbers, it does converge

neg: Diophantine approximation Diophantine approximation In number theory, the field of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria. The first problem was to know how well a real number can be approximated by rational numbers. For this problem, a rational number "a"/"b" is a "good" approximation of a real number "α" if the absolute value of the difference between "a"/"b" and "α" may not decrease if "a"/"b" is replaced by another rational number with a smaller denominator. This problem was solved during the 18th century by means of continued fractions. Knowing

neg: Archimedean property to non-trivial absolute values; with respect to the trivial absolute value, the rational field is a discrete topological space, so complete. The completion with respect to the usual absolute value (from the order) is the field of real numbers. By this construction the field of real numbers is Archimedean both as an ordered field and as a normed field. On the other hand, the completions with respect to the other non-trivial absolute values give the fields of -adic numbers, where is a prime integer number (see below); since the -adic absolute values satisfy the ultrametric property, then the -adic number

neg: Absolute value (algebra) Absolute value (algebra) In algebra, an absolute value (also called a valuation, magnitude, or norm, although "norm" usually refers to a specific kind of absolute value on a field) is a function which measures the "size" of elements in a field or integral domain. More precisely, if "D" is an integral domain, then an absolute value is any mapping | "x" | from "D" to the real numbers R satisfying: It follows from these axioms that | 1 | = 1 and | −1 | = 1. Furthermore, for every positive integer "n", The classical "absolute value" is one in

neg: Sign (mathematics) a function would be called a "positive function" if its values are positive for all arguments of its domain, or a "non-negative function", if all of its values are non-negative. Complex numbers are impossible to order, so they cannot carry the structure of an ordered ring, and, accordingly, cannot be partitioned into positive and negative complex numbers. They do, however, share an attribute with the reals, which is called "absolute value", or "magnitude". Magnitudes are always non-negative real numbers, and to any non-zero number there belongs a positive real number, its absolute value. For example, the absolute value of and

neg: Mandelbrot set takes to use BigNum software to compute a given formula_169. If the difference between the bounds is greater than the number of iterations, it is possible to perform binomial search using BigNum software, successively halving the gap until it becomes more time efficient to find the escape value using floating point hardware. Mandelbrot set The Mandelbrot set is the set of complex numbers formula_1 for which the function formula_2 does not diverge when iterated from formula_3, i.e., for which the sequence formula_4, formula_5, etc., remains bounded in absolute value. Its definition and name are due to Adrien Douady, in tribute

neg: Absolute value = "x x"* and called the norm of x. The real numbers ℝ, complex numbers ℂ, and quaternions ℍ are all composition algebras with norms given by definite quadratic forms. The absolute value in these division algebras is given by the square root of the composition algebra norm. In general the norm of a composition algebra may be a quadratic form that is not definite and has null vectors. However, as in the case of division algebras, when an element "x" has a non-zero norm, then "x" has a multiplicative inverse given by "x"*/"N"("x"). Absolute value In mathematics, the absolute

neg: Mean absolute percentage error Mean absolute percentage error The mean absolute percentage error (MAPE), also known as mean absolute percentage deviation (MAPD), is a measure of prediction accuracy of a forecasting method in statistics, for example in trend estimation, also used as a Loss function for regression problems in Machine Learning. It usually expresses accuracy as a percentage, and is defined by the formula: where "A" is the actual value and "F" is the forecast value. The difference between "A" and "F" is divided by the actual value "A" again. The absolute value in this calculation is summed for every forecasted point in time

neg: Absolute value is, if is an element of an ordered ring "R", then the absolute value of , denoted by , is defined to be: where is the additive inverse of , 0 is the additive identity element, and < and ≥ have the usual meaning with respect to the ordering in the ring. The four fundamental properties of the absolute value for real numbers can be used to generalise the notion of absolute value to an arbitrary field, as follows. A real-valued function on a field is called an "absolute value" (also a "modulus", "magnitude", "value", or "valuation") if it satisfies

neg: Absolute magnitude for different wavelength ranges corresponding to specified filter bands or passbands; for stars a commonly quoted absolute magnitude is the absolute visual magnitude, which uses the visual (V) band of the spectrum (in the UBV photometric system). Absolute magnitudes are denoted by a capital M, with a subscript representing the filter band used for measurement, such as M for absolute magnitude in the V band. The more luminous an object, the smaller the numerical value of its absolute magnitude. A difference of 5 magnitudes between the absolute magnitudes of two objects corresponds to a ratio of 100 in their luminosities,

neg: P-adic number are taken, their value and representation coincides in and . Define a topology on by taking as a basis of open sets all sets of the form where "a" is a non-negative integer and "n" is an integer in [1, "p"]. For example, in the dyadic integers, "U"(1) is the set of odd numbers. "U"("n") is the set of all "p"-adic integers whose difference from "n" has "p"-adic absolute value less than "p". Then is a compactification of , under the derived topology (it is "not" a compactification of with its usual discrete topology). The relative topology on as a

neg: P-adic order more elementary topics, such as the distinction between singly and doubly even numbers. Let be a prime in . The -adic order or -adic valuation for is defined as The -adic order can be extended into the rational numbers. We can define formula_2 Some properties are: Moreover, if formula_5, then where is the infimum (i.e. the smaller of the two). The -adic absolute value on is defined as The -adic absolute value satisfies the following properties: The “symmetry” formula_8 follows from multiplicativity formula_9 and subadditivity formula_10 from the non-Archimedean triangle inequality formula_11. A metric space can be formed on the

neg: P-adic number value on is equivalent either to the Euclidean absolute value, the trivial absolute value, or to one of the -adic absolute values for some prime . Each absolute value (or metric) leads to a different completion of . (With the trivial absolute value, is already complete.) In the algebraic approach, we first define the ring of -adic integers, and then construct the field of fractions of this ring to get the field of -adic numbers. We start with the inverse limit of the rings ("a") such that "a" is in , and if "n" ≤ "m", then "a" ≡ "a"

neg: Absolute value (algebra) of Alexander Ostrowski has it that any field complete with respect to an Archimedean absolute value is isomorphic to either the real or the complex numbers and the valuation is equivalent to the usual one. The Gelfand-Tornheim theorem states that any field with an Archimedean valuation is isomorphic to a subfield of C, the valuation being equivalent to the usual absolute value on C. If "D" is an integral domain with absolute value | "x" |, then we may extend the definition of the absolute value to the field of fractions of "D" by setting On the other hand, if

neg: Bernstein polynomial contribute more than 2"M" times the small probability that the difference exceeds ε. Finally, one observes that the absolute value of the difference between expectations never exceeds the expectation of the absolute value of the difference, and See for instance. Bernstein polynomial In the mathematical field of numerical analysis, a Bernstein polynomial, named after Sergei Natanovich Bernstein, is a polynomial in the Bernstein form, that is a linear combination of Bernstein basis polynomials. A numerically stable way to evaluate polynomials in Bernstein form is de Casteljau's algorithm. Polynomials in Bernstein form were first used by Bernstein in a constructive proof

neg: Euler's four-square identity for prime numbers, after which the more general theorem follows. The sign convention used above corresponds to the signs obtained by multiplying two quaternions. Other sign conventions can be obtained by changing any formula_6 to formula_7, and/or any formula_8 to formula_9. If the formula_6 and formula_8 are real numbers, the identity expresses the fact that the absolute value of the product of two quaternions is equal to the product of their absolute values, in the same way that the Brahmagupta–Fibonacci two-square identity does for complex numbers. This property is the definitive feature of composition algebras. Hurwitz's theorem states that an

neg: Absolute value absolute value for real numbers can be used, with a slight modification, to generalise the notion to an arbitrary vector space. A real-valued function on a vector space over a field , represented as , is called an absolute value, but more usually a norm, if it satisfies the following axioms: For all in , and , in , The norm of a vector is also called its "length" or "magnitude". In the case of Euclidean space , the function defined by is a norm called the Euclidean norm. When the real numbers are considered as the one-dimensional vector space

neg: Ostrowski's theorem have for some , and for . Letting we see that for general positive naturals As per the above remarks we see that formula_34 all rationals, implying the absolute value is equivalent to the -adic one. formula_35 One can also show a stronger conclusion, namely that formula_36 is a nontrivial absolute value if and only if either formula_37 for some formula_38 or formula_39 for some formula_40. Another theorem states that any field, complete with respect to an Archimedean absolute value, is (algebraically and topologically) isomorphic to either the real numbers or the complex numbers. This is sometimes also referred to

neg: P-adic analysis less subtle than classical analysis, since the ultrametric inequality means, for example, that convergence of infinite series of "p"-adic numbers is much simpler. Topological vector spaces over "p"-adic fields show distinctive features; for example aspects relating to convexity and the Hahn–Banach theorem are different. Ostrowski's theorem, due to Alexander Ostrowski (1916), states that every non-trivial absolute value on the rational numbers Q is equivalent to either the usual real absolute value or a -adic absolute value. Mahler's theorem, introduced by Kurt Mahler, expresses continuous "p"-adic functions in terms of polynomials. In any field, one has the following result. Let be

neg: Small number Small number Within a set of positive numbers, a number is small if it is close to zero. A number is smaller if it is less than another number. Within a set of positive and negative numbers there is ambiguity, because being closer to zero does not correspond to being less, but to being less in absolute value. Depending on context a negative number may be called "smaller" if it is closer to zero, or if it is more negative. This article deals with positive numbers, and is also applicable to negative numbers by taking the absolute value. Small numbers

neg: Approximation error difference between the exact value and the approximation. The relative error is the absolute error divided by the magnitude of the exact value. The percent error is the relative error expressed in terms of per 100. These definitions can be extended to the case when formula_5 and formula_6 are "n"-dimensional vectors, by replacing the absolute value with an "n"-norm. As an example, if the exact value is 50 and the approximation is 49.9, then the absolute error is 0.1 and the relative error is 0.1/50 = 0.002 = 0.2%. Another example would be if, in measuring a 6 mL beaker,

neg: Cohen's h the difference, allowing us to decide if the difference is meaningful. Researchers have used Cohen's h as follows. Given a probability or proportion "p", between 0 and 1, its "arcsine transformation" is Given two proportions, formula_2 and formula_3, "h" is defined as the difference between their arcsine transformations. Namely, This is also sometimes called "directional h" because, in addition to showing the magnitude of the difference, it shows which of the two proportions is greater. Often, researchers mean "nondirectional h", which is just the absolute value of the directional h: In R, Cohen's h can be calculated using the codice_1

neg: Complex number This leads to the polar form of complex numbers. The "absolute value" (or "modulus" or "magnitude") of a complex number is If is a real number (that is, if ), then . That is, the absolute value of a real number equals its absolute value as a complex number. By Pythagoras' theorem, the absolute value of complex number is the distance to the origin of the point representing the complex number in the complex plane. The "argument" of (in many applications referred to as the "phase" ) is the angle of the radius "Oz" with the positive real axis, and

neg: Absolute value geometric interpretation of the absolute value of a real number as its distance from 0 can be generalised. The absolute value of a complex number is defined by the Euclidean distance of its corresponding point in the complex plane from the origin. This can be computed using the Pythagorean theorem: for any complex number where and are real numbers, the absolute value or modulus of is denoted and is defined by where Re("z") = "x" and Im("z") = "y" denote the real and imaginary parts of "z", respectively. When the imaginary part is zero, this coincides with the definition of

neg: Absolute value (algebra) real numbers, then we obtain a function from "D" to R ∪ {∞}, with the following properties: Such a function is known as a "valuation" in the terminology of Bourbaki, but other authors use the term "valuation" for "absolute value" and then say "exponential valuation" instead of "valuation". Given an integral domain "D" with an absolute value, we can define the Cauchy sequences of elements of "D" with respect to the absolute value by requiring that for every "r" > 0 there is a positive integer "N" such that for all integers "m", "n" > "N" one has | "x"

neg: Salem number Salem number In mathematics, a Salem number is a real algebraic integer "α" > 1 whose conjugate roots all have absolute value no greater than 1, and at least one of which has absolute value exactly 1. Salem numbers are of interest in Diophantine approximation and harmonic analysis. They are named after Raphaël Salem. Because it has a root of absolute value 1, the minimal polynomial for a Salem number must be reciprocal. This implies that 1/"α" is also a root, and that all other roots have absolute value exactly one. As a consequence α must be a unit in

neg: Thue's lemma an absolute value lower than , and thus that the absolute value of their difference is lower than . This implies that formula_10, and thus the result. The original proof of Thue's lemma is not efficient, in the sense that it does not provide any fast method for computing the solution. The extended Euclidean algorithm, allows us to provide a proof that leads to an efficient algorithm that has the same computational complexity of the Euclidean algorithm. More precisely, given the two integers and appearing in Thue's lemma, the extended Euclidean algorithm computes three sequences of integers , and such

neg: Logarithmic mean Logarithmic mean In mathematics, the logarithmic mean is a function of two non-negative numbers which is equal to their difference divided by the logarithm of their quotient. In symbols: for the positive numbers formula_2. This calculation is applicable in engineering problems involving heat and mass transfer. The logarithmic mean of two numbers is smaller than the arithmetic mean but larger than the geometric mean (unless the numbers are the same, in which case all three means are equal to the numbers): From the mean value theorem the logarithmic mean is obtained as the value of formula_5 by substituting formula_6 for

neg: Average rectified value differ from it whenever a function's absolute value varies locally, as the former then increases disproportionately. The difference is expressed by the form factor Average rectified value In electrical engineering, the average rectified value (ARV) of the quantity is the average of its absolute value. The average of a symmetric alternating value is zero and it is therefore not useful to characterize it. Thus the easiest way to determine a quantitative measurement size is to use the average rectified value. The average rectified value is mainly used to characterize alternating voltage and alternating current. It can be computed by averaging

neg: Value (ethics) more kinds of value systems: The difference between these two types of systems can be seen when people state that they hold one value system yet in practice deviate from it, thus holding a different value system. For example, a religion lists an absolute set of values while the practice of that religion may include exceptions. Implicit exceptions bring about a third type of value system called a formal value system. Whether idealized or realized, this type contains an implicit exception associated with each value: "as long as no higher-priority value is violated". For instance, a person might feel that

neg: Algebraic number field "F". On the other hand, the roots of factors of degree two are pairs of conjugate complex numbers, which allows for two conjugate embeddings into C. Either one of this pair of embeddings can be used to define an absolute value on "F", which is the same for both embeddings since they are conjugate. This absolute value is called a "complex place" of "F". If all roots of "f" above are real (respectively, complex) or, equivalently, any possible embedding "F" ⊂ C is actually forced to be inside R (resp. C), "F" is called totally real (resp. totally complex). To

neg: Negative number sum of a number and its negation is equal to zero: That is, the negation of a positive number is the additive inverse of the number. Using algebra, we may write this principle as an algebraic identity: This identity holds for any positive number . It can be made to hold for all real numbers by extending the definition of negation to include zero and negative numbers. Specifically: For example, the negation of is . In general, The absolute value of a number is the non-negative number with the same magnitude. For example, the absolute value of and the absolute

neg: Rating scale at the ordinal level. Numbers indicate the relative position of items, but not the magnitude of difference. Attitude and opinion scales are usually ordinal; one example is a Likert response scale: Some data are measured at the interval level. Numbers indicate the magnitude of difference between items, but there is no absolute zero point. A good example is a Fahrenheit/Celsius temperature scale where the differences between numbers matter, but placement of zero does not. Some data are measured at the ratio level. Numbers indicate magnitude of difference and there is a fixed zero point. Ratios can be calculated. Examples include

neg: Nomogram statistical analyses. It uses four parameters: "α" (fixed), effect size ("ρ" or "δ"), statistical power, and number of cases "N" (two scales for "α" = .05 (liberal) or .01 (conservative)). The hypothesized effect size in the population can either be expressed as a correlation coefficient ("ρ") or a normalized difference in means ("δ") for a T-test. The normalized difference is equal to the absolute value of the difference between two population means ("μ"₁ − "μ"₂), divided by the pooled standard deviation ("s"). The statistical power desired is estimated by 1 − "β", where "β" is equal to the probability of

neg: Sum of absolute differences the SAD values, the absolute value of the difference between each corresponding pair of pixels is used: the difference between 2 and 2 is 0, 4 and 1 is 3, 7 and 8 is 1, and so forth. Calculating the values of the absolute differences for each pixel, for the three possible template locations, gives the following: For each of these three image patches, the 9 absolute differences are added together, giving SAD values of 20, 25, and 17, respectively. From these SAD values, it could be asserted that the right side of the search image is the most similar

neg: Friendly-index set number of vertices). Given a friendly labeling of the vertices of , one may also label the edges: a given edge is labeled with a 0 if its endpoints and have equal labels, and it is labeled with a 1 if its endpoints have different labels. The friendly index of the labeling is the absolute value of the difference between the number of edges labeled 0 and the number of edges labeled 1. The friendly index set of , denoted , is the set of numbers that can arise as friendly indexes of friendly labelings of . The Dynamic Survey

neg: P-adic order set with a (non-Archimedean, translation-invariant) metric defined by The -adic absolute value is sometimes referred to as the "-adic norm", although it is not actually a norm because it does not satisfy the requirement of homogeneity. The choice of base in the formula makes no difference for most of the properties, but results in the product formula: where the product is taken over all primes and the usual absolute value (Archimedean norm), denoted formula_14. This follows from simply taking the prime factorization: each prime power factor formula_15 contributes its reciprocal to its -adic absolute value, and then the usual Archimedean

neg: Arithmetic underflow by the hardware, or by system software handling the primary underflow condition). This replacement is called "flush to zero". The 1984 edition of IEEE 754 introduced subnormal numbers. The subnormal numbers (including zero) fill the underflow gap with values where the absolute distance between adjacent values is the same as for adjacent values just outside the underflow gap. This enables "gradual underflow" where a nearest subnormal value is used, just as a nearest normal value is used when possible. Even when using gradual underflow, the nearest value may be zero. The absolute distance between adjacent floating point values just outside

neg: Negative number value of are both equal to , and the absolute value of is . In a similar manner to rational numbers, we can extend the natural numbers N to the integers Z by defining integers as an ordered pair of natural numbers ("a", "b"). We can extend addition and multiplication to these pairs with the following rules: We define an equivalence relation ~ upon these pairs with the following rule: This equivalence relation is compatible with the addition and multiplication defined above, and we may define Z to be the quotient set N²/~, i.e. we identify two pairs ("a", "b")

neg: Amplitude measurement on an oscilloscope, the peaks of the waveform being easily identified and measured against the graticule. This remains a common way of specifying amplitude, but sometimes other measures of amplitude are more appropriate. In audio system measurements, telecommunications and other areas where the measurand is a signal that swings above and below a reference value but is not sinusoidal, peak amplitude is often used. If the reference is zero, this is the maximum absolute value of the signal; if the reference is a mean value (DC component), the peak amplitude is the maximum absolute value of the difference from

neg: Electronic mixer add together audio signals such as voice signals, music signals, and sound effects. Multiplicative mixers multiply together two time-varying input signals instantaneously (instant-by-instant). If the two input signals are both sinusoids of specified frequencies f and f, then the output of the mixer will contain two new sinusoids that have the sum f + f frequency and the difference frequency absolute value |f - f|. Note: Any nonlinear electronic block driven by two signals with frequencies f and f would generate intermodulation (mixing) products. A multiplier (which is a nonlinear device) will generate ideally only the sum and difference frequencies,

neg: Gilbreath's conjecture false. Gilbreath observed a pattern while playing with the ordered sequence of prime numbers Computing the absolute value of the difference between term "n"+1 and term "n" in this sequence yields the sequence If the same calculation is done for the terms in this new sequence, and the sequence that is the outcome of this process, and again "ad infinitum" for each sequence that is the output of such a calculation, the following five sequences in this list are What Gilbreath—and François Proth before him—noticed is that the first term in each series of differences appears to be 1. Stating

neg: Stability theory analogous criterion for a continuously differentiable map with a fixed point , expressed in terms of its Jacobian matrix at , . If all eigenvalues of are real or complex numbers with absolute value strictly less than 1 then is a stable fixed point; if at least one of them has absolute value strictly greater than 1 then is unstable. Just as for =1, the case of the largest absolute value being 1 needs to be investigated further — the Jacobian matrix test is inconclusive. The same criterion holds more generally for diffeomorphisms of a smooth manifold. The stability of

neg: Fatou–Lebesgue theorem function "g" on "S" which dominates the sequence in absolute value, meaning that |"f"| ≤ "g" for all natural numbers "n", then all "f" as well as the limit inferior and the limit superior of the "f" are integrable and Here the limit inferior and the limit superior of the "f" are taken pointwise. The integral of the absolute value of these limiting functions is bounded above by the integral of "g". Since the middle inequality (for sequences of real numbers) is always true, the directions of the other inequalities are easy to remember. All "f" as well as the

neg: Relative change and difference of relativity) and is a newsworthy result. The percent error equation, when rewritten by removing the absolute values, becomes: It is important to note that the two values in the numerator do not commute. Therefore, it is vital to preserve the order as above: subtract the theoretical value from the experimental value and not vice versa. A percentage change is a way to express a change in a variable. It represents the relative change between the old value and the new one. For example, if a house is worth $100,000 today and the year after its value goes up to

neg: Exponential sum started by basic work of Hermann Weyl in diophantine approximation. The main thrust of the subject is that a sum is "trivially" estimated by the number "N" of terms. That is, the absolute value by the triangle inequality, since each summand has absolute value 1. In applications one would like to do better. That involves proving some cancellation takes place, or in other words that this sum of complex numbers on the unit circle is not of numbers all with the same argument. The best that is reasonable to hope for is an estimate of the form which signifies, up

neg: Absolute value property may be rephrased as follows: the absolute value is a group homomorphism from the multiplicative group of the complex numbers onto the group under multiplication of positive real numbers. Importantly, the property of subadditivity ("triangle inequality") extends to any finite collection of complex This inequality also applies to infinite families, provided that the infinite series formula_25 is absolutely convergent. If Lebesgue integration is viewed as the continuous analog of summation, then this inequality is analogously obeyed by complex-valued, measurable functions formula_26 when integrated over a measurable subset formula_27: The triangle inequality, as given by formula_30, can be demonstrated by

neg: Complete metric space to the irrational number formula_1. The open interval , again with the absolute value metric, is not complete either. The sequence defined by "x" = is Cauchy, but does not have a limit in the given space. However the closed interval is complete; for example the given sequence does have a limit in this interval and the limit is zero. The space R of real numbers and the space C of complex numbers (with the metric given by the absolute value) are complete, and so is Euclidean space R, with the usual distance metric. In contrast, infinite-dimensional normed vector spaces

neg: Graeffe's method formula_44 in the exponent reduces the absolute value of "c" to the corresponding dyadic root. Since this preserves the magnitude of the (representation of the) initial coefficients, this process was named renormalization. Multiplication of two numbers of this type is straightforward, whereas addition is performed following the factorization formula_45, where formula_46 is chosen as the larger of both numbers, that is, formula_47. Thus The coefficients formula_51 of the final stage "k" of the Graeffe iteration, for some reasonably large value of "k", are represented by pairs formula_52, formula_53. By identifying the corners of the convex envelope of the point set

neg: Concordant pair the other pair contains a higher value of "Y". The Kendall tau distance between two series is the total number of discordant pairs. The Kendall tau rank correlation coefficient, which measures how closely related two series of numbers are, is proportional to the difference between the number of concordant pairs and the number of discordant pairs. An estimate of Goodman and Kruskal's gamma, another measure of rank correlation, is given by the ratio of the difference to the sum of the numbers of concordant and discordant pairs. Somers' "D" is another similar but asymmetric measure given by the ratio of

neg: Relative change and difference remove the absolute values to provide more information about the result. Thus, if an experimental value is less than the theoretical value, the percent error will be negative. This negative result provides additional information about the experimental result. For example, experimentally calculating the speed of light and coming up with a negative percent error says that the experimental value is a velocity that is less than the speed of light. This is a big difference from getting a positive percent error, which means the experimental value is a velocity that is greater than the speed of light (violating the theory

neg: Equation of time the largest absolute error between the two is 2.46 s on day 324 (20 November). For the choice of the appropriate branch of the relation with respect to function continuity a modified version of the arctangent function is helpful. It brings in previous knowledge about the expected value by a parameter. The modified arctangent function is defined as: It produces a value that is as close to as possible. The function rounds to the nearest integer. Applying this yields: The parameter arranges here to set to the zero nearest value which is the desired one. The difference between the MICA

neg: Odor detection threshold nose. It is only in recent years that such compounds were separated on gas chromatographs. "Odor threshold value (OTV)" (also "aroma threshold value (ATV)", "Flavor threshold") is defined as the most minimal concentration of a substance that can be detected by a human nose. Some substances can be detected when their concentration is only few milligrams per 1000 tonnes, which is less than a drop in an Olympic swimming pool. Odor threshold value can be expressed as a concentration in water or concentration in air. Two major types of flavor thresholds can be distinguished: the absolute and the difference threshold.

neg: Floating-point arithmetic precision of 24 bits. In binary single-precision floating-point, this is represented as "s" = 1.10010010000111111011011 with "e" = 1. This has a decimal value of whereas a more accurate approximation of the true value of π is The result of rounding differs from the true value by about 0.03 parts per million, and matches the decimal representation of π in the first 7 digits. The difference is the discretization error and is limited by the machine epsilon. The arithmetical difference between two consecutive representable floating-point numbers which have the same exponent is called a unit in the last place (ULP).

neg: Instrumental temperature record oceans and sea ice regions, with a best estimate of 14 °C (57.2 °F). The estimate is uncertain, but probably lies within 0.5 °C of the true value. Given the difference in uncertainties between this absolute value and any annual anomaly, it's not valid to add them together to imply a precise absolute value for a specific year. The period for which reasonably reliable instrumental records of near-surface temperature exist with quasi-global coverage is generally considered to begin around 1850. Earlier records exist, but with sparser coverage and less standardized instrumentation. The temperature data for the record come from measurements

neg: Prime number way that the real numbers can be formed from the rational numbers and their distances, by adding extra limiting values to form a complete field, the rational numbers with the formula_20-adic distance can be extended to a different complete field, the formula_20-adic numbers. This picture of an order, absolute value, and complete field derived from them can be generalized to algebraic number fields and their valuations (certain mappings from the multiplicative group of the field to a totally ordered additive group, also called orders), absolute values (certain multiplicative mappings from the field to the real numbers, also called norms), and

neg: Degree of isochronous distortion Degree of isochronous distortion The degree of isochronous distortion, in data transmission, is the ratio of the absolute value of the maximum measured difference between the actual and the theoretical intervals separating any two significant instants of modulation (or demodulation), to the unit interval. These instants are not necessarily consecutive. This value is usually expressed as a percentage. The result of the measurement should be qualified by an indication if the period, usually limited, of the observation. For a prolonged modulation (or demodulation), it will be appropriate to consider the probability that an assigned value of the degree of distortion

neg: Degree of isochronous distortion will be exceeded. Degree of isochronous distortion The degree of isochronous distortion, in data transmission, is the ratio of the absolute value of the maximum measured difference between the actual and the theoretical intervals separating any two significant instants of modulation (or demodulation), to the unit interval. These instants are not necessarily consecutive. This value is usually expressed as a percentage. The result of the measurement should be qualified by an indication if the period, usually limited, of the observation. For a prolonged modulation (or demodulation), it will be appropriate to consider the probability that an assigned value of the

neg: Construction of the real numbers adding new points to the metric space in a process called completion. R is defined as the completion of Q with respect to the metric |"x"-"y"|, as will be detailed below (for completions of Q with respect to other metrics, see "p"-adic numbers.) Let "R" be the set of Cauchy sequences of rational numbers. That is, sequences of rational numbers such that for every rational , there exists an integer "N" such that for all natural numbers , . Here the vertical bars denote the absolute value. Cauchy sequences ("x") and ("y") can be added and multiplied as follows: Two

neg: Numerical digit numeral system with an integer base, the number of digits required to express arbitrary numbers is given by the absolute value of the base. For example, the decimal system (base 10) requires ten digits (0 through to 9), whereas the binary system (base 2) has two digits (e.g.: 0 and 1). In a basic digital system, a numeral is a sequence of digits, which may be of arbitrary length. Each position in the sequence has a place value, and each digit has a value. The value of the numeral is computed by multiplying each digit in the sequence by its

neg: P-adic number its elements are equivalence classes of Cauchy sequences, where two sequences are called equivalent if their difference converges to zero. In this way, we obtain a complete metric space which is also a field and contains . It can be shown that in , every element "x" may be written in a unique way as where "k" is some integer such that "a" ≠ "0" and each "a" is in {0, …, }. This series converges to "x" with respect to the metric d. With this absolute value, the field is a local field. Ostrowski's theorem states that each absolute

neg: Earth mass density of , about 1% below the modern value. While the mass of the Earth is implied by stating the Earth's radius and density, it was not usual to state the absolute mass explicitly prior to the introduction of scientific notation using powers of 10 in the later 19th century, because the absolute numbers would have been too awkward. Ritchie (1850) gives the mass of the Earth's atmosphere as "11,456,688,186,392,473,000 lbs." ( = , modern value is ) and states that "compared with the weight of the globe this mighty sum dwindles to insignificance". Absolute figures for the mass of