query: numerical value of magnitude of a physical quantity is

pos: Physical quantity Physical quantity A physical quantity is a physical property of a phenomenon, body, or substance, that can be quantified by measurement. A physical quantity can be expressed as the combination of a magnitude expressed by a number – usually a real number – and a unit: formula_1 where formula_2 is the magnitude and formula_3 is the unit. For example, (the mass of the neutron), or (the speed of light). The same physical quantity formula_4 can be represented equivalently in many unit systems, "i.e." formula_5. Symbols for quantities should be chosen according to the international recommendations of ISO/IEC 80000, the IUPAP

neg: Unit of measurement same phenomenon. Any value of a physical quantity is expressed as a comparison to a unit of that quantity. For example, the value of a physical quantity "Z" is expressed as the product of a unit [Z] and a numerical factor: The multiplication sign is usually left out, just as it is left out between variables in scientific notation of formulas. The conventions used to express quantities is referred to as quantity calculus. In formulas the unit [Z] can be treated as if it were a specific magnitude of a kind of physical dimension: see dimensional analysis for more on

neg: Physical constant on the earlier definitions, including the kilogram, to ensure minimal impact. Whereas the physical quantity indicated by a physical constant does not depend on the unit system used to express the quantity, the numerical values of dimensional physical constants do depend on choice of unit system. The term "physical constant" refers to the physical quantity, and not to the numerical value within any given system of units. For example, the speed of light is defined as having the numerical value of 299,792,458 in SI units, and as having the numerical value of 1 in natural units. While its numerical value

neg: Dot product identity, also known as "Lagrange's formula" may be remembered as "BAC minus CAB", keeping in mind which vectors are dotted together. This formula finds application in simplifying vector calculations in physics. In physics, vector magnitude is a scalar in the physical sense, i.e. a physical quantity independent of the coordinate system, expressed as the product of a numerical value and a physical unit, not just a number. The dot product is also a scalar in this sense, given by the formula, independent of the coordinate system. Examples include: For vectors with complex entries, using the given definition of the dot

neg: Physical quantity A scalar is a physical quantity that has magnitude but no direction. Symbols for physical quantities are usually chosen to be a single letter of the Latin or Greek alphabet, and are printed in italic type. Vectors are physical quantities that possess both magnitude and direction. Symbols for physical quantities that are vectors are in bold type, underlined or with an arrow above. If, e.g., "u" is the speed of a particle, then the straightforward notation for its velocity is u, u, or formula_6. Numerical quantities, even those denoted by letters, are usually printed in roman (upright) type, though sometimes

neg: ISO 31-0 all other quantities in the same category can be expressed in terms of this unit, multiplied by a number called the "numerical value". For example, if we write then ""λ"" is the symbol for the physical quantity (wavelength), "m" is the symbol for the unit (metre), and "6.982 × 10" is the numerical value of the wavelength in metres. More generally, we can write where "A" is the symbol for the quantity, {"A"} symbolizes the numerical value of "A", and ["A"] represents the corresponding unit in which "A" is expressed here. Both the numerical value and the unit symbol are

neg: Quantity Quantity Quantity is a property that can exist as a multitude or magnitude. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in terms of a unit of measurement. Quantity is among the basic classes of things along with quality, substance, change, and relation. Some quantities are such by their inner nature (as number), while others are functioning as states (properties, dimensions, attributes) of things such as heavy and light, long and short, broad and narrow, small and great, or much and little. Two basic divisions of quantity, magnitude and multitude, imply

neg: Cours d'Analyse a variable quantity becomes "infinitely small" when its numerical value decreases indefinitely in such a way as to converge towards the limit zero." On the same page, we find the only explicit example of such a variable to be found in Cauchy, namely On page 22, Cauchy starts the discussion of orders of magnitude of infinitesimals as follows: "Let formula_2 be an infinitely small quantity, that is a variable whose numerical value decreases indefinitely. When the various integer powers of formula_2, namely enter into the same calculation, these various powers are called, respectively, infinitely small of the "first", the "second",

neg: Quantity calculus Quantity calculus Quantity calculus is the formal method for describing the mathematical relations between "abstract" physical quantities. (Here the term "calculus" should be understood in its broader sense of "a system of computation", rather than in the sense of differential calculus and integral calculus.) Its roots can be traced to Fourier's concept of dimensional analysis (1822). The basic axiom of quantity calculus is Maxwell's description of a physical quantity as the product of a "numerical value" and a "reference quantity" (i.e. a "unit quantity" or a "unit of measurement"). De Boer summarized the multiplication, division, addition, association and commutation rules

neg: ISO 31-0 factors, and their product is the quantity. A quantity itself has no inherent particular numerical value or unit; as with any product, there are many different combinations of numerical value and unit that lead to the same quantity (e.g., "A" = 300 ⋅ m = 0.3 ⋅ km = ...). This ambiguity makes the {"A"} and ["A"] notations useless, unless they are used together. The value of a quantity is independent of the unit chosen to represent it. It must be distinguished from the numerical value of the quantity that occurs when the quantity is expressed in a particular unit.

neg: Size Size Size is the magnitude or dimensions of a thing. Size can be measured as length, width, height, diameter, perimeter, area, volume, or mass. In mathematical terms, "size is a concept abstracted from the process of measuring by comparing a longer to a shorter". Size is determined by the process of comparing or measuring objects, which results in the determination of the magnitude of a quantity, such as length or mass, relative to a unit of measurement. Such a magnitude is usually expressed as a numerical value of units on a previously established spatial scale, such as meters or inches.

neg: Physical constant Physical constant A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that is generally believed to be both universal in nature and have constant value in time. It is contrasted with a mathematical constant, which has a fixed numerical value, but does not directly involve any physical measurement. There are many physical constants in science, some of the most widely recognized being the speed of light in vacuum "c", the gravitational constant "G", the Planck constant "h", the electric constant "ε", and the elementary charge "e". Physical constants can take many dimensional forms: the speed

neg: Quantity unidentified amounts, the parts and examples of a mass are indicated with respect to the following: a measure of a mass (two kilos of rice and twenty bottles of milk or ten pieces of paper); a piece or part of a mass (part, element, atom, item, article, drop); or a shape of a container (a basket, box, case, cup, bottle, vessel, jar). Some further examples of quantities are: Quantity Quantity is a property that can exist as a multitude or magnitude. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in terms

neg: Scalar (physics) classical theories, like Newtonian mechanics, this means that rotations or reflections preserve scalars, while in relativistic theories, Lorentz transformations or space-time translations preserve scalars. Since scalars mostly may be treated as special cases of tensors (as is done with "vectors", ...), "physical scalar fields" are a special case of more general fields, like vector fields, spinor fields, and tensor fields. A physical quantity is expressed as the product of a numerical value and a physical unit, not merely a number. The quantity does not depend on the unit (e.g. for distance, 1 km is the same as 1000 m), although

neg: Value-form is market-trade, i.e. a free negotiation between trading partners who are equals in the marketplace. In his story, Marx defines the magnitude of "value" simply as the ratio of a physical quantity of product to a quantity of average labour-time, which is equal to a quantity of gold-money (in other words, a scalar): He admits early on, that the assumption of gold-money is a theoretical simplification, since the buying power of money tokens can vary due to causes that have nothing to do with the production system (within certain limits, X, Y and Z can vary independently of each other);

neg: Velocity Velocity The velocity of an object is the rate of change of its position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction of motion (e.g. to the north). Velocity is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of bodies. Velocity is a physical vector quantity; both magnitude and direction are needed to define it. The scalar absolute value (magnitude) of velocity is called , being a coherent derived unit whose quantity is measured in the SI

neg: Coherence (units of measurement) quantity is a statement that determines the ratio of any two instances of the quantity. The specification of the value of any constant factor is not a part of the definition since it does not affect the ratio. The definition of velocity above satisfies this requirement since it implies that "v"/"v" = ("d"/"d")/("t"/"t"); thus if the ratios of distances and times are determined, then so is the ratio of velocities. A definition of a unit of a physical quantity is a statement that determines the ratio of any instance of the quantity to the unit. This ratio is the numerical

neg: Proton-to-electron mass ratio Proton-to-electron mass ratio In physics, the proton-to-electron mass ratio, μ or β, is simply the rest mass of the proton divided by that of the electron. Because this is a ratio of like-dimensioned physical quantities, it is a dimensionless quantity, a function of the dimensionless physical constants, and has numerical value independent of the system of units, namely: The number enclosed in parentheses is the measurement uncertainty on the last two digits. The value of "μ" is known to about 0.1 parts per billion. "μ" is an important fundamental physical constant because: Astrophysicists have tried to find evidence that "μ"

neg: Cauchy stress tensor itself is a physical quantity and as such, it is independent of the coordinate system chosen to represent it. There are certain invariants associated with every tensor which are also independent of the coordinate system. For example, a vector is a simple tensor of rank one. In three dimensions, it has three components. The value of these components will depend on the coordinate system chosen to represent the vector, but the magnitude of the vector is a physical quantity (a scalar) and is independent of the Cartesian coordinate system chosen to represent the vector. Similarly, every second rank tensor (such

neg: Dimensional analysis the acceleration of gravity is 32 feet per second per second and the initial upward speed is 500 feet per second. It is not even necessary for "t" to be in "seconds". For example, suppose "t" = 0.01 minutes. Then the first term would be The value of a dimensional physical quantity "Z" is written as the product of a unit ["Z"] within the dimension and a dimensionless numerical factor, "n". When like-dimensioned quantities are added or subtracted or compared, it is convenient to express them in consistent units so that the numerical values of these quantities may be directly

neg: Planck constant distinction between "value" and "numerical value" as it applies to frequency and the Planck constant is explained in more detail in this pdf file Link. These two relations are the temporal and spatial component parts of the special relativistic expression using 4-vectors. Classical statistical mechanics requires the existence of (but does not define its value). Eventually, following upon Planck's discovery, it was recognized that physical action cannot take on an arbitrary value. Instead, it must be some multiple of a very small quantity, the "quantum of action", now called the Planck constant. This is the so-called "old quantum theory" developed

neg: Bolometric correction different filters is a physical and not arbitrary quantity, the absolute bolometric magnitude of the Sun is arbitrary, and so the zero-point of the bolometric correction scale that follows from it. This explain why classic references have tabulated apparently mutually incompatible values for these quantities. The bolometric scale historically had varied somewhat in the literature, with the Sun's bolometric correction in V-band varying from -0.19 to -0.07 magnitude. It follows that any value for the absolute bolometric magnitude of the Sun is legitimate, on the condition that once chosen all bolometric corrections are rescaled accordingly. If not, this will induce

neg: Juxtaposition meaning. In geometry, juxtaposition of names of points represents lines or line segments. In physics, juxtaposition is also used for "multiplication" of a numerical value and a physical quantity, and of two physical quantities, for example, three times π would be written as 3π and area equals length times width as "A" = "lw". Juxtaposition Juxtaposition is an act or instance of placing two elements close together or side by side. This is often done in order to compare/contrast the two, to show similarities or differences, etc. Juxtaposition in literary terms is the showing contrast by concepts placed side by

neg: Dimensional analysis for obtaining a quantity in units of Celsius from units of Fahrenheit; one could have started with the equivalence between 100 °C and 212 °F, though this would yield the same formula at the end. Hence, to convert the numerical quantity value of a temperature "T"[F] in degrees Fahrenheit to a numerical quantity value "T"[C] in degrees Celsius, this formula may be used: To convert "T"[C] in degrees Celsius to "T"[F] in degrees Fahrenheit, this formula may be used: Dimensional analysis is most often used in physics and chemistry – and in the mathematics thereof – but finds some applications

neg: Fine-structure constant Fine-structure constant In physics, the fine-structure constant, also known as Sommerfeld's constant, commonly denoted by (the Greek letter "alpha"), is a dimensionless physical constant characterizing the strength of the electromagnetic interaction between elementary charged particles. It is related to the elementary charge , which characterizes the strength of the coupling of an elementary charged particle with the electromagnetic field, by the formula . Being a dimensionless quantity, it has the same numerical value of about in all systems of units. Some equivalent definitions of in terms of other fundamental physical constants are: where: The definition reflects the relationship between and

neg: Scalar (physics) Scalar (physics) A scalar or scalar quantity in physics is a physical quantity that can be described by a single element of a number field such as a real number, often accompanied by units of measurement. A scalar is usually said to be a physical quantity that only has magnitude and no other characteristics. This is in contrast to vectors, tensors, etc. which are described by several numbers that characterize their magnitude, direction, and so on. The concept of a scalar in physics is essentially the same as in mathematics. Formally, a scalar is unchanged by coordinate system transformations. In

neg: Level of measurement that measurement is the estimation of the ratio between a magnitude of a continuous quantity and a unit magnitude of the same kind (Michell, 1997, 1999). A ratio scale possesses a meaningful (unique and non-arbitrary) zero value. Most measurement in the physical sciences and engineering is done on ratio scales. Examples include mass, length, duration, plane angle, energy and electric charge. In contrast to interval scales, ratios are now meaningful because having a non-arbitrary zero point makes it meaningful to say, for example, that one object has "twice the length." Very informally, many ratio scales can be described as specifying

neg: ISO 31-0 The above curly-bracket notation could be extended with a unit-symbol index to clarify this dependency, as in {"λ"} = 6.982 × 10 or equivalently {"λ"} = 698.2. In practice, where it is necessary to refer to the numerical value of a quantity expressed in a particular unit, it is notationally more convenient to simply divide the quantity through that unit, as in or equivalently This is a particularly useful and widely used notation for labelling the axes of graphs or for the headings of table columns, where repeating the unit after each numerical value can be typographically inconvenient. See Sect.

neg: Quantum superposition the states which have definite values for this quantity. Every physical quantity has a Hermitian linear operator associated to it, and the states where the value of this physical quantity is definite are the eigenstates of this linear operator. The linear combination of two or more eigenstates results in quantum superposition of two or more values of the quantity. If the quantity is measured, the value of the physical quantity will be random, with a probability equal to the square of the coefficient of the superposition in the linear combination. Immediately after the measurement, the state will be given by

neg: Random variable variables. Random variable In probability and statistics, a random variable, random quantity, aleatory variable, or stochastic variable is a variable whose possible values are outcomes of a random phenomenon. More specifically, a random variable is defined as a function that maps the outcomes of unpredictable processes to numerical quantities (labels), typically real numbers. In this sense, it is a procedure for assigning a numerical quantity to each physical outcome. Contrary to its name, this procedure itself is neither random nor variable. Rather, the underlying process providing the input to this procedure yields random (possibly non-numerical) output that the procedure maps

neg: Random variable Random variable In probability and statistics, a random variable, random quantity, aleatory variable, or stochastic variable is a variable whose possible values are outcomes of a random phenomenon. More specifically, a random variable is defined as a function that maps the outcomes of unpredictable processes to numerical quantities (labels), typically real numbers. In this sense, it is a procedure for assigning a numerical quantity to each physical outcome. Contrary to its name, this procedure itself is neither random nor variable. Rather, the underlying process providing the input to this procedure yields random (possibly non-numerical) output that the procedure maps to

neg: Cepstrum uses only the information of the magnitude of the spectrum. Many texts define the process as FT → abs() → log → IFT, i.e., that the cepstrum is the "inverse Fourier transform of the log-magnitude Fourier spectrum". (the difference between squaring or taking the absolute value amounts to an overall factor of 2). The "kepstrum", which stands for "Kolmogorov-equation power-series time response", is similar to the cepstrum and has the same relation to it as expected value has to statistical average, i.e. cepstrum is the empirically measured quantity, while kepstrum is the theoretical quantity. It was in use before the

neg: Intensive and extensive properties Intensive and extensive properties Physical properties of materials and systems can often be categorized as being either intensive or extensive, according to how the property changes when the size (or extent) of the system changes. According to IUPAC, an intensive quantity is one whose magnitude is independent of the size of the system whereas an extensive quantity is one whose magnitude is additive for subsystems. This reflects the corresponding mathematical ideas of mean and measure, respectively. An intensive property is a bulk property, meaning that it is a local physical property of a system that does not depend on the

neg: Tensor numerical value of the mass in kg is still given by integral of formula_57. Thus formula_58 (in units of ). More generally, if the Cartesian coordinates undergo a linear transformation, then the numerical value of the density must change by a factor of the reciprocal of the absolute value of the determinant of the coordinate transformation, so that the integral remains invariant, by the change of variables formula for integration. Such a quantity that scales by the reciprocal of the absolute value of the determinant of the coordinate transition map is called a scalar density. To model a non-constant density,

neg: Numerical resistivity field with the grid, and numerical method. In general, numerical resistivity will not behave isotropically, and there can be different effective numerical resistivities in different parts of the computational domain. For current (2005) simulations of the solar corona and inner heliosphere, this numerical effect can be several orders of magnitude larger than the physical resistivity of the plasma. Numerical resistivity Numerical resistivity is a problem in computer simulations of ideal magnetohydrodynamics (MHD). It is a form of numerical diffusion. In near-ideal MHD systems, the magnetic field can diffuse only very slowly through the plasma or fluid of the system; it

neg: Accuracy and precision "precise". Related terms include "bias" (non-random or directed effects caused by a factor or factors unrelated to the independent variable) and "error" (random variability). The terminology is also applied to indirect measurements—that is, values obtained by a computational procedure from observed data. In addition to accuracy and precision, measurements may also have a measurement resolution, which is the smallest change in the underlying physical quantity that produces a response in the measurement. In numerical analysis, accuracy is also the nearness of a calculation to the true value; while precision is the resolution of the representation, typically defined by the number

neg: Suzhou numerals represented by the Chinese ideograph. The digits are positional. The full numerical notations are written in two lines to indicate numerical value, order of magnitude, and unit of measurement. Following the rod numeral system, the digits of the Suzhou numerals are always written horizontally from left to right, even when used within vertically written documents. The first line contains the numerical values, in this example, "" stands for "4022". The second line consists of Chinese characters that represents the order of magnitude and unit of measurement of the first digit in the numerical representation. In this case "" which stands

neg: Measurement uncertainty uncertainty is the measurement uncertainty relative to the magnitude of a particular single choice for the value for the measured quantity, when this choice is nonzero. This particular single choice is usually called the measured value, which may be optimal in some well-defined sense (e.g., a mean, median, or mode). Thus, the relative measurement uncertainty is the measurement uncertainty divided by the absolute value of the measured value, when the measured value is not zero. The purpose of measurement is to provide information about a quantity of interest – a measurand. For example, the measurand might be the size of

neg: Self-similarity it can be continually magnified 3x without changing shape. The non-trivial similarity evident in fractals is distinguished by their fine structure, or detail on arbitrarily small scales. As a counterexample, whereas any portion of a straight line may resemble the whole, further detail is not revealed. A time developing phenomenon is said to exhibit self-similarity if the numerical value of certain observable quantity formula_1 measured at different times are different but the corresponding dimensionless quantity at given value of formula_2 remain invariant. It happens if the quantity formula_1 exhibits dynamic scaling. The idea is just an extension of the idea

neg: Unit of measurement is the same length regardless of temperature, but a metal bar will be exactly one metre long only at a certain temperature. There are certain rules that have to be used when dealing with units: Conversion of units involves comparison of different standard physical values, either of a single physical quantity or of a physical quantity and a combination of other physical quantities. Starting with: just replace the original unit formula_9 with its meaning in terms of the desired unit formula_10, e.g. if formula_11, then: Now formula_13 and formula_14 are both numerical values, so just calculate their product. Or, which

neg: Quantum superposition in some basis formula_84, a list of numbers at each value of n for the vector which has value q for the physical quantity. Now form the outer product of the vectors by multiplying all the vector components and add them with coefficients to make the matrix where the sum extends over all possible values of q. This matrix is necessarily symmetric because it is formed from the orthogonal states, and has eigenvalues q. The matrix A is called the observable associated to the physical quantity. It has the property that the eigenvalues and eigenvectors determine the physical quantity and

neg: International System of Quantities define quantities used in scientific disciplines such as mechanics (e.g., pressure), light, acoustics (e.g., sound pressure), electromagnetism, information technology (e.g., storage capacity), chemistry, mathematics (e.g., Fourier transform), and physiology. A base quantity is a physical quantity in a subset of a given system of quantities that is chosen by convention, where no quantity in the set can be expressed in terms of the others. The ISQ defines seven base quantities. The symbols for them, as for other quantities, are written in italics. The dimension of a physical quantity does not include magnitude or units. The conventional symbolic representation of the

neg: Fabric sound evaluation system on by seven different attributes, clearness, highness, loudness, pleasantness, roughness, sharpness and softness, in the format of free-modulus magnitude estimation (FMME). A FMME is a method of magnitude estimation. In this example, after listening to the different sounds, participants were asked to give a numerical value to the first sound they heard. Then, based on that value given, they were asked to give a value to the next value. Once all of the values have been given, an average of multiple participants’ can be collated. The end result is the data being presented in a rational scale. The psychological characteristics

neg: Approximation error Approximation error The approximation error in some data is the discrepancy between an exact value and some approximation to it. An approximation error can occur because: In the mathematical field of numerical analysis, the numerical stability of an algorithm indicates how the error is propagated by the algorithm. One commonly distinguishes between the relative error and the absolute error. Given some value "v" and its approximation "v", the absolute error is where the vertical bars denote the absolute value. If formula_2 the relative error is and the percent error is In words, the absolute error is the magnitude of the

neg: Classical Hamiltonian quaternions defined "tensor" as a positive numerical quantity, or, more properly, signless number. A tensor can be thought of as a positive scalar. The "tensor" can be thought of as representing a "stretching factor." Hamilton introduced the term tensor in his first book, Lectures on Quaternions, based on lectures he gave shortly after his invention of the quaternions: Each quaternion has a tensor, which is a measure of its magnitude (in the same way as the length of a vector is a measure of a vectors' magnitude). When a quaternion is defined as the quotient of two vectors, its tensor is

neg: Electric-field integral equation real physical quantity. After all, if we can freely choose formula_26 to be anything, then formula_29 is not unique. One may ask: what is the "true" value of formula_29 measured in an experiment? If formula_29 is not unique, then the only logical answer must be that we can never measure the value of formula_29. On this basis, it is often stated that it is not a real physical quantity and it is believed that the fields formula_33 and formula_34 are the true physical quantities. However, there is at least one experiment in which value of the formula_33 and formula_34 are

neg: Stevens's power law generally involves subjects altering the magnitude of one physical quantity, such as the brightness of a light, so that its perceived intensity is equal to the perceived intensity of another type of quantity, such as warmth or pressure. Stevens generally collected magnitude estimation data from multiple observers, averaged the data across subjects, and then fitted a power function to the data. Because the fit was generally reasonable, he concluded the power law was correct. A principal criticism has been that Stevens's approach provides neither a direct test of the power law itself nor the underlying assumptions of the magnitude estimation/production

neg: Stroop effect to incongruent ones (i.e., a up-pointing arrow located below the fixation sign). A similar effect, the Simon effect, uses non-spatial stimuli. The Numerical Stroop effect demonstrates the close relationship between numerical values and physical sizes. Digits symbolize numerical values but they also have physical sizes. A digit can be presented as big or small (e.g., 5 vs. 5), irrespective of its numerical value. Comparing digits in incongruent trials (e.g., 3 5) is slower than comparing digits in congruent trials (e.g., 5 3) and the difference in reaction time is termed the numerical Stroop effect. The effect of irrelevant numerical values

neg: Scalar (physics) 3-vector) to comprise a relativistic 4-vector. Similarly, energy density must be combined with momentum density and pressure into the stress–energy tensor. Examples of scalar quantities in relativity include electric charge, spacetime interval (e.g., proper time and proper length), and invariant mass. Scalar (physics) A scalar or scalar quantity in physics is a physical quantity that can be described by a single element of a number field such as a real number, often accompanied by units of measurement. A scalar is usually said to be a physical quantity that only has magnitude and no other characteristics. This is in contrast to

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: Apparent magnitude band. As the amount of light actually received by a telescope is reduced by transmission through the Earth's atmosphere, any measurement of apparent magnitude is corrected for what it would have been as seen from above the atmosphere. The dimmer an object appears, the higher the numerical value given to its apparent magnitude, with a difference of 5 magnitudes corresponding to a brightness factor of exactly 100. Therefore, the apparent magnitude , in the spectral band , would be given by which is more commonly expressed in terms of common (base-10) logarithms as where is the observed flux density using

neg: Planck units ratio of two like-dimensioned quantities is a dimensionless quantity. If, by a shorthand convention, it is understood that all physical quantities are expressed in terms of Planck units, the ratios above may be expressed simply with the symbols of physical quantity, without being scaled explicitly by their corresponding unit: In order for this last equation to be valid (without "G" present), "F", "m", "m", and "r" are understood to be the dimensionless numerical values of these quantities measured in terms of Planck units. This is why Planck units or any other use of natural units should be employed with care.

neg: Euclidean vector the vector (0, 5) (in 2 dimensions with the positive "y"-axis as 'up'). Another quantity represented by a vector is force, since it has a magnitude and direction and follows the rules of vector addition. Vectors also describe many other physical quantities, such as linear displacement, displacement, linear acceleration, angular acceleration, linear momentum, and angular momentum. Other physical vectors, such as the electric and magnetic field, are represented as a system of vectors at each point of a physical space; that is, a vector field. Examples of quantities that have magnitude and direction but fail to follow the rules of

neg: Preparation (physics) Preparation (physics) Generally, measuring instruments are meant to get the (pre-)existent value of a physical quantity of a system with as little change of, or interaction with the system as possible. — If a measuring instrument fixes a physical quantity of a physical system for further use (post-existent), then the measurement process is called preparation. For example, a graduated cylinder filled to a definite mark prepares a certain volume of the liquid for further use. Though preparation measures a physical quantity — at least by fixing it — it often destroys previous states of the physical system. This is usually

neg: Dimensional analysis the same dimensions on its left and right sides, a property known as "dimensional homogeneity". Checking for dimensional homogeneity is a common application of dimensional analysis, serving as a plausibility check on derived equations and computations. It also serves as a guide and constraint in deriving equations that may describe a physical system in the absence of a more rigorous derivation. A lot of parameters and the measurements(m) in the physical sciences and engineering are expressed as a concrete number – a numerical quantity(q) and a corresponding dimensional unit. Often a quantity is expressed in terms of several other quantities;

neg: Qualitative reasoning Qualitative reasoning Qualitative Reasoning (QR) is an area of research within Artificial Intelligence (AI) that automates reasoning about continuous aspects of the physical world, such as space, time, and quantity, for the purpose of problem solving and planning using qualitative rather than quantitative information. Precise numerical values or quantities are avoided, and qualitative values are used instead (e.g., high, low, zero, rising, falling, etc.). Qualitative reasoning creates non-numerical descriptions of physical systems and their behavior, preserving important behavioral properties and qualitative distinctions. The goal of qualitative reasoning research is to develop representation and reasoning methods that enable computer programs to

neg: Preparation (physics) experienced in quantum mechanics. Preparation (physics) Generally, measuring instruments are meant to get the (pre-)existent value of a physical quantity of a system with as little change of, or interaction with the system as possible. — If a measuring instrument fixes a physical quantity of a physical system for further use (post-existent), then the measurement process is called preparation. For example, a graduated cylinder filled to a definite mark prepares a certain volume of the liquid for further use. Though preparation measures a physical quantity — at least by fixing it — it often destroys previous states of the physical

neg: Unit of measurement Unit of measurement A unit of measurement is a definite magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity. Any other quantity of that kind can be expressed as a multiple of the unit of measurement. For example, a length is a physical quantity. The metre is a unit of length that represents a definite predetermined length. When we say 10 metres (or 10 m), we actually mean 10 times the definite predetermined length called "metre". Measurement is a process of determining how

neg: Quantum electrodynamics diagram. As well as the visual shorthand for the actions Feynman introduces another kind of shorthand for the numerical quantities called probability amplitudes. The probability is the square of the absolute value of total probability amplitude, formula_1. If a photon moves from one place and time formula_2 to another place and time formula_3, the associated quantity is written in Feynman's shorthand as formula_4. The similar quantity for an electron moving from formula_5 to formula_6 is written formula_7. The quantity that tells us about the probability amplitude for the emission or absorption of a photon he calls "j". This is related

neg: Introduction to the mathematics of general relativity line, which is a one-dimensional object. A tensor extends this concept to additional dimensions. A two-dimensional tensor would be called a second-order tensor. This can be viewed as a set of related vectors, moving in multiple directions on a plane. Vectors are fundamental in the physical sciences. They can be used to represent any quantity that has both a magnitude and direction, such as velocity, the magnitude of which is speed. For example, the velocity "5 meters per second upward" could be represented by the vector (in 2 dimensions with the positive axis as 'up'). Another quantity represented by a

neg: Euclidean vector the vector for "F" is −15 N. In either case, the magnitude of the vector is 15 N. Likewise, the vector representation of a displacement Δ"s" of 4 meters would be 4 m or −4 m, depending on its direction, and its magnitude would be 4 m regardless. Vectors are fundamental in the physical sciences. They can be used to represent any quantity that has magnitude, has direction, and which adheres to the rules of vector addition. An example is velocity, the magnitude of which is speed. For example, the velocity "5 meters per second upward" could be represented by

neg: Absolute value "absolute value" include "numerical value" and "magnitude". In programming languages and computational software packages, the absolute value of "x" is generally represented by abs("x"), or a similar expression. The vertical bar notation also appears in a number of other mathematical contexts: for example, when applied to a set, it denotes its cardinality; when applied to a matrix, it denotes its determinant. Vertical bars denote the absolute value only for algebraic objects for which the notion of an absolute value is defined, notably an element of a normed division algebra like a real number, complex number, quaternion. A closely related but

neg: Deviation (statistics) Deviation (statistics) In mathematics and statistics, deviation is a measure of difference between the observed value of a variable and some other value, often that variable's mean. The sign of the deviation (positive or negative), reports the direction of that difference (the deviation is positive when the observed value exceeds the reference value). The magnitude of the value indicates the size of the difference. A deviation that is a difference between an observed value and the "true value" of a quantity of interest (such as a population mean) is an error and a deviation that is the difference between the

neg: Dimension (metadata) a value measured in degrees Fahrenheit there is a value measured in degrees Celsius with the same quantity, and vice versa. Quantity preserving one-to-one correspondences are the well-known equations Cº = (5/9)*(Fº − 32) and Fº = (9/5)*(Cº) + 32. Units of measure are not limited to physical categories. Examples of physical categories are: linear measure, area, volume, mass, velocity, time duration.Examples of non-physical categories are: currency, quality indicator, colour intensity. Quantities may be grouped together into categories of quantities which are mutually comparable. Lengths, diameters, distances, heights, wavelengths and so on would constitute such a category. Mutually comparable quantities

neg: AB magnitude "f", the monochromatic AB magnitude is: or This exact definition is defined in magnitudes relative to the cgs units of erg s cm Hz: Inverting this leads to the true definition of the numerical value "3631 Jy" often cited: Actual measurements are always made across some continuous range of wavelengths. The "bandpass" AB magnitude is defined so that the zero point corresponds to a bandpass-averaged spectral flux density of about 3631 Jansky: where "e(ν)" is the "equal-energy" filter response function and the "(hν)" term assumes that the detector is a photon-counting device such as a CCD or photomultiplier. (Filter responses

neg: Dimensionless physical constant Dimensionless physical constant In physics, a dimensionless physical constant, sometimes called a fundamental physical constant, is a physical constant that is dimensionless. It has no units attached and has a numerical value that is independent of the system of units used. Perhaps the best-known example is the fine-structure constant, α, which has an approximate value of . The term "fundamental physical constant" is often used to refer to the dimensionless constants, but has also been used (primarily by NIST and CODATA) to refer to certain universal dimensioned physical constants, such as the speed of light "c", vacuum permittivity "ε", Planck

neg: Round number stand for a value or values near to the nominal value expressed. For instance, a round number such as 600 might be used to refer to a value whose magnitude is actually 592, because the actual value is more cumbersome to express exactly. Likewise, a round number may be used to refer to a range of values near the nominal value that expresses imprecision about a quantity. Thus, a value reported as 600 might actually represent any value near 600, possibly as low as 550 or as high as 650, all of which would "round" to 600. In decimal notation,

neg: Law of value their value, where the magnitude of their value is determined by the average quantity of human labour which is currently socially necessary to produce them (see labor theory of value and value-form). In itself, this theorem is fairly simple to understand and intuitively it makes sense to many working people. However, theorizing its implications is a much more complex task as it kept Marx busy across more than two decades. When Marx talked about "value relationships" or "value proportions" (German: "Wertverhältnisse"), he did not mean "the money" or "the price". Instead, he meant the value proportions that exist between products

neg: Dimensionless physical constant either derive their values from the mathematics of the theory, or accept their values as empirical. Dimensionless physical constant In physics, a dimensionless physical constant, sometimes called a fundamental physical constant, is a physical constant that is dimensionless. It has no units attached and has a numerical value that is independent of the system of units used. Perhaps the best-known example is the fine-structure constant, α, which has an approximate value of . The term "fundamental physical constant" is often used to refer to the dimensionless constants, but has also been used (primarily by NIST and CODATA) to refer to

neg: Pressure force per unit area (the pressure) is the same. We can shrink the size of our "container" down to a very small point (becoming less true as we approach the atomic scale), and the pressure will still have a single value at that point. Therefore, pressure is a scalar quantity, not a vector quantity. It has magnitude but no direction sense associated with it. Pressure force acts in all directions at a point inside a gas. At the surface of a gas, the pressure force acts perpendicular (at right angle) to the surface. A closely related quantity is the stress

neg: Sign bit representations, the latter of which does not correspond to the mathematical concept of a negative number. In the two's complement representation, the sign bit has the weight where is the number of bits. In the ones' complement representation, the most negative value is , but there are two representations of zero, one for each value of the sign bit. In a sign-and-magnitude representation of numbers, the value of the sign bit determines whether the numerical value is positive or negative. Floating point numbers, such as IEEE format, IBM format, VAX format, and even the format used by the Zuse Z1

neg: Labor theory of value in the second sense to represent "exchange value". Since the term "value" is understood in the LTV as denoting something created by labor, and its "magnitude" as something proportional to the quantity of labor performed, it is important to explain how the labor process both preserves value and adds new value in the commodities it creates. The value of a commodity increases in proportion to the duration and intensity of labor performed on average for its production. Part of what the LTV means by "socially necessary" is that the value only increases in proportion to this labor as it is

neg: Dyscalculia in their numerical magnitude representation by as much as five years. However, the lack of longitudinal studies still leaves the question open as to whether the deficient numerical magnitude representation is a delayed development or impairment. Rousselle & Noël propose that dyscalculia is caused by the inability to map preexisting representations of numerical magnitude onto symbolic Indian digits. Evidence for this hypothesis is based on research studies that have found that individuals with dyscalculia are proficient on tasks that measure knowledge of non-symbolic numerical magnitude (i.e., non-symbolic comparison tasks) but show an impaired ability to process symbolic representations of number

neg: Vega of 0.314″ for the star system 61 Cygni, Struve revised his value for Vega's parallax to nearly double the original estimate. This change cast further doubt on Struve's data. Thus most astronomers at the time, including Struve, credited Bessel with the first published parallax result. However, Struve's initial result was actually close to the currently accepted value of 0.129″, as determined by the Hipparcos astrometry satellite. The brightness of a star, as seen from Earth, is measured with a standardized, logarithmic scale. This apparent magnitude is a numerical value that decreases in value with increasing brightness of the star. The

neg: Scattering parameters of the output power-wave to the input power-wave, and it equals the square-root of the power gain. This is a real-value (or scalar) quantity, the phase information being dropped. The scalar logarithmic (decibel or dB) expression for gain (g) is: This is more commonly used than scalar linear gain and a positive quantity is normally understood as simply a "gain", while a negative quantity is a "negative gain" (a "loss"), equivalent to its magnitude in dB. For example, at 100 MHz, a 10 m length of cable may have a gain of -1 dB, equal to a loss of 1

neg: Rounding Rounding Rounding a numerical value means replacing it by another value that is approximately equal but has a shorter, simpler, or more explicit representation; for example, replacing $ with $, or the fraction 312/937 with 1/3, or the expression with . Rounding is often done to obtain a value that is easier to report and communicate than the original. Rounding can also be important to avoid misleadingly precise reporting of a computed number, measurement or estimate; for example, a quantity that was computed as but is known to be accurate only to within a few hundred units is better stated

neg: Adjoint state method Adjoint state method The adjoint state method is a numerical method for efficiently computing the gradient of a function or operator in a numerical optimization problem. It has applications in geophysics, seismic imaging and more recently in neural networks. The adjoint state space is chosen to simplify the physical interpretation of equation constraints. It may take the form of a Hilbert space. Adjoint state techniques allow the use of integration by parts, resulting in a form which explicitly contains the physically interesting quantity. An adjoint state equation is introduced, including a new unknown variable. The adjoint method formulates the gradient

neg: Numerical Stroop effect damage to the left intraparietal sulcus. Numerical Stroop effect In Psychology, the numerical Stroop effect (related to the standard Stroop effect) demonstrates the relationship between numerical values and physical sizes. When digits are presented visually, they can be physically large or small, irrespective of their actual values. Congruent pairs occur when size and value correspond (e.g., large 5 small 3) while incongruent pairs occur when size and value are incompatible (e.g., large 3 small 5). It was found that when people are asked to compare digits, their reaction time tends to be slower in the case of incongruent pairs. This

neg: Numerical Stroop effect Numerical Stroop effect In Psychology, the numerical Stroop effect (related to the standard Stroop effect) demonstrates the relationship between numerical values and physical sizes. When digits are presented visually, they can be physically large or small, irrespective of their actual values. Congruent pairs occur when size and value correspond (e.g., large 5 small 3) while incongruent pairs occur when size and value are incompatible (e.g., large 3 small 5). It was found that when people are asked to compare digits, their reaction time tends to be slower in the case of incongruent pairs. This reaction time difference between congruent and

neg: Circular orbit to velocity) causes change in direction. If it is constant in magnitude and changing in direction with the velocity, we get a circular motion. For this centripetal acceleration we have where: The formula is dimensionless, describing a ratio true for all units of measure applied uniformly across the formula. If the numerical value of formula_5 is measured in meters per second per second, then the numerical values for formula_2 will be in meters per second, formula_3 in meters, and formula_4 in radians per second. The relative velocity is constant: where: The orbit equation in polar coordinates, which in general gives

neg: Planck units Planck mass, length divided by the Planck length, etc.) and the only quantities that we ultimately measure in physical experiments or in our perception of reality are dimensionless numbers. When one commonly measures a length with a ruler or tape-measure, that person is actually counting tick marks on a given standard or is measuring the length relative to that given standard, which is a dimensionless value. It is no different for physical experiments, as all physical quantities are measured relative to some other like-dimensioned quantity. We can notice a difference if some dimensionless physical quantity such as fine-structure constant, "α",

neg: Transformation problem concrete forms of that labour; there is nothing left but what is common to them all; all are reduced to one and the same sort of labour, human labour in the abstract." "A use value, or useful article, therefore, has value only because human labour in the abstract has been embodied or materialised in it. How, then, is the magnitude of this value to be measured? Plainly, by the quantity of the value-creating substance, the labour, contained in the article." As labour produces in this sense more than its own value, the direct-labour input is called variable capital and denoted

neg: Finite difference method into a uniform grid (see image to the right). Note that this means that finite-difference methods produce sets of discrete numerical approximations to the derivative, often in a "time-stepping" manner. An expression of general interest is the local truncation error of a method. Typically expressed using Big-O notation, local truncation error refers to the error from a single application of a method. That is, it is the quantity formula_8 if formula_9 refers to the exact value and formula_10 to the numerical approximation. The remainder term of a Taylor polynomial is convenient for analyzing the local truncation error. Using the Lagrange

neg: Rotation matrix numerical error, we may construct a symmetric matrix, and find the eigenvector, , of its largest magnitude eigenvalue. (If is truly a rotation matrix, that value will be 1.) The quaternion so obtained will correspond to the rotation matrix closest to the given matrix . If the matrix is nonsingular, its columns are linearly independent vectors; thus the Gram–Schmidt process can adjust them to be an orthonormal basis. Stated in terms of numerical linear algebra, we convert to an orthogonal matrix, , using QR decomposition. However, we often prefer a closest to , which this method does not accomplish. For

neg: Precision (computer science) Precision (computer science) In computer science, the precision of a numerical quantity is a measure of the detail in which the quantity is expressed. This is usually measured in bits, but sometimes in decimal digits. It is related to precision in mathematics, which describes the number of digits that are used to express a value. Some of the standardized precision formats are Of these, octuple-precision format is rarely used. The single- and double-precision formats are most widely used and supported on nearly all platforms. The use of half-precision format has been increasing especially in the field of machine learning since

neg: Capital, Volume I 1 coat or = 10 lb. tea or = 40 lb. coffee or = 1 quarter of corn or = 2 ounces of gold or = ½ ton of iron or = etc." Marx explains that with this example of the expanded form of value the linen "is now expressed in terms of innumerable other members of the world of commodities. Every other commodity now becomes a mirror of linen's value." At this point, the particular use-value of linen becomes unimportant, but rather it is the magnitude of value (determined by socially necessary labor time) possessed in a quantity of

neg: Pyxplot Mathematical expressions involving quantities with physical units automatically calculate the dimensions of the resulting quantity; for example, squaring a distance will automatically produce an area. Data files read in in one unit (for example inches), can be output in another unit (for example meters). All of Pyxplot's numerical analysis commands, such as curve-fitting, Fourier transforms and histogram generation also natively support quantities with physical units. Pyxplot's vector graphics drawing commands allow positions on the page and rotation angles to be specified by quantities with appropriate physical units. Pyxplot is extensively documented online. Its website includes a large Users' Manual and

neg: Numerical tower but to coerce most reals to an integer could well result in a problem (for example, the real 1/3 does not equal any integer). The Scheme programming language defines all its arithmetic within this model, as do most other Lisp dialects. Some implementations may extend or adapt the tower. Kawa, a Scheme implementation for JVM, extends it with a Quantity type that is even more generic than Number. Smalltalk is another programming language that follows this model, but it has a Magnitude as superclass of Number. Another popular variant is having both exact and inexact versions of the tower or

neg: Avogadro constant the mass number 12 (6 protons, 6 neutrons), has a similar number of carbon atoms, . The Avogadro number is a dimensionless quantity, and has the same numerical value of the Avogadro constant when given in base units. In contrast, the Avogadro constant has the dimension of reciprocal amount of substance. The Avogadro constant can also be expressed as , which can be used to convert from volume per molecule in cubic ångströms to molar volume in millilitres per mole. Pending revisions in the base set of SI units necessitated redefinitions of the concepts of chemical quantity. The Avogadro number,

neg: Production (economics) input, and it can be described by means of the production function. It refers to a series of events in production in which production inputs of different quality and quantity are combined into products of different quality and quantity. Products can be physical goods, immaterial services and most often combinations of both. The characteristics created into the product by the producer imply surplus value to the consumer, and on the basis of the market price this value is shared by the consumer and the producer in the marketplace. This is the mechanism through which surplus value originates to the consumer

neg: Katal units of concentration per second, as moles per liter per second. Rather, the katal is used to express catalytic activity, which is a property of the catalyst. The katal is invariant of the measurement procedure, but the measured numerical value is not; the value depends on the experimental conditions. Therefore, to define the quantity of a catalyst in katals, the rate of conversion of a defined chemical reaction is specified as moles reacted per second. One katal of trypsin, for example, is that amount of trypsin which breaks one mole of peptide bonds in one second under specified conditions. The

neg: Division by zero can be given in this setting. It follows from the properties of the number system we are using (that is, integers, rationals, reals, etc.), if then the equation is equivalent to . Assuming that is a number , then it must be that . However, the single number would then have to be determined by the equation , but every number satisfies this equation, so we cannot assign a numerical value to . The concept that explains division in algebra is that it is the inverse of multiplication. For example, since 2 is the value for which the unknown quantity

neg: Isoelastic function the constant elasticity demand function, in which "p" is the price of a product and "D"("p") is the resulting quantity demanded by consumers. For most goods the elasticity "r" (the responsiveness of quantity demanded to price) is negative, so it can be convenient to write the constant elasticity demand function with a negative sign on the exponent, in order for the coefficient formula_1 to take on a positive value: where formula_8 is now interpreted as the unsigned magnitude of the responsiveness. The constant elasticity function is also used in the theory of choice under risk aversion, which usually assumes that

neg: Significant figures a tax return. As an illustration, the decimal quantity 12.345 can be expressed with various numbers of significant digits or decimal places. If insufficient precision is available then the number is rounded in some manner to fit the available precision. The following table shows the results for various total precisions and decimal places. Another example for 0.012345: The representation of a positive number "x" to a precision of "p" significant digits has a numerical value that is given by the formula: For negative numbers, the formula can be used on the absolute value; for zero, no transformation is necessary. Note

neg: Statics is the action of one body on another. A force is either a push or a pull. A force tends to move a body in the direction of its action. The action of a force is characterized by its magnitude, by the direction of its action, and by its point of application. Thus, force is a vector quantity, because its effect depends on the direction as well as on the magnitude of the action. Forces are classified as either contact or body forces. A contact force is produced by direct physical contact; an example is the force exerted on a

neg: Pseudovector be a polar vector, v is known to be a pseudovector, and v is defined to be their sum, . If the universe is transformed by a rotation matrix "R", then v is transformed to Therefore, v is neither a polar vector nor a pseudovector. For an improper rotation, v does not in general even keep the same magnitude: If the magnitude of v were to describe a measurable physical quantity, that would mean that the laws of physics would not appear the same if the universe was viewed in a mirror. In fact, this is exactly what happens in

neg: Hertzsprung–Russell diagram thereby obtain absolute magnitudes for those stars. There are several forms of the Hertzsprung–Russell diagram, and the nomenclature is not very well defined. All forms share the same general layout: stars of greater luminosity are toward the top of the diagram, and stars with higher surface temperature are toward the left side of the diagram. The original diagram displayed the spectral type of stars on the horizontal axis and the absolute visual magnitude on the vertical axis. The spectral type is not a numerical quantity, but the sequence of spectral types is a monotonic series that reflects the stellar surface

neg: Matrix mechanics so it follows that and that the time derivative of "U" is zero—it is conserved. The eigenvalues of unitary matrices are pure phases, so that the value of a unitary conserved quantity is a complex number of unit magnitude, not a real number. Another way of saying this is that a unitary matrix is the exponential of "i" times a Hermitian matrix, so that the additive conserved real quantity, the phase, is only well-defined up to an integer multiple of "2π". Only when the unitary symmetry matrix is part of a family that comes arbitrarily close to the identity are

neg: Quantum superposition in quantum mechanics, there is a list of all the states where that property has some value. These states are necessarily perpendicular to each other using the Euclidean notion of perpendicularity which comes from sums-of-squares length, except that they also must not be i multiples of each other. This list of perpendicular states has an associated value which is the value of the physical property. The superposition principle guarantees that any state can be written as a combination of states of this form with complex coefficients. Write each state with the value q of the physical quantity as a vector

neg: Conserved quantity Conserved quantity In mathematics, a conserved quantity of a dynamical system is a function of the dependent variables whose value remains constant along each trajectory of the system. Not all systems have conserved quantities, and conserved quantities are not unique, since one can always apply a function to a conserved quantity, such as adding a number. Since many laws of physics express some kind of conservation, conserved quantities commonly exist in mathematical models of physical systems. For example, any classical mechanics model will have energy as a conserved quantity so long as the forces involved are conservative. For a first