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64-400: Quantity or amount is a property that can exist as a multitude or magnitude , which illustrate discontinuity and continuity . Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value multiple of a unit of measurement . Mass , time , distance , heat , and angle are among the familiar examples of quantitative properties. Quantity is among

128-460: A ⟩ {\displaystyle |\psi _{a}\rangle } are unit vectors , and the eigenspace of a {\displaystyle a} is one-dimensional), then the eigenvalue a {\displaystyle a} is returned with probability | ⟨ ψ a | ϕ ⟩ | 2 {\displaystyle |\langle \psi _{a}|\phi \rangle |^{2}} , by

192-417: A ⟩ {\displaystyle |\psi _{a}\rangle } is an eigenket ( eigenvector ) of the observable A ^ {\displaystyle {\hat {A}}} , with eigenvalue a {\displaystyle a} , and exists in a Hilbert space . Then A ^ | ψ a ⟩ = a | ψ

256-421: A ⟩ . {\displaystyle {\hat {A}}|\psi _{a}\rangle =a|\psi _{a}\rangle .} This eigenket equation says that if a measurement of the observable A ^ {\displaystyle {\hat {A}}} is made while the system of interest is in the state | ψ a ⟩ {\displaystyle |\psi _{a}\rangle } , then

320-404: A self-adjoint operator A ^ {\displaystyle {\hat {A}}} that acts on the state of the quantum system. The eigenvalues of operator A ^ {\displaystyle {\hat {A}}} correspond to the possible values that the dynamical variable can be observed as having. For example, suppose | ψ

384-621: A volumetric ratio ; its value remains independent of the specific units of volume used, such as in milliliters per milliliter (mL/mL). The number one is recognized as a dimensionless base quantity . Radians serve as dimensionless units for angular measurements , derived from the universal ratio of 2π times the radius of a circle being equal to its circumference. Counting Counting sometimes involves numbers other than one; for example, when counting money, counting out change, "counting by twos" (2, 4, 6, 8, 10, 12, ...), or "counting by fives" (5, 10, 15, 20, 25, ...). There

448-448: A (finite) set, the answer is the same. In a broader context, the theorem is an example of a theorem in the mathematical field of (finite) combinatorics —hence (finite) combinatorics is sometimes referred to as "the mathematics of counting." Many sets that arise in mathematics do not allow a bijection to be established with {1, 2, ..., n } for any natural number n ; these are called infinite sets , while those sets for which such

512-465: A bijection does exist (for some n ) are called finite sets . Infinite sets cannot be counted in the usual sense; for one thing, the mathematical theorems which underlie this usual sense for finite sets are false for infinite sets. Furthermore, different definitions of the concepts in terms of which these theorems are stated, while equivalent for finite sets, are inequivalent in the context of infinite sets. The notion of counting may be extended to them in

576-642: A bijection with the natural numbers, and these sets are called " uncountable ." Sets for which there exists a bijection between them are said to have the same cardinality , and in the most general sense counting a set can be taken to mean determining its cardinality. Beyond the cardinalities given by each of the natural numbers, there is an infinite hierarchy of infinite cardinalities, although only very few such cardinalities occur in ordinary mathematics (that is, outside set theory that explicitly studies possible cardinalities). Counting, mostly of finite sets, has various applications in mathematics. One important principle

640-419: A bijection with the original set is not excluded. For instance, the set of all integers (including negative numbers) can be brought into bijection with the set of natural numbers, and even seemingly much larger sets like that of all finite sequences of rational numbers are still (only) countably infinite. Nevertheless, there are sets, such as the set of real numbers , that can be shown to be "too large" to admit

704-409: A collection of variables , each assuming a set of values. These can be a set of a single quantity, referred to as a scalar when represented by real numbers, or have multiple quantities as do vectors and tensors , two kinds of geometric objects. The mathematical usage of a quantity can then be varied and so is situationally dependent. Quantities can be used as being infinitesimal , arguments of

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768-400: A dependence of measurement results on the order in which measurements of observables A ^ {\displaystyle {\hat {A}}} and B ^ {\displaystyle {\hat {B}}} are performed. A measurement of A ^ {\displaystyle {\hat {A}}} alters the quantum state in a way that

832-403: A function , variables in an expression (independent or dependent), or probabilistic as in random and stochastic quantities. In mathematics, magnitudes and multitudes are also not only two distinct kinds of quantity but furthermore relatable to each other. Number theory covers the topics of the discrete quantities as numbers: number systems with their kinds and relations. Geometry studies

896-436: A great number, many, several (for count names); a bit of, a little, less, a great deal (amount) of, much (for mass names); all, plenty of, a lot of, enough, more, most, some, any, both, each, either, neither, every, no". For the complex case of 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);

960-416: A measurement is applied, the state description by a single vector may be destroyed, being replaced by a statistical ensemble . The irreversible nature of measurement operations in quantum physics is sometimes referred to as the measurement problem and is described mathematically by quantum operations . By the structure of quantum operations, this description is mathematically equivalent to that offered by

1024-573: A noun of multitude standing either for a single entity or for the individuals making the whole. An amount in general is expressed by a special class of words called identifiers, indefinite and definite and quantifiers, definite and indefinite. The amount may be expressed by: singular form and plural from, ordinal numbers before a count noun singular (first, second, third...), the demonstratives; definite and indefinite numbers and measurements (hundred/hundreds, million/millions), or cardinal numbers before count nouns. The set of language quantifiers covers "a few,

1088-583: A particular structure that was first explicitly characterized by Hölder (1901) as a set of axioms that define such features as identities and relations between magnitudes. In science, quantitative structure is the subject of empirical investigation and cannot be assumed to exist a priori for any given property. The linear continuum represents the prototype of continuous quantitative structure as characterized by Hölder (1901) (translated in Michell & Ernst, 1996). A fundamental feature of any type of quantity

1152-571: A physically meaningful observable. Also, not all physical observables are associated with non-trivial self-adjoint operators. For example, in quantum theory, mass appears as a parameter in the Hamiltonian, not as a non-trivial operator. In the case of transformation laws in quantum mechanics, the requisite automorphisms are unitary (or antiunitary ) linear transformations of the Hilbert space V . Under Galilean relativity or special relativity ,

1216-537: 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: Dimensionless quantities , or quantities of dimension one, are quantities implicitly defined in a manner that prevents their aggregation into units of measurement . Typically expressed as ratios that align with another system, these quantities do not necessitate explicitly defined units . For instance, alcohol by volume (ABV) represents

1280-727: A quantitative science; chemistry, biology and others are increasingly so. Their progress is chiefly achieved due to rendering the abstract qualities of material entities into physical quantities, by postulating that all material bodies marked by quantitative properties or physical dimensions are subject to some measurements and observations. Setting the units of measurement, physics covers such fundamental quantities as space (length, breadth, and depth) and time, mass and force, temperature, energy, and quanta . A distinction has also been made between intensive quantity and extensive quantity as two types of quantitative property, state or relation. The magnitude of an intensive quantity does not depend on

1344-489: A year after learning these skills for a child to understand what they mean and why the procedures are performed. In the meantime, children learn how to name cardinalities that they can subitize . In mathematics, the essence of counting a set and finding a result n , is that it establishes a one-to-one correspondence (or bijection) of the subject set with the subset of positive integers {1, 2, ..., n }. A fundamental fact, which can be proved by mathematical induction ,

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1408-434: Is a physical property or physical quantity that can be measured . In classical mechanics , an observable is a real -valued "function" on the set of all possible system states, e.g., position and momentum . In quantum mechanics , an observable is an operator , or gauge , where the property of the quantum state can be determined by some sequence of operations . For example, these operations might involve submitting

1472-399: Is a syntactic category , along with person and gender . The quantity is expressed by identifiers, definite and indefinite, and quantifiers , definite and indefinite, as well as by three types of nouns : 1. count unit nouns or countables; 2. mass nouns , uncountables, referring to the indefinite, unidentified amounts; 3. nouns of multitude ( collective nouns ). The word ‘number’ belongs to

1536-602: Is a third interval, etc., and going up seven notes is an octave . Learning to count is an important educational/developmental milestone in most cultures of the world. Learning to count is a child's very first step into mathematics, and constitutes the most fundamental idea of that discipline. However, some cultures in Amazonia and the Australian Outback do not count, and their languages do not have number words. Many children at just 2 years of age have some skill in reciting

1600-511: Is archaeological evidence suggesting that humans have been counting for at least 50,000 years. Counting was primarily used by ancient cultures to keep track of social and economic data such as the number of group members, prey animals, property, or debts (that is, accountancy ). Notched bones were also found in the Border Caves in South Africa, which may suggest that the concept of counting

1664-440: Is divisible into two or more constituent parts, of which each is by nature a one and a this . A quantum is a plurality if it is numerable, a magnitude if it is measurable. Plurality means that which is divisible potentially into non-continuous parts, magnitude that which is divisible into continuous parts; of magnitude, that which is continuous in one dimension is length; in two breadth, in three depth. Of these, limited plurality

1728-522: Is incompatible with the subsequent measurement of B ^ {\displaystyle {\hat {B}}} and vice versa. Observables corresponding to commuting operators are called compatible observables . For example, momentum along say the x {\displaystyle x} and y {\displaystyle y} axis are compatible. Observables corresponding to non-commuting operators are called incompatible observables or complementary variables . For example,

1792-428: Is number, limited length is a line, breadth a surface, depth a solid. In his Elements , Euclid developed the theory of ratios of magnitudes without studying the nature of magnitudes, as Archimedes, but giving the following significant definitions: A magnitude is a part of a magnitude, the less of the greater, when it measures the greater; A ratio is a sort of relation in respect of size between two magnitudes of

1856-407: Is still useful for some things. Refer also to the fencepost error , which is a type of off-by-one error . Modern mathematical English language usage has introduced another difficulty, however. Because an exclusive counting is generally tacitly assumed, the term "inclusive" is generally used in reference to a set which is actually counted exclusively. For example; How many numbers are included in

1920-452: Is that if two sets X and Y have the same finite number of elements, and a function f : X → Y is known to be injective , then it is also surjective , and vice versa. A related fact is known as the pigeonhole principle , which states that if two sets X and Y have finite numbers of elements n and m with n > m , then any map f : X → Y is not injective (so there exist two distinct elements of X that f sends to

1984-419: Is that no bijection can exist between {1, 2, ..., n } and {1, 2, ..., m } unless n = m ; this fact (together with the fact that two bijections can be composed to give another bijection) ensures that counting the same set in different ways can never result in different numbers (unless an error is made). This is the fundamental mathematical theorem that gives counting its purpose; however you count

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2048-569: Is that the relationships of equality or inequality can in principle be stated in comparisons between particular magnitudes, unlike quality, which is marked by likeness, similarity and difference, diversity. Another fundamental feature is additivity. Additivity may involve concatenation, such as adding two lengths A and B to obtain a third A + B. Additivity is not, however, restricted to extensive quantities but may also entail relations between magnitudes that can be established through experiments that permit tests of hypothesized observable manifestations of

2112-830: The Born rule . A crucial difference between classical quantities and quantum mechanical observables is that some pairs of quantum observables may not be simultaneously measurable, a property referred to as complementarity . This is mathematically expressed by non- commutativity of their corresponding operators, to the effect that the commutator [ A ^ , B ^ ] := A ^ B ^ − B ^ A ^ ≠ 0 ^ . {\displaystyle \left[{\hat {A}},{\hat {B}}\right]:={\hat {A}}{\hat {B}}-{\hat {B}}{\hat {A}}\neq {\hat {0}}.} This inequality expresses

2176-764: The ides ; more generally, dates are specified as inclusively counted days up to the next named day. In the Christian liturgical calendar , Quinquagesima (meaning 50) is 49 days before Easter Sunday. When counting "inclusively", the Sunday (the start day) will be day 1 and therefore the following Sunday will be the eighth day . For example, the French phrase for " fortnight " is quinzaine (15 [days]), and similar words are present in Greek (δεκαπενθήμερο, dekapenthímero ), Spanish ( quincena ) and Portuguese ( quinzena ). In contrast,

2240-532: The mathematical formulation of quantum mechanics , up to a phase constant , pure states are given by non-zero vectors in a Hilbert space V . Two vectors v and w are considered to specify the same state if and only if w = c v {\displaystyle \mathbf {w} =c\mathbf {v} } for some non-zero c ∈ C {\displaystyle c\in \mathbb {C} } . Observables are given by self-adjoint operators on V . Not every self-adjoint operator corresponds to

2304-444: The relative state interpretation where the original system is regarded as a subsystem of a larger system and the state of the original system is given by the partial trace of the state of the larger system. In quantum mechanics, dynamical variables A {\displaystyle A} such as position, translational (linear) momentum , orbital angular momentum , spin , and total angular momentum are each associated with

2368-488: The American mathematical psychologist R. Duncan Luce and statistician John Tukey (1964). Magnitude (how much) and multitude (how many), the two principal types of quantities, are further divided as mathematical and physical. In formal terms, quantities—their ratios, proportions, order and formal relationships of equality and inequality—are studied by mathematics. The essential part of mathematical quantities consists of having

2432-477: The English word "fortnight" itself derives from "a fourteen-night", as the archaic " sennight " does from "a seven-night"; the English words are not examples of inclusive counting. In exclusive counting languages such as English, when counting eight days "from Sunday", Monday will be day 1 , Tuesday day 2 , and the following Monday will be the eighth day . For many years it was a standard practice in English law for

2496-471: The additive relations of magnitudes. Another feature is continuity, on which Michell (1999, p. 51) says of length, as a type of quantitative attribute, "what continuity means is that if any arbitrary length, a, is selected as a unit, then for every positive real number, r , there is a length b such that b = r a". A further generalization is given by the theory of conjoint measurement , independently developed by French economist Gérard Debreu (1960) and by

2560-598: The basic classes of things along with quality , substance , change, and relation. Some quantities are such by their inner nature (as number), while others function 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. Under the name of multitude comes what is discontinuous and discrete and divisible ultimately into indivisibles, such as: army, fleet, flock, government, company, party, people, mess (military), chorus, crowd , and number ; all which are cases of collective nouns . Under

2624-478: The case of infinite sets this can even apply in situations where it is impossible to give an example. The domain of enumerative combinatorics deals with computing the number of elements of finite sets, without actually counting them; the latter usually being impossible because infinite families of finite sets are considered at once, such as the set of permutations of {1, 2, ..., n } for any natural number n . Observable In physics , an observable

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2688-412: The continuous (studied by geometry and later calculus ). The theory fits reasonably well elementary or school mathematics but less well the abstract topological and algebraic structures of modern mathematics. Establishing quantitative structure and relationships between different quantities is the cornerstone of modern science, especially but not restricted to physical sciences. Physics is fundamentally

2752-438: The count list (that is, saying "one, two, three, ..."). They can also answer questions of ordinality for small numbers, for example, "What comes after three ?". They can even be skilled at pointing to each object in a set and reciting the words one after another. This leads many parents and educators to the conclusion that the child knows how to use counting to determine the size of a set. Research suggests that it takes about

2816-499: The difference in usage between the terms "inclusive counting" and "inclusive" or "inclusively", and one must recognize that it's not uncommon for the former term to be loosely used for the latter process. Inclusive counting is usually encountered when dealing with time in Roman calendars and the Romance languages . In the ancient Roman calendar , the nones (meaning "nine") is 8 days before

2880-433: The eigenvalues are real ; however, the converse is not necessarily true. As a consequence, only certain measurements can determine the value of an observable for some state of a quantum system. In classical mechanics, any measurement can be made to determine the value of an observable. The relation between the state of a quantum system and the value of an observable requires some linear algebra for its description. In

2944-431: The end of each interval. For inclusive counting, unit intervals are counted beginning with the start of the first interval and ending with end of the last interval. This results in a count which is always greater by one when using inclusive counting, as compared to using exclusive counting, for the same set. Apparently, the introduction of the number zero to the number line resolved this difficulty; however, inclusive counting

3008-579: The four fingers and the three bones in each finger ( phalanges ) to count to twelve. Other hand-gesture systems are also in use, for example the Chinese system by which one can count to 10 using only gestures of one hand. With finger binary it is possible to keep a finger count up to 1023 = 2 − 1 . Various devices can also be used to facilitate counting, such as tally counters and abacuses . Inclusive/exclusive counting are two different methods of counting. For exclusive counting, unit intervals are counted at

3072-422: The issues of spatial magnitudes: straight lines, curved lines, surfaces and solids, all with their respective measurements and relationships. A traditional Aristotelian realist philosophy of mathematics , stemming from Aristotle and remaining popular until the eighteenth century, held that mathematics is the "science of quantity". Quantity was considered to be divided into the discrete (studied by arithmetic) and

3136-438: The mathematics of frames of reference is particularly simple, considerably restricting the set of physically meaningful observables. In quantum mechanics, measurement of observables exhibits some seemingly unintuitive properties. Specifically, if a system is in a state described by a vector in a Hilbert space , the measurement process affects the state in a non-deterministic but statistically predictable way. In particular, after

3200-432: The name of magnitude comes what is continuous and unified and divisible only into smaller divisibles, such as: matter, mass, energy, liquid, material —all cases of non-collective nouns. Along with analyzing its nature and classification , the issues of quantity involve such closely related topics as dimensionality, equality, proportion, the measurements of quantities, the units of measurements, number and numbering systems,

3264-459: The observed value of that particular measurement must return the eigenvalue a {\displaystyle a} with certainty. However, if the system of interest is in the general state | ϕ ⟩ ∈ H {\displaystyle |\phi \rangle \in {\mathcal {H}}} (and | ϕ ⟩ {\displaystyle |\phi \rangle } and | ψ

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3328-480: The phrase "from a date" to mean "beginning on the day after that date": this practice is now deprecated because of the high risk of misunderstanding. Similar counting is involved in East Asian age reckoning , in which newborns are considered to be 1 at birth. Musical terminology also uses inclusive counting of intervals between notes of the standard scale: going up one note is a second interval, going up two notes

3392-409: The position and momentum along the same axis are incompatible. Incompatible observables cannot have a complete set of common eigenfunctions . Note that there can be some simultaneous eigenvectors of A ^ {\displaystyle {\hat {A}}} and B ^ {\displaystyle {\hat {B}}} , but not enough in number to constitute

3456-409: The ratio of magnitudes of any quantity, whether volume, mass, heat and so on, is a number. Following this, Newton then defined number, and the relationship between quantity and number, in the following terms: By number we understand not so much a multitude of unities, as the abstracted ratio of any quantity to another quantity of the same kind, which we take for unity. Continuous quantities possess

3520-441: The same element of Y ); this follows from the former principle, since if f were injective, then so would its restriction to a strict subset S of X with m elements, which restriction would then be surjective, contradicting the fact that for x in X outside S , f ( x ) cannot be in the image of the restriction. Similar counting arguments can prove the existence of certain objects without explicitly providing an example. In

3584-453: The same kind. For Aristotle and Euclid, relations were conceived as whole numbers (Michell, 1993). John Wallis later conceived of ratios of magnitudes as real numbers : When a comparison in terms of ratio is made, the resultant ratio often [namely with the exception of the 'numerical genus' itself] leaves the genus of quantities compared, and passes into the numerical genus, whatever the genus of quantities compared may have been. That is,

3648-404: The sense of establishing (the existence of) a bijection with some well-understood set. For instance, if a set can be brought into bijection with the set of all natural numbers, then it is called " countably infinite ." This kind of counting differs in a fundamental way from counting of finite sets, in that adding new elements to a set does not necessarily increase its size, because the possibility of

3712-426: The set that ranges from 3 to 8, inclusive? The set is counted exclusively, once the range of the set has been made certain by the use of the word "inclusive". The answer is 6; that is 8-3+1, where the +1 range adjustment makes the adjusted exclusive count numerically equivalent to an inclusive count, even though the range of the inclusive count does not include the number eight unit interval. So, it's necessary to discern

3776-468: The size, or extent, of the object or system of which the quantity is a property, whereas magnitudes of an extensive quantity are additive for parts of an entity or subsystems. Thus, magnitude does depend on the extent of the entity or system in the case of extensive quantity. Examples of intensive quantities are density and pressure , while examples of extensive quantities are energy , volume , and mass . In human languages, including English , number

3840-402: The space in question. In quantum mechanics , observables manifest as self-adjoint operators on a separable complex Hilbert space representing the quantum state space . Observables assign values to outcomes of particular measurements , corresponding to the eigenvalue of the operator. If these outcomes represent physically allowable states (i.e. those that belong to the Hilbert space)

3904-405: The system to various electromagnetic fields and eventually reading a value. Physically meaningful observables must also satisfy transformation laws that relate observations performed by different observers in different frames of reference . These transformation laws are automorphisms of the state space , that is bijective transformations that preserve certain mathematical properties of

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3968-424: The types of numbers and their relations to each other as numerical ratios. In mathematics, the concept of quantity is an ancient one extending back to the time of Aristotle and earlier. Aristotle regarded quantity as a fundamental ontological and scientific category. In Aristotle's ontology , quantity or quantum was classified into two different types, which he characterized as follows: Quantum means that which

4032-511: Was known to humans as far back as 44,000 BCE. The development of counting led to the development of mathematical notation , numeral systems , and writing . Verbal counting involves speaking sequential numbers aloud or mentally to track progress. Generally such counting is done with base 10 numbers: "1, 2, 3, 4", etc. Verbal counting is often used for objects that are currently present rather than for counting things over time, since following an interruption counting must resume from where it

4096-490: Was left off, a number that has to be recorded or remembered. Counting a small set of objects, especially over time, can be accomplished efficiently with tally marks : making a mark for each number and then counting all of the marks when done tallying. Tallying is base 1 counting. Finger counting is convenient and common for small numbers. Children count on fingers to facilitate tallying and for performing simple mathematical operations. Older finger counting methods used

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