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99-400: Area is the measure of a region 's size on a surface . The area of a plane region or plane area refers to the area of a shape or planar lamina , while surface area refers to the area of an open surface or the boundary of a three-dimensional object . Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or

198-451: A σ {\displaystyle \sigma } -algebra over X . {\displaystyle X.} A set function μ {\displaystyle \mu } from Σ {\displaystyle \Sigma } to the extended real number line is called a measure if the following conditions hold: If at least one set E {\displaystyle E} has finite measure, then

297-399: A definite integral : The formula for the area enclosed by an ellipse is related to the formula of a circle; for an ellipse with semi-major and semi-minor axes x and y the formula is: Most basic formulas for surface area can be obtained by cutting surfaces and flattening them out (see: developable surfaces ). For example, if the side surface of a cylinder (or any prism )

396-429: A greatest element μ sf . {\displaystyle \mu _{\text{sf}}.} We say the semifinite part of μ {\displaystyle \mu } to mean the semifinite measure μ sf {\displaystyle \mu _{\text{sf}}} defined in the above theorem. We give some nice, explicit formulas, which some authors may take as definition, for

495-514: A least measure μ 0 − ∞ . {\displaystyle \mu _{0-\infty }.} Also, we have μ = μ sf + μ 0 − ∞ . {\displaystyle \mu =\mu _{\text{sf}}+\mu _{0-\infty }.} We say the 0 − ∞ {\displaystyle \mathbf {0-\infty } } part of μ {\displaystyle \mu } to mean

594-404: A square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power  2 , and is denoted by a superscript 2; for instance, the square of 3 may be written as 3 , which is the number 9. In some cases when superscripts are not available, as for instance in programming languages or plain text files,

693-466: A complex number is always a nonnegative real number, that is zero if and only if the complex number is zero. It is easier to compute than the absolute value (no square root), and is a smooth real-valued function . Because of these two properties, the absolute square is often preferred to the absolute value for explicit computations and when methods of mathematical analysis are involved (for example optimization or integration ). For complex vectors ,

792-409: A corresponding unit of area, namely the area of a square with the given side length. Thus areas can be measured in square metres (m), square centimetres (cm), square millimetres (mm), square kilometres (km), square feet (ft), square yards (yd), square miles (mi), and so forth. Algebraically, these units can be thought of as the squares of the corresponding length units. The SI unit of area

891-548: A countable union of measurable sets of finite measure. Analogously, a set in a measure space is said to have a σ-finite measure if it is a countable union of sets with finite measure. For example, the real numbers with the standard Lebesgue measure are σ-finite but not finite. Consider the closed intervals [ k , k + 1 ] {\displaystyle [k,k+1]} for all integers k ; {\displaystyle k;} there are countably many such intervals, each has measure 1, and their union

990-512: A field F with involution. The square function z is the "norm" of the composition algebra C {\displaystyle \mathbb {C} } , where the identity function forms a trivial involution to begin the Cayley–Dickson constructions leading to bicomplex, biquaternion, and bioctonion composition algebras. On complex numbers , the square function z → z 2 {\displaystyle z\to z^{2}}

1089-411: A formula in which the variables that are quantified by ∀ or ∃ represent elements, not sets), is true for every real closed field, and conversely every property of the first-order logic, which is true for a specific real closed field is also true for the real numbers. There are several major uses of the square function in geometry. The name of the square function shows its importance in the definition of

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1188-485: A measure except that instead of requiring countable additivity we require only finite additivity. Historically, this definition was used first. It turns out that in general, finitely additive measures are connected with notions such as Banach limits , the dual of L ∞ {\displaystyle L^{\infty }} and the Stone–Čech compactification . All these are linked in one way or another to

1287-655: A measure on A . {\displaystyle {\cal {A}}.} We say μ {\displaystyle \mu } is semifinite to mean that for all A ∈ μ pre { + ∞ } , {\displaystyle A\in \mu ^{\text{pre}}\{+\infty \},} P ( A ) ∩ μ pre ( R > 0 ) ≠ ∅ . {\displaystyle {\cal {P}}(A)\cap \mu ^{\text{pre}}(\mathbb {R} _{>0})\neq \emptyset .} Semifinite measures generalize sigma-finite measures, in such

1386-1556: A measure. If E 1 {\displaystyle E_{1}} and E 2 {\displaystyle E_{2}} are measurable sets with E 1 ⊆ E 2 {\displaystyle E_{1}\subseteq E_{2}} then μ ( E 1 ) ≤ μ ( E 2 ) . {\displaystyle \mu (E_{1})\leq \mu (E_{2}).} For any countable sequence E 1 , E 2 , E 3 , … {\displaystyle E_{1},E_{2},E_{3},\ldots } of (not necessarily disjoint) measurable sets E n {\displaystyle E_{n}} in Σ : {\displaystyle \Sigma :} μ ( ⋃ i = 1 ∞ E i ) ≤ ∑ i = 1 ∞ μ ( E i ) . {\displaystyle \mu \left(\bigcup _{i=1}^{\infty }E_{i}\right)\leq \sum _{i=1}^{\infty }\mu (E_{i}).} If E 1 , E 2 , E 3 , … {\displaystyle E_{1},E_{2},E_{3},\ldots } are measurable sets that are increasing (meaning that E 1 ⊆ E 2 ⊆ E 3 ⊆ … {\displaystyle E_{1}\subseteq E_{2}\subseteq E_{3}\subseteq \ldots } ) then

1485-1701: A monotonically non-decreasing sequence converging to t . {\displaystyle t.} The monotonically non-increasing sequences { x ∈ X : f ( x ) > t n } {\displaystyle \{x\in X:f(x)>;t_{n}\}} of members of Σ {\displaystyle \Sigma } has at least one finitely μ {\displaystyle \mu } -measurable component, and { x ∈ X : f ( x ) ≥ t } = ⋂ n { x ∈ X : f ( x ) > t n } . {\displaystyle \{x\in X:f(x)\geq t\}=\bigcap _{n}\{x\in X:f(x)>t_{n}\}.} Continuity from above guarantees that μ { x ∈ X : f ( x ) ≥ t } = lim t n ↑ t μ { x ∈ X : f ( x ) > t n } . {\displaystyle \mu \{x\in X:f(x)\geq t\}=\lim _{t_{n}\uparrow t}\mu \{x\in X:f(x)>t_{n}\}.} The right-hand side lim t n ↑ t F ( t n ) {\displaystyle \lim _{t_{n}\uparrow t}F\left(t_{n}\right)} then equals F ( t ) = μ { x ∈ X : f ( x ) > t } {\displaystyle F(t)=\mu \{x\in X:f(x)>t\}} if t {\displaystyle t}

1584-941: A null set. One defines μ ( Y ) {\displaystyle \mu (Y)} to equal μ ( X ) . {\displaystyle \mu (X).} If f : X → [ 0 , + ∞ ] {\displaystyle f:X\to [0,+\infty ]} is ( Σ , B ( [ 0 , + ∞ ] ) ) {\displaystyle (\Sigma ,{\cal {B}}([0,+\infty ]))} -measurable, then μ { x ∈ X : f ( x ) ≥ t } = μ { x ∈ X : f ( x ) > t } {\displaystyle \mu \{x\in X:f(x)\geq t\}=\mu \{x\in X:f(x)>t\}} for almost all t ∈ [ − ∞ , ∞ ] . {\displaystyle t\in [-\infty ,\infty ].} This property

1683-425: A rectangle with length l and width w , the formula for the area is: That is, the area of the rectangle is the length multiplied by the width. As a special case, as l = w in the case of a square, the area of a square with side length s is given by the formula: The formula for the area of a rectangle follows directly from the basic properties of area, and is sometimes taken as a definition or axiom . On

1782-409: A ring that is equal to its own square is called an idempotent . In any ring, 0 and 1 are idempotents. There are no other idempotents in fields and more generally in integral domains . However, the ring of the integers modulo   n has 2 idempotents, where k is the number of distinct prime factors of  n . A commutative ring in which every element is equal to its square (every element

1881-948: A sense, semifinite once its 0 − ∞ {\displaystyle 0-\infty } part (the wild part) is taken away. Theorem (Luther decomposition)  —  For any measure μ {\displaystyle \mu } on A , {\displaystyle {\cal {A}},} there exists a 0 − ∞ {\displaystyle 0-\infty } measure ξ {\displaystyle \xi } on A {\displaystyle {\cal {A}}} such that μ = ν + ξ {\displaystyle \mu =\nu +\xi } for some semifinite measure ν {\displaystyle \nu } on A . {\displaystyle {\cal {A}}.} In fact, among such measures ξ , {\displaystyle \xi ,} there exists

1980-474: A shape can be measured by comparing the shape to squares of a fixed size. In the International System of Units (SI), the standard unit of area is the square metre (written as m), which is the area of a square whose sides are one metre long. A shape with an area of three square metres would have the same area as three such squares. In mathematics , the unit square is defined to have area one, and

2079-449: A special case of semifinite measures and a generalization of sigma-finite measures. Let X {\displaystyle X} be a set, let A {\displaystyle {\cal {A}}} be a sigma-algebra on X , {\displaystyle X,} and let μ {\displaystyle \mu } be a measure on A . {\displaystyle {\cal {A}}.} A measure

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2178-458: A sphere was first obtained by Archimedes in his work On the Sphere and Cylinder . The formula is: where r is the radius of the sphere. As with the formula for the area of a circle, any derivation of this formula inherently uses methods similar to calculus . Measure (mathematics) The intuition behind this concept dates back to ancient Greece , when Archimedes tried to calculate

2277-449: A square, is not considered to be a quadratic residue. Every finite field of this type has exactly ( p − 1)/2 quadratic residues and exactly ( p − 1)/2 quadratic non-residues. The quadratic residues form a group under multiplication. The properties of quadratic residues are widely used in number theory . More generally, in rings, the square function may have different properties that are sometimes used to classify rings. Zero may be

2376-437: A way that some big theorems of measure theory that hold for sigma-finite but not arbitrary measures can be extended with little modification to hold for semifinite measures. (To-do: add examples of such theorems; cf. the talk page.) The zero measure is sigma-finite and thus semifinite. In addition, the zero measure is clearly less than or equal to μ . {\displaystyle \mu .} It can be shown there

2475-418: Is a commutative semigroup , then one has In the language of quadratic forms , this equality says that the square function is a "form permitting composition". In fact, the square function is the foundation upon which other quadratic forms are constructed which also permit composition. The procedure was introduced by L. E. Dickson to produce the octonions out of quaternions by doubling. The doubling method

2574-438: Is a greatest measure with these two properties: Theorem (semifinite part)  —  For any measure μ {\displaystyle \mu } on A , {\displaystyle {\cal {A}},} there exists, among semifinite measures on A {\displaystyle {\cal {A}}} that are less than or equal to μ , {\displaystyle \mu ,}

2673-488: Is a measure space with a probability measure. For measure spaces that are also topological spaces various compatibility conditions can be placed for the measure and the topology. Most measures met in practice in analysis (and in many cases also in probability theory ) are Radon measures . Radon measures have an alternative definition in terms of linear functionals on the locally convex topological vector space of continuous functions with compact support . This approach

2772-1036: Is a point of continuity of F . {\displaystyle F.} Since F {\displaystyle F} is continuous almost everywhere, this completes the proof. Measures are required to be countably additive. However, the condition can be strengthened as follows. For any set I {\displaystyle I} and any set of nonnegative r i , i ∈ I {\displaystyle r_{i},i\in I} define: ∑ i ∈ I r i = sup { ∑ i ∈ J r i : | J | < ∞ , J ⊆ I } . {\displaystyle \sum _{i\in I}r_{i}=\sup \left\lbrace \sum _{i\in J}r_{i}:|J|<;\infty ,J\subseteq I\right\rbrace .} That is, we define

2871-410: Is a square" has been generalized to the notion of a real closed field , which is an ordered field such that every non-negative element is a square and every polynomial of odd degree has a root. The real closed fields cannot be distinguished from the field of real numbers by their algebraic properties: every property of the real numbers, which may be expressed in first-order logic (that is expressed by

2970-420: Is a twofold cover in the sense that each non-zero complex number has exactly two square roots. The square of the absolute value of a complex number is called its absolute square , squared modulus , or squared magnitude . It is the product of the complex number with its complex conjugate , and equals the sum of the squares of the real and imaginary parts of the complex number. The absolute square of

3069-466: Is a unique t 0 ∈ { − ∞ } ∪ [ 0 , + ∞ ) {\displaystyle t_{0}\in \{-\infty \}\cup [0,+\infty )} such that F {\displaystyle F} is infinite to the left of t {\displaystyle t} (which can only happen when t 0 ≥ 0 {\displaystyle t_{0}\geq 0} ) and finite to

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3168-427: Is an example of a quadratic form. It demonstrates a quadratic relation of the moment of inertia to the size ( length ). There are infinitely many Pythagorean triples , sets of three positive integers such that the sum of the squares of the first two equals the square of the third. Each of these triples gives the integer sides of a right triangle. The square function is defined in any field or ring . An element in

3267-445: Is approximately triangular in shape, and the sectors can be rearranged to form an approximate parallelogram. The height of this parallelogram is r , and the width is half the circumference of the circle, or π r . Thus, the total area of the circle is π r : Though the dissection used in this formula is only approximate, the error becomes smaller and smaller as the circle is partitioned into more and more sectors. The limit of

3366-474: Is called a measurable space , and the members of Σ {\displaystyle \Sigma } are called measurable sets . A triple ( X , Σ , μ ) {\displaystyle (X,\Sigma ,\mu )} is called a measure space . A probability measure is a measure with total measure one – that is, μ ( X ) = 1. {\displaystyle \mu (X)=1.} A probability space

3465-418: Is cut lengthwise, the surface can be flattened out into a rectangle. Similarly, if a cut is made along the side of a cone , the side surface can be flattened out into a sector of a circle, and the resulting area computed. The formula for the surface area of a sphere is more difficult to derive: because a sphere has nonzero Gaussian curvature , it cannot be flattened out. The formula for the surface area of

3564-519: Is equivalent to the statement that the ideal of null sets is κ {\displaystyle \kappa } -complete. A measure space ( X , Σ , μ ) {\displaystyle (X,\Sigma ,\mu )} is called finite if μ ( X ) {\displaystyle \mu (X)} is a finite real number (rather than ∞ {\displaystyle \infty } ). Nonzero finite measures are analogous to probability measures in

3663-465: Is false without the assumption that at least one of the E n {\displaystyle E_{n}} has finite measure. For instance, for each n ∈ N , {\displaystyle n\in \mathbb {N} ,} let E n = [ n , ∞ ) ⊆ R , {\displaystyle E_{n}=[n,\infty )\subseteq \mathbb {R} ,} which all have infinite Lebesgue measure, but

3762-432: Is idempotent) is called a Boolean ring ; an example from computer science is the ring whose elements are binary numbers , with bitwise AND as the multiplication operation and bitwise XOR as the addition operation. In a totally ordered ring , x ≥ 0 for any x . Moreover, x = 0  if and only if  x = 0 . In a supercommutative algebra where 2 is invertible, the square of any odd element equals zero. If A

3861-460: Is known as Heron's formula for the area of a triangle in terms of its sides, and a proof can be found in his book, Metrica , written around 60 CE. It has been suggested that Archimedes knew the formula over two centuries earlier, and since Metrica is a collection of the mathematical knowledge available in the ancient world, it is possible that the formula predates the reference given in that work. In 300 BCE Greek mathematician Euclid proved that

3960-414: Is measurable. A measure can be extended to a complete one by considering the σ-algebra of subsets Y {\displaystyle Y} which differ by a negligible set from a measurable set X , {\displaystyle X,} that is, such that the symmetric difference of X {\displaystyle X} and Y {\displaystyle Y} is contained in

4059-523: Is necessarily of finite variation , hence complex measures include finite signed measures but not, for example, the Lebesgue measure . Measures that take values in Banach spaces have been studied extensively. A measure that takes values in the set of self-adjoint projections on a Hilbert space is called a projection-valued measure ; these are used in functional analysis for the spectral theorem . When it

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4158-406: Is necessary to distinguish the usual measures which take non-negative values from generalizations, the term positive measure is used. Positive measures are closed under conical combination but not general linear combination , while signed measures are the linear closure of positive measures. Another generalization is the finitely additive measure , also known as a content . This is the same as

4257-427: Is related to the definition of determinants in linear algebra , and is a basic property of surfaces in differential geometry . In analysis , the area of a subset of the plane is defined using Lebesgue measure , though not every subset is measurable if one supposes the axiom of choice. In general, area in higher mathematics is seen as a special case of volume for two-dimensional regions. Area can be defined through

4356-462: Is said to be s-finite if it is a countable sum of finite measures. S-finite measures are more general than sigma-finite ones and have applications in the theory of stochastic processes . If the axiom of choice is assumed to be true, it can be proved that not all subsets of Euclidean space are Lebesgue measurable ; examples of such sets include the Vitali set , and the non-measurable sets postulated by

4455-1000: Is semifinite then μ = μ sf . {\displaystyle \mu =\mu _{\text{sf}}.} Every 0 − ∞ {\displaystyle 0-\infty } measure that is not the zero measure is not semifinite. (Here, we say 0 − ∞ {\displaystyle 0-\infty } measure to mean a measure whose range lies in { 0 , + ∞ } {\displaystyle \{0,+\infty \}} : ( ∀ A ∈ A ) ( μ ( A ) ∈ { 0 , + ∞ } ) . {\displaystyle (\forall A\in {\cal {A}})(\mu (A)\in \{0,+\infty \}).} ) Below we give examples of 0 − ∞ {\displaystyle 0-\infty } measures that are not zero measures. Measures that are not semifinite are very wild when restricted to certain sets. Every measure is, in

4554-558: Is spatial distribution of mass (see for example, gravity potential ), or another non-negative extensive property , conserved (see conservation law for a list of these) or not. Negative values lead to signed measures, see "generalizations" below. Measure theory is used in machine learning. One example is the Flow Induced Probability Measure in GFlowNet. Let μ {\displaystyle \mu } be

4653-820: Is such that μ { x ∈ X : f ( x ) > t } = + ∞ {\displaystyle \mu \{x\in X:f(x)>t\}=+\infty } then monotonicity implies μ { x ∈ X : f ( x ) ≥ t } = + ∞ , {\displaystyle \mu \{x\in X:f(x)\geq t\}=+\infty ,} so that F ( t ) = G ( t ) , {\displaystyle F(t)=G(t),} as required. If μ { x ∈ X : f ( x ) > t } = + ∞ {\displaystyle \mu \{x\in X:f(x)>t\}=+\infty } for all t {\displaystyle t} then we are done, so assume otherwise. Then there

4752-399: Is taken by Bourbaki (2004) and a number of other sources. For more details, see the article on Radon measures . Some important measures are listed here. Other 'named' measures used in various theories include: Borel measure , Jordan measure , ergodic measure , Gaussian measure , Baire measure , Radon measure , Young measure , and Loeb measure . In physics an example of a measure

4851-471: Is the entire real line. Alternatively, consider the real numbers with the counting measure , which assigns to each finite set of reals the number of points in the set. This measure space is not σ-finite, because every set with finite measure contains only finitely many points, and it would take uncountably many such sets to cover the entire real line. The σ-finite measure spaces have some very convenient properties; σ-finiteness can be compared in this respect to

4950-432: Is the original number. No square root can be taken of a negative number within the system of real numbers , because squares of all real numbers are non-negative . The lack of real square roots for the negative numbers can be used to expand the real number system to the complex numbers , by postulating the imaginary unit i , which is one of the square roots of −1. The property "every non-negative real number

5049-467: Is the square metre, which is considered an SI derived unit . Calculation of the area of a square whose length and width are 1 metre would be: 1 metre × 1 metre = 1 m and so, a rectangle with different sides (say length of 3 metres and width of 2 metres) would have an area in square units that can be calculated as: 3 metres × 2 metres = 6 m. This is equivalent to 6 million square millimetres. Other useful conversions are: In non-metric units,

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5148-644: Is used in connection with Lebesgue integral . Both F ( t ) := μ { x ∈ X : f ( x ) > t } {\displaystyle F(t):=\mu \{x\in X:f(x)>t\}} and G ( t ) := μ { x ∈ X : f ( x ) ≥ t } {\displaystyle G(t):=\mu \{x\in X:f(x)\geq t\}} are monotonically non-increasing functions of t , {\displaystyle t,} so both of them have at most countably many discontinuities and thus they are continuous almost everywhere, relative to

5247-454: The Cartesian coordinates ( x i , y i ) {\displaystyle (x_{i},y_{i})} ( i =0, 1, ..., n -1) of whose n vertices are known, the area is given by the surveyor's formula : where when i = n -1, then i +1 is expressed as modulus n and so refers to 0. The most basic area formula is the formula for the area of a rectangle . Given

5346-544: The Hausdorff paradox and the Banach–Tarski paradox . For certain purposes, it is useful to have a "measure" whose values are not restricted to the non-negative reals or infinity. For instance, a countably additive set function with values in the (signed) real numbers is called a signed measure , while such a function with values in the complex numbers is called a complex measure . Observe, however, that complex measure

5445-487: The Lindelöf property of topological spaces. They can be also thought of as a vague generalization of the idea that a measure space may have 'uncountable measure'. Let X {\displaystyle X} be a set, let A {\displaystyle {\cal {A}}} be a sigma-algebra on X , {\displaystyle X,} and let μ {\displaystyle \mu } be

5544-426: The area : it comes from the fact that the area of a square with sides of length   l is equal to l . The area depends quadratically on the size: the area of a shape n  times larger is n  times greater. This holds for areas in three dimensions as well as in the plane: for instance, the surface area of a sphere is proportional to the square of its radius, a fact that is manifested physically by

5643-506: The area of a circle . But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel , Henri Lebesgue , Nikolai Luzin , Johann Radon , Constantin Carathéodory , and Maurice Fréchet , among others. Let X {\displaystyle X} be a set and Σ {\displaystyle \Sigma }

5742-431: The axiom of choice . Contents remain useful in certain technical problems in geometric measure theory ; this is the theory of Banach measures . A charge is a generalization in both directions: it is a finitely additive, signed measure. (Cf. ba space for information about bounded charges, where we say a charge is bounded to mean its range its a bounded subset of R .) Square (algebra) In mathematics ,

5841-475: The hectare is still commonly used to measure land: Other uncommon metric units of area include the tetrad , the hectad , and the myriad . The acre is also commonly used to measure land areas, where An acre is approximately 40% of a hectare. On the atomic scale, area is measured in units of barns , such that: The barn is commonly used in describing the cross-sectional area of interaction in nuclear physics . In South Asia (mainly Indians), although

5940-675: The intersection of the sets E n {\displaystyle E_{n}} is measurable; furthermore, if at least one of the E n {\displaystyle E_{n}} has finite measure then μ ( ⋂ i = 1 ∞ E i ) = lim i → ∞ μ ( E i ) = inf i ≥ 1 μ ( E i ) . {\displaystyle \mu \left(\bigcap _{i=1}^{\infty }E_{i}\right)=\lim _{i\to \infty }\mu (E_{i})=\inf _{i\geq 1}\mu (E_{i}).} This property

6039-453: The inverse-square law describing how the strength of physical forces such as gravity varies according to distance. The square function is related to distance through the Pythagorean theorem and its generalization, the parallelogram law . Euclidean distance is not a smooth function : the three-dimensional graph of distance from a fixed point forms a cone , with a non-smooth point at

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6138-406: The standard deviation of a set of values, or a random variable . The deviation of each value  x i from the mean   x ¯ {\displaystyle {\overline {x}}} of the set is defined as the difference x i − x ¯ {\displaystyle x_{i}-{\overline {x}}} . These deviations are squared, then

6237-417: The surveyor's formula for the area of any polygon with known vertex locations by Gauss in the 19th century. The development of integral calculus in the late 17th century provided tools that could subsequently be used for computing more complicated areas, such as the area of an ellipse and the surface areas of various curved three-dimensional objects. For a non-self-intersecting ( simple ) polygon,

6336-964: The union of the sets E n {\displaystyle E_{n}} is measurable and μ ( ⋃ i = 1 ∞ E i )   =   lim i → ∞ μ ( E i ) = sup i ≥ 1 μ ( E i ) . {\displaystyle \mu \left(\bigcup _{i=1}^{\infty }E_{i}\right)~=~\lim _{i\to \infty }\mu (E_{i})=\sup _{i\geq 1}\mu (E_{i}).} If E 1 , E 2 , E 3 , … {\displaystyle E_{1},E_{2},E_{3},\ldots } are measurable sets that are decreasing (meaning that E 1 ⊇ E 2 ⊇ E 3 ⊇ … {\displaystyle E_{1}\supseteq E_{2}\supseteq E_{3}\supseteq \ldots } ) then

6435-554: The 5th century BCE, Hippocrates of Chios was the first to show that the area of a disk (the region enclosed by a circle) is proportional to the square of its diameter, as part of his quadrature of the lune of Hippocrates , but did not identify the constant of proportionality . Eudoxus of Cnidus , also in the 5th century BCE, also found that the area of a disk is proportional to its radius squared. Subsequently, Book I of Euclid's Elements dealt with equality of areas between two-dimensional figures. The mathematician Archimedes used

6534-561: The Lebesgue measure. If t < 0 {\displaystyle t<0} then { x ∈ X : f ( x ) ≥ t } = X = { x ∈ X : f ( x ) > t } , {\displaystyle \{x\in X:f(x)\geq t\}=X=\{x\in X:f(x)>t\},} so that F ( t ) = G ( t ) , {\displaystyle F(t)=G(t),} as desired. If t {\displaystyle t}

6633-406: The amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analogue of the length of a curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept). Two different regions may have the same area (as in squaring the circle ); by synecdoche , "area" sometimes is used to refer to the region, as in a " polygonal area ". The area of

6732-466: The area of a cyclic quadrilateral (a quadrilateral inscribed in a circle) in terms of its sides. In 1842, the German mathematicians Carl Anton Bretschneider and Karl Georg Christian von Staudt independently found a formula, known as Bretschneider's formula , for the area of any quadrilateral. The development of Cartesian coordinates by René Descartes in the 17th century allowed the development of

6831-499: The area of a triangle is half that of a parallelogram with the same base and height in his book Elements of Geometry . In 499 Aryabhata , a great mathematician - astronomer from the classical age of Indian mathematics and Indian astronomy , expressed the area of a triangle as one-half the base times the height in the Aryabhatiya . In the 7th century CE, Brahmagupta developed a formula, now known as Brahmagupta's formula , for

6930-423: The area of any other shape or surface is a dimensionless real number . There are several well-known formulas for the areas of simple shapes such as triangles , rectangles , and circles . Using these formulas, the area of any polygon can be found by dividing the polygon into triangles . For shapes with curved boundary, calculus is usually required to compute the area. Indeed, the problem of determining

7029-531: The area of plane figures was a major motivation for the historical development of calculus . For a solid shape such as a sphere , cone, or cylinder, the area of its boundary surface is called the surface area . Formulas for the surface areas of simple shapes were computed by the ancient Greeks , but computing the surface area of a more complicated shape usually requires multivariable calculus . Area plays an important role in modern mathematics. In addition to its obvious importance in geometry and calculus, area

7128-415: The areas of the approximate parallelograms is exactly π r , which is the area of the circle. This argument is actually a simple application of the ideas of calculus . In ancient times, the method of exhaustion was used in a similar way to find the area of the circle, and this method is now recognized as a precursor to integral calculus . Using modern methods, the area of a circle can be computed using

7227-406: The condition of non-negativity is dropped, and μ {\displaystyle \mu } takes on at most one of the values of ± ∞ , {\displaystyle \pm \infty ,} then μ {\displaystyle \mu } is called a signed measure . The pair ( X , Σ ) {\displaystyle (X,\Sigma )}

7326-416: The conversion between two square units is the square of the conversion between the corresponding length units. the relationship between square feet and square inches is where 144 = 12 = 12 × 12. Similarly: In addition, conversion factors include: There are several other common units for area. The are was the original unit of area in the metric system , with: Though the are has fallen out of use,

7425-516: The countries use SI units as official, many South Asians still use traditional units. Each administrative division has its own area unit, some of them have same names, but with different values. There's no official consensus about the traditional units values. Thus, the conversions between the SI units and the traditional units may have different results, depending on what reference that has been used. Some traditional South Asian units that have fixed value: In

7524-447: The dot product can be defined involving the conjugate transpose , leading to the squared norm . Squares are ubiquitous in algebra, more generally, in almost every branch of mathematics, and also in physics where many units are defined using squares and inverse squares: see below . Least squares is the standard method used with overdetermined systems . Squaring is used in statistics and probability theory in determining

7623-637: The following hold: ⋃ α ∈ λ X α ∈ Σ {\displaystyle \bigcup _{\alpha \in \lambda }X_{\alpha }\in \Sigma } μ ( ⋃ α ∈ λ X α ) = ∑ α ∈ λ μ ( X α ) . {\displaystyle \mu \left(\bigcup _{\alpha \in \lambda }X_{\alpha }\right)=\sum _{\alpha \in \lambda }\mu \left(X_{\alpha }\right).} The second condition

7722-417: The image of this function is called a square , and the inverse images of a square are called square roots . The notion of squaring is particularly important in the finite fields Z / p Z formed by the numbers modulo an odd prime number p . A non-zero element of this field is called a quadratic residue if it is a square in Z / p Z , and otherwise, it is called a quadratic non-residue. Zero, while

7821-422: The intersection is empty. A measurable set X {\displaystyle X} is called a null set if μ ( X ) = 0. {\displaystyle \mu (X)=0.} A subset of a null set is called a negligible set . A negligible set need not be measurable, but every measurable negligible set is automatically a null set. A measure is called complete if every negligible set

7920-433: The interval [0, +∞) . On the negative numbers, numbers with greater absolute value have greater squares, so the square is a monotonically decreasing function on (−∞,0] . Hence, zero is the (global) minimum of the square function. The square x of a number x is less than x (that is x < x ) if and only if 0 < x < 1 , that is, if x belongs to the open interval (0,1) . This implies that

8019-405: The left. If the triangle is moved to the other side of the trapezoid, then the resulting figure is a rectangle. It follows that the area of the parallelogram is the same as the area of the rectangle: However, the same parallelogram can also be cut along a diagonal into two congruent triangles, as shown in the figure to the right. It follows that the area of each triangle is half the area of

8118-864: The measure μ 0 − ∞ {\displaystyle \mu _{0-\infty }} defined in the above theorem. Here is an explicit formula for μ 0 − ∞ {\displaystyle \mu _{0-\infty }} : μ 0 − ∞ = ( sup { μ ( B ) − μ sf ( B ) : B ∈ P ( A ) ∩ μ sf pre ( R ≥ 0 ) } ) A ∈ A . {\displaystyle \mu _{0-\infty }=(\sup\{\mu (B)-\mu _{\text{sf}}(B):B\in {\cal {P}}(A)\cap \mu _{\text{sf}}^{\text{pre}}(\mathbb {R} _{\geq 0})\})_{A\in {\cal {A}}}.} Localizable measures are

8217-416: The notations x ^2 ( caret ) or x **2 may be used in place of x . The adjective which corresponds to squaring is quadratic . The square of an integer may also be called a square number or a perfect square . In algebra , the operation of squaring is often generalized to polynomials , other expressions , or values in systems of mathematical values other than the numbers. For instance,

8316-432: The other hand, if geometry is developed before arithmetic , this formula can be used to define multiplication of real numbers . Most other simple formulas for area follow from the method of dissection . This involves cutting a shape into pieces, whose areas must sum to the area of the original shape. For an example, any parallelogram can be subdivided into a trapezoid and a right triangle , as shown in figure to

8415-418: The parallelogram: Similar arguments can be used to find area formulas for the trapezoid as well as more complicated polygons . The formula for the area of a circle (more properly called the area enclosed by a circle or the area of a disk ) is based on a similar method. Given a circle of radius r , it is possible to partition the circle into sectors , as shown in the figure to the right. Each sector

8514-563: The requirement μ ( ∅ ) = 0 {\displaystyle \mu (\varnothing )=0} is met automatically due to countable additivity: μ ( E ) = μ ( E ∪ ∅ ) = μ ( E ) + μ ( ∅ ) , {\displaystyle \mu (E)=\mu (E\cup \varnothing )=\mu (E)+\mu (\varnothing ),} and therefore μ ( ∅ ) = 0. {\displaystyle \mu (\varnothing )=0.} If

8613-872: The right. Arguing as above, μ { x ∈ X : f ( x ) ≥ t } = + ∞ {\displaystyle \mu \{x\in X:f(x)\geq t\}=+\infty } when t < t 0 . {\displaystyle t<t_{0}.} Similarly, if t 0 ≥ 0 {\displaystyle t_{0}\geq 0} and F ( t 0 ) = + ∞ {\displaystyle F\left(t_{0}\right)=+\infty } then F ( t 0 ) = G ( t 0 ) . {\displaystyle F\left(t_{0}\right)=G\left(t_{0}\right).} For t > t 0 , {\displaystyle t>t_{0},} let t n {\displaystyle t_{n}} be

8712-404: The semifinite part: Since μ sf {\displaystyle \mu _{\text{sf}}} is semifinite, it follows that if μ = μ sf {\displaystyle \mu =\mu _{\text{sf}}} then μ {\displaystyle \mu } is semifinite. It is also evident that if μ {\displaystyle \mu }

8811-421: The sense that any finite measure μ {\displaystyle \mu } is proportional to the probability measure 1 μ ( X ) μ . {\displaystyle {\frac {1}{\mu (X)}}\mu .} A measure μ {\displaystyle \mu } is called σ-finite if X {\displaystyle X} can be decomposed into

8910-409: The square function is an even function . The squaring operation defines a real function called the square function or the squaring function . Its domain is the whole real line , and its image is the set of nonnegative real numbers. The square function preserves the order of positive numbers: larger numbers have larger squares. In other words, the square is a monotonic function on

9009-419: The square of an integer is never less than the original number x . Every positive real number is the square of exactly two numbers, one of which is strictly positive and the other of which is strictly negative. Zero is the square of only one number, itself. For this reason, it is possible to define the square root function, which associates with a non-negative real number the non-negative number whose square

9108-503: The square of some non-zero elements. A commutative ring such that the square of a non zero element is never zero is called a reduced ring . More generally, in a commutative ring, a radical ideal is an ideal  I such that x 2 ∈ I {\displaystyle x^{2}\in I} implies x ∈ I {\displaystyle x\in I} . Both notions are important in algebraic geometry , because of Hilbert's Nullstellensatz . An element of

9207-443: The square of the linear polynomial x + 1 is the quadratic polynomial ( x + 1) = x + 2 x + 1 . One of the important properties of squaring, for numbers as well as in many other mathematical systems, is that (for all numbers x ), the square of x is the same as the square of its additive inverse − x . That is, the square function satisfies the identity x = (− x ) . This can also be expressed by saying that

9306-606: The sum of the r i {\displaystyle r_{i}} to be the supremum of all the sums of finitely many of them. A measure μ {\displaystyle \mu } on Σ {\displaystyle \Sigma } is κ {\displaystyle \kappa } -additive if for any λ < κ {\displaystyle \lambda <\kappa } and any family of disjoint sets X α , α < λ {\displaystyle X_{\alpha },\alpha <\lambda }

9405-404: The tip of the cone. However, the square of the distance (denoted d or r ), which has a paraboloid as its graph, is a smooth and analytic function . The dot product of a Euclidean vector with itself is equal to the square of its length: v ⋅ v = v . This is further generalised to quadratic forms in linear spaces via the inner product . The inertia tensor in mechanics

9504-409: The tools of Euclidean geometry to show that the area inside a circle is equal to that of a right triangle whose base has the length of the circle's circumference and whose height equals the circle's radius, in his book Measurement of a Circle . (The circumference is 2 π r , and the area of a triangle is half the base times the height, yielding the area π r for the disk.) Archimedes approximated

9603-509: The use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers. It can be proved that such a function exists. An approach to defining what is meant by "area" is through axioms . "Area" can be defined as a function from a collection M of a special kinds of plane figures (termed measurable sets) to the set of real numbers, which satisfies the following properties: It can be proved that such an area function actually exists. Every unit of length has

9702-419: The value of π (and hence the area of a unit-radius circle) with his doubling method , in which he inscribed a regular triangle in a circle and noted its area, then doubled the number of sides to give a regular hexagon , then repeatedly doubled the number of sides as the polygon's area got closer and closer to that of the circle (and did the same with circumscribed polygons ). Heron of Alexandria found what

9801-479: Was formalized by A. A. Albert who started with the real number field R {\displaystyle \mathbb {R} } and the square function, doubling it to obtain the complex number field with quadratic form x + y , and then doubling again to obtain quaternions. The doubling procedure is called the Cayley–Dickson construction , and has been generalized to form algebras of dimension 2 over

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