What is the quotient of 35 and 5 in algebraic expression?


The quotient of 35 and 5 is seven . Quotient means divided by .

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In mathematics, a quotient (from Latin: ) is the result of division. For example, when dividing 6 by 3, the quotient is 2, while 6 is called the dividend, and 3 the divisor. The quotient further is expressed as the number of times the divisor divides into the dividend, e.g. 3 divides 2 times into 6. A quotient can also mean just the integer part of the result of dividing two integers. For example, the quotient of 13 and 5 would be 2 while the remainder would be 3. For more, see the Euclidean division. In more abstract branches of mathematics, the word quotient is often used to describe sets, spaces, or algebraic structures whose elements are the equivalence classes of some equivalence relation on another set, space, or algebraic structure. See: The quotient rule is a method for finding derivatives in calculus. Quotients also come up in certain tests, like the IQ or SAT test, which stands for intelligence quotient. In recent decades, as more emphasis has been placed on full personal development, other similar quotients have appeared. These include moral quotient, emotional quotient, adversity quotient, social quotient, creativity quotient, etc.
In arithmetic, the Euclidean division is the conventional process of division of two integers producing a quotient and a remainder. There is a theorem stating that the quotient and remainder exist, are unique, and have certain properties. Integer division algorithms compute the quotient and remainder given two integers, the most well-known such algorithm being long division. The integer division algorithm is an important ingredient for other algorithms, such as the Euclidean algorithm for finding the greatest common divisor of two integers. Suppose that a pie has 9 slices and they are to be divided evenly among 4 people. Using Euclidean division, 9 divided by 4 is 2 with remainder 1. In other words, each person receives 2 slices of pie, and there is 1 slice left over. This can be confirmed using multiplication, the inverse of division: if each of the 4 people received 2 slices, then 4 × 2 = 8 slices were given out in all. Adding the 1 slice remaining, the result is 9 slices. In summary: 9 = 4 × 2 + 1. In general, if the number of slices is denoted a and the number of people is b, one can divide the pie evenly among the people such that each person receives q slices (the quotient) and some number of slices r < b are left over (the remainder). Regardless, the equation a = bq + r holds. If 9 slices were divided among 3 people instead of 4, each would receive 3 and no slices would be left over. In this case the remainder is zero, and it is said that 3 evenly divides 9, or that 3 divides 9. Euclidean division can also be extended to negative integers using the same formula; for example −9 = 4 × (−3) + 3, so −9 divided by 4 is −3 with remainder 3. The remainder is the only one of the four numbers that can never be negative. Given two integers a and b, with b ≠ 0, there exist unique integers q and r such that a = bq + r and 0 ≤ r < |b|, where |b| denotes the absolute value of b. The four integers that appear in this theorem have been given a name: a is called the dividend, b is called the divisor, q is called the quotient and r is called the remainder. The computation of the quotient and the remainder from the dividend and the divisor is called division or, in case of ambiguity, Euclidean division. The theorem is frequently referred to as the division algorithm, although it is a theorem and not an algorithm, because its proof as given below also provides a simple division algorithm for computing q and r. Division is not defined in the case where b = 0; see division by zero. The proof consists of two parts — first, the proof of the existence of q and r, and second, the proof of the uniqueness of q and r. Consider first the case b < 0. Setting b' = −b and q' = −q, the equation a = bq + r may be rewritten a = b'q' + r and the inequality 0 ≤ r < |b| may be rewritten 0 ≤ r < |b'|. This reduces the existence for the case b < 0 to that of the case b > 0. Similarly, if a < 0 and b > 0, setting a' = −a, q' = −q − 1 and r' = b − r, the equation a = bq + r may be rewritten a' = bq' + r' and the inequality 0 ≤ r < b may be rewritten 0 ≤ r' < b. Thus the proof of the existence is reduced to the case a ≥ 0 and b > 0 and we consider only this case in the remainder of the proof. Let q1 and r1, both nonnegative, such that a = bq1 + r1, for example q1 = 0 and r1 = a. If r1 < b, we are done. Otherwise q2 = q1 + 1 and r2 = r1 − b satisfy a = bq2 + r2 and 0 < r2 < r1. Repeating this process one gets eventually q = qk and r = rk such that a = bq + r and 0 ≤ r < b. This proves the existence and also gives a simple division algorithm to compute the quotient and the remainder. However this algorithm needs q steps and is thus not efficient. Suppose there exists q, q' , r, r' with 0 ≤ r, r' < |b| such that a = bq + r and a = bq' + r' . Adding the two inequalities 0 ≤ r < |b| and −|b| < −r' ≤ 0 yields −|b| < r − r' < |b|, that is |r − r' | < |b|. Subtracting the two equations yields: b(q'  − q) = (r − r' ). Thus |b| divides |r − r' |. If |r − r' | ≠ 0 this implies |b| < |r − r' |, contradicting previous inequality. Thus, r = r' and b(q'  − q) = 0. As b ≠ 0, this implies q = q' , proving uniqueness. Some proofs of the algorithm rely on the Well-ordering principle. Usually, an existence proof does not provide an algorithm to compute the existing object, but the above proof provides immediately an algorithm (see Division algorithm#Division_by_repeated_subtraction). However this is not a very efficient method, as it requires as many steps as the size of the quotient. This is related to the fact that it only uses addition, subtraction and comparison of the integers, without involving multiplication, nor any particular representation of the integers, such as decimal notation. In terms of decimal notation, long division provides a much more efficient division algorithm. Its generalization to binary notation allows to use it in a computer. However, for large inputs, algorithms that reduce division to multiplication, like Newton–Raphson one, are usually preferred, because they need a time which is proportional to the time of the multiplication needed to verify the result, independently of the multiplication algorithm which is used. Euclidean domains are defined as integral domains which support the following generalization of Euclidean division: Given an element a and a non-zero element b in a Euclidean domain R equipped with a Euclidean function d, there exist q and r in R such that and either or . Unlike in the integer case, q and r need not be unique. Examples of Euclidean domains include fields, polynomial rings in one variable over a field, and the Gaussian integers. The Euclidean division of polynomials has been the object of specific developments. See Polynomial long division, Polynomial greatest common divisor#Euclidean division and Polynomial greatest common divisor#Pseudo-remainder sequences. The 1st generalized division algorithm: Given integers m, a, d with m>0, there exist unique integers q and r with d \le r < m+d such that a = mq+r. Especially, if  d=- \left\lfloor \frac{m}{2} \right\rfloor then  - \left\lfloor \frac{m}{2} \right\rfloor  \le r < m-\left\lfloor \frac{m}{2} \right\rfloor . In this case, r is called the least absolute remainder. As an application of this generalization, the original Euclidean algorithm for integers can be slightly sped up. The 2nd generalized division algorithm: Given integers m, a with m>0, and let R^{-1} \in \mathbb Z_m ^* be the multiplicative inverse of R \in \mathbb Z_m ^*. Then there exist unique integers q and r with 0 \le r < m such that  a = mq+R^{-1} \cdot r . This result generalizes Hensel's odd division (1900), and its proof can be found in. The value r in the 2nd generalization corresponds to the N-residue defined in Montgomery reduction.
In mathematics, specifically group theory, a quotient group (or factor group) is a group obtained by aggregating similar elements of a larger group using an equivalence relation. For example, the cyclic group of naddition modulo can be obtained from the integers by identifying elements that differ by a multiple of n and defining a group structure that operates on each such class (known as a congruence class) as a single entity. In a quotient of a group, the equivalence class of the identity element is always a normal subgroup of the original group, and the other equivalence classes are the cosets of this normal subgroup. The resulting quotient is written , where G is the original group and N is the normal subgroup. (This is pronounced "G mod N," where "mod" is short for modulo.) Much of the importance of quotient groups is derived from their relation to homomorphisms. The first isomorphism theorem states that the image of any group G under a homomorphism is always isomorphic to a quotient of G. Specifically, the image of G under a homomorphism is isomorphic to where ker(φ) denotes the kernel of φ. The dual notion of a quotient group is a subgroup, these being the two primary ways of forming a smaller group from a larger one. Any normal subgroup has a corresponding quotient group, formed from the larger group by eliminating the distinction between elements of the subgroup. In category theory, quotient groups are examples of quotient objects, which are dual to subobjects. For other examples of quotient objects, see quotient ring, quotient space (linear algebra), quotient space (topology), and quotient set. In the following discussion, we will use a binary operation on the subsets of G: if two subsets S and T of G are given, we define their product as . This operation is associative and has as identity element the singleton {e}, where e is the identity element of G. Thus, the set of all subsets of G forms a monoid under this operation. In terms of this operation we can first explain what a quotient group is, and then explain what a normal subgroup is: It is fully determined by the subset containing e. A normal subgroup of G is the set containing e in any such partition. The subsets in the partition are the cosets of this normal subgroup. A subgroup N of a group G is normal if and only if the coset equality aN = Na holds for all a in G. In terms of the binary operation on subsets defined above, a normal subgroup of G is a subgroup that commutes with every subset of G and is denoted . A subgroup that permutes with every subgroup of G is called a permutable subgroup. Let N be a normal subgroup of a group G. We define the set G/N to be the set of all left cosets of N in G, i.e., . The group operation on G/N is the product of subsets defined above. In other words, for each aN and bN in G/N, the product of aN and bN is (aN)(bN). This operation is closed, because (aN)(bN) really is a left coset: The normality of N is used in this equation. Because of the normality of N, the left cosets and right cosets of N in G are equal, and so G/N could be defined as the set of right cosets of N in G. Because the operation is derived from the product of subsets of G, the operation is well-defined (does not depend on the particular choice of representatives), associative, and has identity element N. The inverse of an element aN of G/N is a−1N. For example, consider the group with addition modulo 6: Let The quotient group is: The basic argument above is still valid if G/N is defined to be the set of all right cosets. The reason G/N is called a quotient group comes from division of integers. When dividing 12 by 3 one obtains the answer 4 because one can regroup 12 objects into 4 subcollections of 3 objects. The quotient group is the same idea, however we end up with a group for a final answer instead of a number because groups have more structure than an arbitrary collection of objects. To elaborate, when looking at G/N with N a normal subgroup of G, the group structure is used to form a natural "regrouping". These are the cosets of N in G. Because we started with a group and normal subgroup, the final quotient contains more information than just the number of cosets (which is what regular division yields), but instead has a group structure itself. Consider the group of integers Z (under addition) and the subgroup 2Z consisting of all even integers. This is a normal subgroup, because Z is abelian. There are only two cosets: the set of even integers and the set of odd integers; therefore, the quotient group Z/2Z is the cyclic group with two elements. This quotient group is isomorphic with the set with addition modulo 2; informally, it is sometimes said that Z/2Z equals the set with addition modulo 2. A slight generalization of the last example. Once again consider the group of integers Z under addition. Let n be any positive integer. We will consider the subgroup nZ of Z consisting of all multiples of n. Once again nZ is normal in Z because Z is abelian. The cosets are the collection {nZ, 1+nZ, ..., (n−2)+nZ, (n−1)+nZ}. An integer k belongs to the coset r+nZ, where r is the remainder when dividing k by n. The quotient Z/nZ can be thought of as the group of "remainders" modulo n. This is a cyclic group of order n. The twelfth roots of unity, which are points on the unit circle, form a multiplicative abelian group G, shown on the picture on the right as colored balls with the number at each point giving its complex argument. Consider its subgroup N made of the fourth roots of unity, shown as red balls. This normal subgroup splits the group into three cosets, shown in red, green and blue. One can check that the cosets form a group of three elements (the product of a red element with a blue element is blue, the inverse of a blue element is green, etc.). Thus, the quotient group G/N is the group of three colors, which turns out to be the cyclic group with three elements. Consider the group of real numbers R under addition, and the subgroup Z of integers. The cosets of Z in R are all sets of the form a+Z, with a real number. Adding such cosets is done by adding the corresponding real numbers, and subtracting 1 if the result is greater than or equal to 1. The quotient group R/Z is isomorphic to the circle group S1, the group of complex numbers of absolute value 1 under multiplication, or correspondingly, the group of rotations in 2D about the origin, i.e., the special orthogonal group SO(2). An isomorphism is given by (see Euler's identity). If G is the group of invertible 3 × 3 real matrices, and N is the subgroup of 3 × 3 real matrices with determinant 1, then N is normal in G (since it is the kernel of the determinant homomorphism). The cosets of N are the sets of matrices with a given determinant, and hence G/N is isomorphic to the multiplicative group of non-zero real numbers. Consider the abelian group (that is, the set with addition modulo 4), and its subgroup . The quotient group is . This is a group with identity element , and group operations such as }. Both the subgroup and the quotient group are isomorphic with Z2. Consider the multiplicative group G=\mathbf{Z}^*_{n^2}. The set N of nth residues is a multiplicative subgroup isomorphic to \mathbf{Z}^*_{n}. Then N is normal in G and the factor group G/N has the cosets N, (1+n)N, (1+n)2N, ..., (1+n)−1nN. The Pallier cryptosystem is based on the conjecture that it is difficult to determine the coset of a random element of G without knowing the factorization of n. The quotient group is isomorphic to the trivial group (the group with one element), and is isomorphic to G. The order of , by definition the number of elements, is equal to , the index of N in G. If G is finite, the index is also equal to the order of G divided by the order of N. Note that may be finite, although both G and N are infinite (e.g. ). There is a "natural" surjective group homomorphism , sending each element g of G to the coset of N to which g belongs, that is: . The mapping π is sometimes called the canonical projection of G onto . Its kernel is N. There is a bijective correspondence between the subgroups of G that contain N and the subgroups of ; if H is a subgroup of G containing N, then the corresponding subgroup of is π(H). This correspondence holds for normal subgroups of G and as well, and is formalized in the lattice theorem. Several important properties of quotient groups are recorded in the fundamental theorem on homomorphisms and the isomorphism theorems. If G is abelian, nilpotent or solvable, then so is . If G is cyclic or finitely generated, then so is . If N is contained in the center of G, then G is called the central extension of the quotient group. If H is a subgroup in a finite group G, and the order of H is one half of the order of G, then H is guaranteed to be a normal subgroup, so exists and is isomorphic to C2. This result can also be stated as "any subgroup of index 2 is normal", and in this form it applies also to infinite groups. To extend the above, if p is the smallest prime number dividing the order of a finite group, G, then if has order p, H must be a normal subgroup of G. Every finitely generated group is isomorphic to a quotient of a free group. Sometimes, but not necessarily, a group G can be reconstructed from and N, as a direct product or semidirect product. The problem of determining when this is the case is known as the extension problem. An example where it is not possible is as follows. is isomorphic to Z2, and also, but the only semidirect product is the direct product, because Z2 has only the trivial automorphism. Therefore Z4, which is different from , cannot be reconstructed. If G is a Lie group and N is a normal Lie subgroup of G, the quotient is also a Lie group. In this case, the original group G has the structure of a fiber bundle (specifically, a -bundleNprincipal ), with base space and fiber N. For a non-normal Lie subgroup N, the space of left cosets is not a group, but simply a differentiable manifold on which G acts. The result is known as a homogeneous space.
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring or residue class ring, is a construction quite similar to the factor groups of group theory and the quotient spaces of linear algebra. One starts with a ring R and a two-sided ideal I in R, and constructs a new ring, the quotient ring R/I, essentially by requiring that all elements of I be zero in R. Intuitively, the quotient ring R/I is a "simplified version" of R where the elements of I are "ignored". Quotient rings are distinct from the so-called 'quotient field', or field of fractions, of an integral domain as well as from the more general 'rings of quotients' obtained by localization. Given a ring R and a two-sided ideal I in R, we may define an equivalence relation ~ on R as follows: Using the ideal properties, it is not difficult to check that ~ is a congruence relation. In case a ~ b, we say that a and b are congruent modulo I. The equivalence class of the element a in R is given by This equivalence class is also sometimes written as a mod I and called the "residue class of a modulo I". The set of all such equivalence classes is denoted by R/I; it becomes a ring, the factor ring or quotient ring of R modulo I, if one defines (Here one has to check that these definitions are well-defined. Compare coset and quotient group.) The zero-element of R/I is (0 + I) = I, and the multiplicative identity is (1 + I). The map p from R to R/I defined by p(a) = a + I is a surjective ring homomorphism, sometimes called the natural quotient map or the canonical homomorphism. The quotients R[X]/(X), R[X]/(X + 1), and R[X]/(X − 1) are all isomorphic to R and gain little interest at first. But note that R[X]/(X2) is called the dual number plane in geometric algebra. It consists only of linear binomials as "remainders" after reducing an element of R[X] by X2. This alternative complex plane arises as a subalgebra whenever the algebra contains a real line and a nilpotent. Furthermore, the ring quotient R[X]/(X2 − 1) does split into R[X]/(X + 1) and R[X]/(X − 1), so this ring is often viewed as the direct sum R  \oplus R. Nevertheless, an alternative complex number z = x + y j is suggested by j as a root of X2 − 1, compared to i as root of X2 + 1 = 0. This plane of split-complex numbers normalizes the direct sum R \oplus R by providing a basis {1, j } for 2-space where the identity of the algebra is at unit distance from the zero. With this basis a unit hyperbola may be compared to the unit circle of the ordinary complex plane. Hamilton’s quaternions of 1843 can be cast as R[X,Y]/(X2 + 1, Y2 + 1, XY + YX). If Y2 − 1 is substituted for Y2 + 1, then one obtains the ring of split-quaternions. Substituting minus for plus in both the quadratic binomials also results in split-quaternions. The anti-commutative property YX = −XY implies that XY has for its square The three types of biquaternions can also be written as quotients by conscripting the three-indeterminate ring R[X,Y,Z] and constructing appropriate ideals. Clearly, if R is a commutative ring, then so is R/I; the converse however is not true in general. The natural quotient map p has I as its kernel; since the kernel of every ring homomorphism is a two-sided ideal, we can state that two-sided ideals are precisely the kernels of ring homomorphisms. The intimate relationship between ring homomorphisms, kernels and quotient rings can be summarized as follows: the ring homomorphisms defined on R/I are essentially the same as the ring homomorphisms defined on R that vanish (i.e. are zero) on I. More precisely: given a two-sided ideal I in R and a ring homomorphism f : RS whose kernel contains I, then there exists precisely one ring homomorphism g : R/IS with gp = f (where p is the natural quotient map). The map g here is given by the well-defined rule g([a]) = f(a) for all a in R. Indeed, this universal property can be used to define quotient rings and their natural quotient maps. As a consequence of the above, one obtains the fundamental statement: every ring homomorphism f : RS induces a ring isomorphism between the quotient ring R/ker(f) and the image im(f). (See also: fundamental theorem on homomorphisms.) The ideals of R and R/I are closely related: the natural quotient map provides a bijection between the two-sided ideals of R that contain I and the two-sided ideals of R/I (the same is true for left and for right ideals). This relationship between two-sided ideal extends to a relationship between the corresponding quotient rings: if M is a two-sided ideal in R that contains I, and we write M/I for the corresponding ideal in R/I (i.e. M/I = p(M)), the quotient rings R/M and (R/I)/(M/I) are naturally isomorphic via the (well-defined!) mapping a + M ↦ (a+I) + M/I. In commutative algebra and algebraic geometry, the following statement is often used: If R ≠ {0} is a commutative ring and I is a maximal ideal, then the quotient ring R/I is a field; if I is only a prime ideal, then R/I is only an integral domain. A number of similar statements relate properties of the ideal I to properties of the quotient ring R/I. The Chinese remainder theorem states that, if the ideal I is the intersection (or equivalently, the product) of pairwise coprime ideals I1,...,Ik, then the quotient ring R/I is isomorphic to the product of the quotient rings R/Ip, p=1,...,k.
Bite force quotient (BFQ) is the regression of the quotient of an animal's bite force divided by its body mass. Table sources (unless otherwise stated):
The Wilson quotient W(p) is defined as: If p is a prime number, the quotient is an integer by Wilson's theorem; moreover, if p is composite, the quotient is not an integer. If p divides W(p), it is called a Wilson prime. The integer values of W(p) are (sequence in OEIS):
In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by "collapsing" N to zero. The space obtained is called a quotient space and is denoted V/N (read V mod N or V by N). Formally, the construction is as follows (Halmos 1974, §21-22). Let V be a vector space over a field K, and let N be a subspace of V. We define an equivalence relation ~ on V by stating that x ~ y if x − yN. That is, x is related to y if one can be obtained from the other by adding an element of N. From this definition, one can deduce that any element of N is related to the zero vector; more precisely all the vectors in N get mapped into the equivalence class of the zero vector. The equivalence class of x is often denoted since it is given by The quotient space V/N is then defined as V/~, the set of all equivalence classes over V by ~. Scalar multiplication and addition are defined on the equivalence classes by It is not hard to check that these operations are well-defined (i.e. do not depend on the choice of representative). These operations turn the quotient space V/N into a vector space over K with N being the zero class, [0]. The mapping that associates to v ∈ V the equivalence class [v] is known as the quotient map. Let X = R2 be the standard Cartesian plane, and let Y be a line through the origin in X. Then the quotient space X/Y can be identified with the space of all lines in X which are parallel to Y. That is to say that, the elements of the set X/Y are lines in X parallel to Y. This gives one way in which to visualize quotient spaces geometrically. Another example is the quotient of Rn by the subspace spanned by the first m standard basis vectors. The space Rn consists of all n-tuples of real numbers (x1,…,xn). The subspace, identified with Rm, consists of all n-tuples such that the last n-m entries are zero: (x1,…,xm,0,0,…,0). Two vectors of Rn are in the same congruence class modulo the subspace if and only if they are identical in the last nm coordinates. The quotient space Rn/ Rm is isomorphic to Rmn in an obvious manner. More generally, if V is an (internal) direct sum of subspaces U and W, then the quotient space V/U is naturally isomorphic to W (Halmos 1974, Theorem 22.1). An important example of a functional quotient space is a spacepL. There is a natural epimorphism from V to the quotient space V/U given by sending x to its equivalence class [x]. The kernel (or nullspace) of this epimorphism is the subspace U. This relationship is neatly summarized by the short exact sequence If U is a subspace of V, the dimension of V/U is called the codimension of U in V. Since a basis of V may be constructed from a basis A of U and a basis B of V/U by adding a representative of each element of B to A, the dimension of V is the sum of the dimensions of U and V/U. If V is finite-dimensional, it follows that the codimension of U in V is the difference between the dimensions of V and U (Halmos 1974, Theorem 22.2): Let T : VW be a linear operator. The kernel of T, denoted ker(T), is the set of all xV such that Tx = 0. The kernel is a subspace of V. The first isomorphism theorem of linear algebra says that the quotient space V/ker(T) is isomorphic to the image of V in W. An immediate corollary, for finite-dimensional spaces, is the rank-nullity theorem: the dimension of V is equal to the dimension of the kernel (the nullity of T) plus the dimension of the image (the rank of T). The cokernel of a linear operator T : VW is defined to be the quotient space W/im(T). If X is a Banach space and M is a closed subspace of X, then the quotient X/M is again a Banach space. The quotient space is already endowed with a vector space structure by the construction of the previous section. We define a norm on X/M by The quotient space X/M is complete with respect to the norm, so it is a Banach space. Let C[0,1] denote the Banach space of continuous real-valued functions on the interval [0,1] with the sup norm. Denote the subspace of all functions fC[0,1] with f(0) = 0 by M. Then the equivalence class of some function g is determined by its value at 0, and the quotient space C[0,1] / M is isomorphic to R. If X is a Hilbert space, then the quotient space X/M is isomorphic to the orthogonal complement of M. The quotient of a locally convex space by a closed subspace is again locally convex (Dieudonné 1970, 12.14.8). Indeed, suppose that X is locally convex so that the topology on X is generated by a family of seminorms {pα|α∈A} where A is an index set. Let M be a closed subspace, and define seminorms qα by on X/M Then X/M is a locally convex space, and the topology on it is the quotient topology. If, furthermore, X is metrizable, then so is X/M. If X is a Fréchet space, then so is X/M (Dieudonné 1970, 12.11.3).

Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas arithmetic deals with specified numbers, algebra introduces quantities without fixed values, known as variables. This use of variables entails a use of algebraic notation and an understanding of the general rules of the operators introduced in arithmetic. Unlike abstract algebra, elementary algebra is not concerned with algebraic structures outside the realm of real and complex numbers.

The use of variables to denote quantities allows general relationships between quantities to be formally and concisely expressed, and thus enables solving a broader scope of problems. Most quantitative results in science and mathematics are expressed as algebraic equations.

Topology Quotient

In mathematics, an algebraic expression is an expression built up from constants, variables, and a finite number of algebraic operations (addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number). For example, 3x^2 - 2xy + c is an algebraic expression. Since taking the square root is the same as raising to the power \tfrac{1}{2},

is also an algebraic expression.

The Western Aphasia Battery or WAB is an instrument for assessing the language function of adults, able to discern the presence, degree, and type of aphasia. Another such test is the Boston Diagnostic Aphasia Examination. The aphasia quotient (AQ) is the summary score that indicates overall severity of language impairment.

The WAB–R, a full battery of 8 subtests (32 short tasks), maintains the structure and overall content and clinical value of the current measure while creating these improvements:

In mathematics a stack or 2-sheaf is, roughly speaking, a sheaf that takes values in categories rather than sets. Stacks are used to formalise some of the main constructions of descent theory, and to construct fine moduli stacks when fine moduli spaces do not exist.

Descent theory is concerned with generalisations of situations where geometrical objects (such as vector bundles on topological spaces) can be "glued together" when they are isomorphic (in a compatible way) when restricted to intersections of the sets in an open covering of a space. In more general set-up the restrictions are replaced with general pull-backs, and fibred categories form the right framework to discuss the possibility of such "glueing". The intuitive meaning of a stack is that it is a fibred category such that "all possible glueings work". The specification of glueings requires a definition of coverings with regard to which the glueings can be considered. It turns out that the general language for describing these coverings is that of a Grothendieck topology. Thus a stack is formally given as a fibred category over another base category, where the base has a Grothendieck topology and where the fibred category satisfies a few axioms that ensure existence and uniqueness of certain glueings with respect to the Grothendieck topology.


In algebra, which is a broad division of mathematics, abstract algebra is a common name for the sub-area that studies algebraic structures in their own right. Such structures include groups, rings, fields, modules, vector spaces, and algebras. The specific term abstract algebra was coined at the beginning of the 20th century to distinguish this area from the other parts of algebra. The term modern algebra has also been used to denote abstract algebra.

Two mathematical subject areas that study the properties of algebraic structures viewed as a whole are universal algebra and category theory. Algebraic structures, together with the associated homomorphisms, form categories. Category theory is a powerful formalism for studying and comparing different algebraic structures.

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