In mathematics, a set is said to be closed under some operation if performance of that operation on members of the set always produces a member of the set. For example, the real numbers are closed under subtraction, but the natural numbers are not: 3 and 7 are both natural numbers, but the result of 3 − 7 is not.

Similarly, a set is said to be closed under a collection of operations if it is closed under each of the operations individually.

A set that is closed under an operation or collection of operations is said to satisfy a closure property. Often a closure property is introduced as an axiom, which is then usually called the axiom of closure. Note that modern set-theoretic definitions usually define operations as maps between sets, so adding closure to a structure as an axiom is superfluous, though it still makes sense to ask whether subsets are closed. For example, the set of real numbers is closed under subtraction, where (as mentioned above) its subset of natural numbers is not.

When a set S is not closed under some operations, one can usually find the smallest set containing S that is closed. This smallest closed set is called the closure of S (with respect to these operations). For example, the closure under subtraction of the set of natural numbers, viewed as a subset of the real numbers, is the set of integers. An important example is that of topological closure. The notion of closure is generalized by Galois connection, and further by monads.

Note that the set S must be a subset of a closed set in order for the closure operator to be defined. In the preceding example, it is important that the reals are closed under subtraction; in the domain of the natural numbers subtraction is not always defined.

The two uses of the word "closure" should not be confused. The former usage refers to the property of being closed, and the latter refers to the smallest closed set containing one that isn't closed. In short, the closure of a set satisfies a closure property.

Closed sets

A set is closed under an operation if that operation returns a member of the set when evaluated on members of the set. Sometimes the requirement that the operation be valued in a set is explicitly stated, in which case it is known as the axiom of closure. For example, one may define a group as a set with a binary product operator obeying several axioms, including an axiom that the product of any two elements of the group is again an element. However the modern definition of an operation makes this axiom superfluous; an n-ary operator on S is just a subset of Sⁿ⁺¹. By its very definition, an operator on a set cannot have values outside the set.

Nevertheless, the closure property of an operator on a set still has some utility. Closure on a set does not necessarily imply closure on all subsets. Thus a subgroup of a group is a subset on which the binary product and the unary operation of inversion satisfy the closure axiom.

An operation of a different sort is that of finding the limit points of a subset of a topological space (if the space is first-countable, it suffices to restrict consideration to the limits of sequences but in general one must consider at least limits of nets). A set that is closed under this operation is usually just referred to as a closed set in the context of topology. Without any further qualification, the phrase usually means closed in this sense. Closed intervals like [1,2] = {x: 1 ≤ x ≤ 2} are closed in this sense.

A partially ordered set is downward closed (and also called a lower set) if for every element of the set all smaller elements are also in it; this applies for example for the real intervals (-∞, p) and (-∞, p], and for an ordinal number p represented as interval [ 0, p); every downward closed set of ordinal numbers is itself an ordinal number.

Upward closed and upper set are defined similarly.

P closures of binary relations

The notion of a closure can be generalized for an arbitrary binary relation R ⊆ S×S, and an arbitrary property P in the following way: the P closure of R is the least relation Q ⊆ S×S that contains R (i.e. R ⊆ Q) and for which property P holds (i.e. P(Q) is true). For instance, one can define the symmetric closure as the least symmetric relation containing R. This generalization is often encountered in the theory of rewriting systems, where one often uses more "wordy" notions such as the reflexive transitive closure R^*—the smallest preorder containing R, or the reflexive transitive symmetric closure R^≡—the smallest equivalence relation containing R. For arbitrary P and R, the P closure of R need not exist. In the above examples, these exist because reflexivity, transitivity and symmetry are closed under arbitrary intersections. In such cases, the P closure can be directly defined as the intersection of all sets with property P containing R.^[1]

Closure operator

Given an operation on a set X, one can define the closure C(S) of a subset S in X to be the smallest subset closed under that operation that contains S as a subset. For example, the closure of a subset of a group is the subgroup generated by that set.

The closure of sets with respect to some operation defines a closure operator on the subsets of X. The closed sets can be determined from the closure operator; a set is closed if it is equal to its own closure. Typical structural properties of all closure operations are:

• The closure is increasing or extensive: the closure of an object contains the object.
• The closure is idempotent: the closure of the closure equals the closure.
• The closure is monotone, that is, if X is contained in Y, then also C(X) is contained in C(Y).

An object that is its own closure is called closed. By idempotency, an object is closed if and only if it is the closure of some object.

These three properties define an abstract closure operator. Typically, an abstract closure acts on the class of all subsets of a set.

Examples

• In topology and related branches, the relevant operation is taking limits. The topological closure of a set is the corresponding closure operator. The Kuratowski closure axioms characterize this operator.
• In linear algebra, the linear span of a set X of vectors is the closure of that set; it is the smallest subset of the vector space that includes X and is closed under the operation of linear combination. This subset is a subspace.
• In matroid theory, the closure of X is the largest superset of X that has the same rank as X.
• In set theory, the transitive closure of a binary relation.
• In algebra, the algebraic closure of a field.
• In commutative algebra, closure operations for ideals, as integral closure and tight closure.
• In geometry, the convex hull of a set S of points is the smallest convex set of which S is a subset.
• In the theory of formal languages, the Kleene closure of a language can be described as the set of strings that can be made by concatenating zero or more strings from that language.
• In group theory, the normal closure of a set of group elements is the smallest normal subgroup containing the set.

A vinculum is a horizontal line placed over a mathematical expression, used to indicate that it is to be considered grouped together. Vinculum is Latin for "bond", "fetter", "chain", or "tie", which is roughly suggestive of some of the uses of the symbol.

Uses

In a repeating decimal, the vinculum is used to indicate the group of repeating digits:

It is used as part of the notation of a radical to indicate the radicand whose root is being indicated. In the next case, the quantity ab + 2 is the radicand, and thus has a vinculum over it:

It is used to show the repeating terms in a periodic continued fraction. Quadratic irrational numbers are the only numbers that have these.

It can be used in signed-digit representation to represent negative digits, such as the following example in balanced ternary:

The vinculum is sometimes used in Boolean algebra, where it serves to indicate a group of expressions whose logical result is to be negated, as in:

It can even be used as a notation to indicate a group (bracket smaller to parenthesis):

meaning to add b and c first and then subtract the result from a.

In particle physics, the vinculum is used to indicate antiparticles. For example, p and p are the symbols for proton and antiproton, respectively.

The vinculum should not be confused with a similar-looking vector notation, e.g. "vector from A to B", or "vector named a".

In mathematics, a quotient is the result of a division. For example, when dividing 6 by 3, the quotient is 2, while 6 is called the dividend, and 3 the divisor. The quotient can also be expressed as the number of times the divisor divides into the dividend.

A quotient can also mean just the integer part of the result of dividing two integers. For example, the quotient of 13 ÷ 5 would be 2 while the remainder would be 3. For more, see the division algorithm.

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:

• quotient set
• quotient group
• quotient ring
• quotient space (linear algebra)
• quotient space of a topological space
• quotient object
• right quotient and left quotient (operations on formal languages)

The quotient rule is a method for finding derivatives in calculus.

Quotients also come up in certain tests, like the IQ test, which stands for intelligence quotient. In this case, one's quotient is basically one's score^{[clarification needed]}. 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 mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder.

Explanation

The name "divisor" comes from the arithmetic operation of division: if a / b = c then a is the dividend, b the divisor, and c the quotient.

In general, m | n (read as "m divides n") for non-zero integers m and n iff there exists an integer k such that n = km. Thus, divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. (For example, there are six divisors of four, 1, 2, 4, −1, −2, −4, but only the positive ones would usually be mentioned, i.e. 1, 2, and 4.)

1 and −1 divide (are divisors of) every integer, every integer (and its negation) is a divisor of itself, and every integer is a divisor of 0, except by convention 0 itself (see also division by zero). Numbers divisible by 2 are called even and numbers not divisible by 2 are called odd.

1, −1, n and −n are known as the trivial divisors of n. A divisor of n that is not a trivial divisor is known as a non-trivial divisor. A number with at least one non-trivial divisor is known as a composite number, while the units -1 and 1 and prime numbers have no non-trivial divisors.

There are divisibility rules which allow one to recognize certain divisors of a number from the number's digits.

Examples

• 7 is a divisor of 42 because 42 / 7 = 6, so we can say 7 | 42. It can also be said that 42 is divisible by 7, 42 is a multiple of 7, 7 divides 42, or 7 is a factor of 42.
• The non-trivial divisors of 6 are 2, −2, 3, −3.
• The positive divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42.
• The set of all positive divisors of 60, A = {1,2,3,4,5,6,10,12,15,20,30,60}, partially ordered by divisibility, has the Hasse diagram:

Further notions and facts

There are some elementary rules:

• If a | b and b | c, then a | c. This is the transitive relation.
• If a | b and b | a, then a = b or a = − b.
• If a | b and c | b, then it is NOT always true that (a + c) | b (eg 2|6 and 3|6 but 5 does not divide 6). However, when a | b and a | c, then a | (b + c) is true, as is a | (b − c).^[1]

The vertical bar used is a Unicode "Divides" character, code point U+2223. Its negated symbol is ∤. In an ASCII-only environment, the standard vertical bar "|", which is slightly shorter, is often used.

If a | bc, and gcd(a,b) = 1, then a | c. This is called Euclid's lemma.

If p is a prime number and p | ab then p | a or p | b (or both).

A positive divisor of n which is different from n is called a proper divisor or an aliquot part of n. A number that does not evenly divide n but leaves a remainder is called an aliquant part of n.

An integer n > 1 whose only proper divisor is 1 is called a prime number. Equivalently, a prime number is a positive integer which has exactly two positive factors: 1 and itself.

Any positive divisor of n is a product of prime divisors of n raised to some power. This is a consequence of the fundamental theorem of arithmetic.

If a number equals the sum of its proper divisors, it is said to be a perfect number. Numbers less than the sum of their proper divisors are said to be abundant, while numbers greater than that sum are said to be deficient.

The total number of positive divisors of n is a multiplicative function d(n) (e.g. ). The sum of the positive divisors of n is another multiplicative function σ(n) (e.g. ). Both of these functions are examples of divisor functions.

If the prime factorization of n is given by

then the number of positive divisors of n is

and each of the divisors has the form

where for each

It can be shown that for any natural n the inequality holds.

Also it can be shown ^[2] that

One interpretation of this result is that a randomly chosen positive integer n has an expected number of divisors of about lnn.

Divisibility of numbers

The relation of divisibility turns the set N of non-negative integers into a partially ordered set, in fact into a complete distributive lattice. The largest element of this lattice is 0 and the smallest is 1. The meet operation ^ is given by the greatest common divisor and the join operation v by the least common multiple. This lattice is isomorphic to the dual of the lattice of subgroups of the infinite cyclic group Z.

In mathematics, especially in elementary arithmetic, division (÷) is the arithmetic operation that is the inverse of multiplication.

Specifically, if c times b equals a, written:

where b is not zero, then a divided by b equals c, written:

For instance,

since

.

In the above expression, a is called the dividend, b the divisor and c the quotient.

Conceptually, division describes two distinct but related settings. Partitioning involves taking a set of size a and forming b groups that are equal in size. The size of each group formed, c, is the quotient of a and b. Quotative division involves taking a set of size a and forming groups of size b. The number of groups of this size that can be formed, c, is the quotient of a and b.^[1]

Teaching division usually leads to the concept of fractions being introduced to students. Unlike addition, subtraction, and multiplication, the set of all integers is not closed under division. Dividing two integers may result in a remainder. To complete the division of the remainder, the number system is extended to include fractions or rational numbers as they are more generally called.

Notation

Division is often shown in algebra and science by placing the dividend over the divisor with a horizontal line, also called a vinculum or fraction bar, between them. For example, a divided by b is written

This can be read out loud as "a divided by b", "a by b" or "a over b". A way to express division all on one line is to write the dividend, or numerator then a slash, then the divisor, or denominator like this:

This is the usual way to specify division in most computer programming languages since it can easily be typed as a simple sequence of ASCII characters.

A typographical variation, which is halfway between these two forms, uses a solidus (fraction slash) but elevates the dividend, and lowers the divisor:

^a⁄_b

Any of these forms can be used to display a fraction. A fraction is a division expression where both dividend and divisor are integers (although typically called the numerator and denominator), and there is no implication that the division needs to be evaluated further.

A second way to show division is to use the obelus (or division sign), common in arithmetic, in this manner:

This form is infrequent except in elementary arithmetic. The obelus is also used alone to represent the division operation itself, as for instance as a label on a key of a calculator.

In some non-English-speaking cultures, "a divided by b" is written a : b. However, in English usage the colon is restricted to expressing the related concept of ratios (then "a is to b").

Computing division

Division is often introduced through the notion of "sharing out" a set of objects, for example a pile of sweets, into a number of equal portions. Distributing the objects several at a time in each round of sharing to each portion leads to the idea of "chunking", i.e. division by repeated subtraction.

More systematic and more efficient (but also more formalised and more rule-based, and more removed from an overall holistic picture of what division is achieving), a person who knows the multiplication tables can divide two integers using pencil and paper and the method of short division, if the divisor is simple, or long division for larger integer divisors. If the dividend has a fractional part (expressed as a decimal fraction), we can continue the algorithm past the ones place as far as desired. If the divisor has a fractional part, we can restate the problem by moving the decimal to the right in both numbers until the divisor has no fraction.

Modern computers compute division by methods that are faster than long division: see Division (digital).

A person can calculate division with an abacus by repeatedly placing the dividend on the abacus, and then subtracting the divisor the offset of each digit in the result, counting the number of divisions possible at each offset.

A person can use logarithm tables to divide two numbers, by subtracting the two numbers' logarithms, then looking up the antilogarithm of the result.

A person can calculate division with a slide rule by aligning the divisor on the C scale with the dividend on the D scale. The quotient can be found on the D scale where it is aligned with the left index on the C scale. The user is responsible, however, for mentally keeping track of the decimal point.

In modular arithmetic, some numbers have a multiplicative inverse with respect to the modulus. We can calculate division by multiplication in such a case. This approach is useful in computers that do not have a fast division instruction.

Division algorithm

The division algorithm is mathematical theorem that precisely expresses the outcome of the usual process of division of integers. In particular, the theorem asserts that integers called the quotient q and remainder r always exist and that they are uniquely determined by the dividend a and divisor d, with d ≠ 0. Formally, the theorem is stated as follows: There exist unique integers q and r such that a = qd + r and 0 ≤ r < | d |, where | d | denotes the absolute value of d.

Division of integers

Division of integers is not closed. Apart from division by zero being undefined, the quotient will not be an integer unless the dividend is an integer multiple of the divisor; for example 26 cannot be divided by 10 to give an integer. In such a case there are four possible approaches.

1. Say that 26 cannot be divided by 10; division becomes a partial function.
2. Give the answer as a decimal fraction or a mixed number, so or This is the approach usually taken in mathematics.
3. Give the answer as an integer quotient and a remainder, so
4. Give the integer quotient as the answer, so This is sometimes called integer division.

One has to be careful when performing division of integers in a computer program. Some programming languages, such as C, will treat division of integers as in case 4 above, so the answer will be an integer. Other languages, such as MATLAB, will first convert the integers to real numbers, and then give a real number as the answer, as in case 2 above.

Names and symbols used for integer division include div, /, \, and %. Definitions vary regarding integer division when the quotient is negative: rounding may be toward zero or toward −∞.

Divisibility rules can sometimes be used to quickly determine whether one integer divides exactly into another.

Division of rational numbers

The result of dividing two rational numbers is another rational number when the divisor is not 0. We may define division of two rational numbers p/q and r/s by

All four quantities are integers, and only p may be 0. This definition ensures that division is the inverse operation of multiplication.

Division of real numbers

Division of two real numbers results in another real number when the divisor is not 0. It is defined such a/b = c if and only if a = cb and b ≠ 0.

Division by zero

Division of any number by zero (where the divisor is zero) is not defined. This is because zero multiplied by any finite number will always result in a product of zero. Entry of such an equation into most calculators will result in an error message being issued.

Division of complex numbers

Dividing two complex numbers results in another complex number when the divisor is not 0, defined thus:

All four quantities are real numbers. r and s may not both be 0.

Division for complex numbers expressed in polar form is simpler than the definition above:

Again all four quantities are real numbers. r may not be 0.

Division of polynomials

One can define the division operation for polynomials. Then, as in the case of integers, one has a remainder. See polynomial long division or synthetic division.

Division of matrices

One can define a division operation for matrices. The usual way to do this is to define A / B = AB⁻¹, where B⁻¹ denotes the inverse of B, but it is far more common to write out AB⁻¹ explicitly to avoid confusion.

Left and right division

Because matrix multiplication is not commutative, one can also define a left division or so-called backslash-division as A \ B = A⁻¹B. For this to be well defined, B⁻¹ need not exist, however A⁻¹ does need to exist. To avoid confusion, division as defined by A / B = AB⁻¹ is sometimes called right division or slash-division in this context.

Note that with left and right division defined this way, A/(BC) is in general not the same as (A/B)/C and nor is (AB)\C the same as A\(B\C), but A/(BC) = (A/C)/B and (AB)\C = B\(A\C).

Matrix division and pseudoinverse

To avoid problems when A⁻¹ and/or B⁻¹ do not exist, division can also be defined as multiplication with the pseudoinverse, i.e., A / B = AB⁺ and A \ B = A⁺B, where A⁺ and B⁺ denote the pseudoinverse of A and B.

Division in abstract algebra

In abstract algebras such as matrix algebras and quaternion algebras, fractions such as are typically defined as or where b is presumed to be an invertible element (i.e. there exists a multiplicative inverse b ^{− 1} such that bb ^{− 1} = b ^{− 1}b = 1 where 1 is the multiplicative identity). In an integral domain where such elements may not exist, division can still be performed on equations of the form ab = ac or ba = ca by left or right cancellation, respectively. More generally "division" in the sense of "cancellation" can be done in any ring with the aforementioned cancellation properties. If such a ring is finite, then by an application of the pigeonhole principle, every nonzero element of the ring is invertible, so division by any nonzero element is possible in such a ring. To learn about when algebras (in the technical sense) have a division operation, refer to the page on division algebras. In particular Bott periodicity can be used to show that any real normed division algebra must be isomorphic to either the real numbers R, the complex numbers C, the quaternions H, or the octonions O.

Division and calculus

The derivative of the quotient of two functions is given by the quotient rule:

There is no general method to integrate the quotient of two functions.