"additive inverse, often called its ""opposite"", If is in canonical form, the same is true for its opposite. A nonzero rational number has a multiplicative inverse, also called its ""reciprocal"", If is in canonical form, then the canonical form of its reciprocal is either or , depending on the sign of . If both and are nonzero, the division rule is Thus, dividing by is equivalent to multiplying by the reciprocal of : If is a non-negative integer, then The result is in canonical form if the same is true for . In particular, If , then If is in"
"One half One half is the irreducible fraction resulting from dividing one by two or the fraction resulting from dividing any number by its double. Multiplication by one half is equivalent to division by two, or ""halving""; conversely, division by one half is equivalent to multiplication by two, or ""doubling"". One half appears often in mathematical equations, recipes, measurements, etc. Half can also be said to be one part of something divided into two equal parts. For instance, the area ""S"" of a triangle is computed One half also figures in the formula for calculating figurate numbers, such as triangular"
"Quotient In arithmetic, a quotient (from ""how many times"", pronounced ) is the quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as a fraction or a ratio. For example, when dividing twenty (the ""dividend"") by three (the ""divisor""), the ""quotient"" is six and two thirds. In this sense, a quotient is the ratio of a dividend to its divisor. The quotient is most frequently encountered as two numbers, or two variables, divided by a horizontal line. The words ""dividend"" and ""divisor"" refer to each individual part, while the"
"inverse of . This approach is often the most efficient one. The Euclidean division is the mathematical formulation of the outcome of the usual process of division of integers. It asserts that, given two integers, ""a"", the ""dividend"", and ""b"", the ""divisor"", such that ""b"" ≠ 0, there are unique integers ""q"", the ""quotient"", and ""r"", the remainder, such that ""a"" = ""bq"" + ""r"" and 0 ≤ ""r"" < , where denotes the absolute value of ""b"". Division of integers is not closed. Apart from division by zero being undefined, the quotient is not an integer unless the dividend"
"part of the first number that remains, when in the course of computing the quotient, no further full chunk of the size of the second number can be allocated. For a modification of this division to yield only one single result, the natural numbers must be extended to rational numbers or real numbers. In these enlarged number systems, division is the inverse operation to multiplication, that is means , as long as is not zero—if , then this is a division by zero, which is not defined. Both forms of divisions appear in various algebraic structures. Those in which a"
"cosets, while a quotient space may be formed in a similar process by breaking a vector space into a number of similar linear subspaces. Quotient In arithmetic, a quotient (from ""how many times"", pronounced ) is the quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as a fraction or a ratio. For example, when dividing twenty (the ""dividend"") by three (the ""divisor""), the ""quotient"" is six and two thirds. In this sense, a quotient is the ratio of a dividend to its divisor. The quotient is most frequently"
"for finding the greatest common divisor of two integers, and modular arithmetic, for which only remainders are considered. The operation consisting of computing only the remainder is called the ""modulo operation"". Given two integers and , with , there exist unique integers and such that and where denotes the absolute value of . The four integers that appear in this theorem have been given names: is called the dividend, is called the divisor, is called the quotient and is called the remainder. The computation of the quotient and the remainder from the dividend and the divisor is called division or,"
"Division (mathematics) Division is one of the four basic operations of arithmetic, the others being addition, subtraction, and multiplication. At an elementary level the division of two natural numbers is –among other possible interpretations– the process of calculating the number of times one number is contained within another one. This number of times is not always an integer, and this led to two different concepts. The division with remainder or Euclidean division of two natural numbers provides a ""quotient"", which is the number of times the second one is contained in the first one, and a ""remainder"", which is the"
"Partition (number theory) In number theory and combinatorics, a partition of a positive integer ""n"", also called an integer partition, is a way of writing ""n"" as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. (If order matters, the sum becomes a composition.) For example, 4 can be partitioned in five distinct ways: The order-dependent composition 1 + 3 is the same partition as 3 + 1, while the two distinct compositions 1 + 2 + 1 and 1 + 1 + 2 represent the same partition"
"a fraction, the number of equal parts being described is the numerator (from Latin ', ""counter"" or ""numberer""), and the type or variety of the parts is the denominator (from Latin ', ""thing that names or designates""). As an example, the fraction amounts to eight parts, each of which is of the type named ""fifth."" In terms of division, the numerator corresponds to the dividend, and the denominator corresponds to the divisor. Informally, the numerator and denominator may be distinguished by placement alone but in formal contexts they are always separated by a fraction bar. The fraction bar may be"
"rather than regular derivation: in English, for example, compare """"one half"""" against other regular formations like """"one sixth"""". It is acceptable to write one half as a hyphenated word, ""one-half"" or non-hyphenated, ""one half"". One half One half is the irreducible fraction resulting from dividing one by two or the fraction resulting from dividing any number by its double. Multiplication by one half is equivalent to division by two, or ""halving""; conversely, division by one half is equivalent to multiplication by two, or ""doubling"". One half appears often in mathematical equations, recipes, measurements, etc. Half can also be said to"
"into the dividend; for example, leaves a remainder of one, as 10 is not a multiple of three. Sometimes this remainder is added to the quotient as a fractional part, so is equal to or , but in the context of integer division, where numbers have no fractional part, the remainder is kept separately or discarded. When the remainder is kept as a fraction, it leads to a rational number. The set of all rational numbers is created by every possible division using integers. In modern mathematical terms, this is known as ""extending the system"". Unlike multiplication and addition, Division"
"of equal parts, is known as ""partition"" to educators. The basic concept behind partition is ""sharing"". In sharing a whole entity becomes an integer number of equal parts. What quotition concerns is explained by removing the word ""integer"" in the last sentence. Allow ""number"" to be ""any fraction"" and you have quotition instead of partition. Quotition and partition In arithmetic, quotition and partition are two ways of viewing fractions and division. In quotition division one asks how many parts there are; in partition division one asks what the size of each part is. For example, the expression can be construed"
"is a common factor of the left denominator and right numerator and is divided out of both. Since a whole number can be rewritten as itself divided by 1, normal fraction multiplication rules can still apply. When multiplying mixed numbers, it is considered preferable to convert the mixed number into an improper fraction. For example: In other words, formula_17 is the same as formula_112, making 11 quarters in total (because 2 cakes, each split into quarters makes 8 quarters total) and 33 quarters is formula_113, since 8 cakes, each made of quarters, is 32 quarters in total. To divide a"
"Euclidean division In arithmetic, Euclidean division is the process of division of two integers, which produces a quotient and a remainder smaller than the divisor. Its main property is that the quotient and remainder exist and are unique, under some conditions. Because of this uniqueness, ""Euclidean division"" is often considered without referring to any method of computation, and without explicitly computing the quotient and the remainder. The methods of computation are called integer division algorithms, the best known being long division. Euclidean division, and algorithms to compute it, are fundamental for many questions concerning integers, such as the Euclidean algorithm"
"into the first three digits 110 of the dividend one time, so a ""1"" is written on the top line. This result is multiplied by the divisor, and subtracted from the first three digits of the dividend; the next digit (a ""1"") is included to obtain a new three-digit sequence: The procedure is then repeated with the new sequence, continuing until the digits in the dividend have been exhausted: Thus, the quotient of 11011 divided by 101 is 101, as shown on the top line, while the remainder, shown on the bottom line, is 10. In decimal, 27 divided by"
"the ""remainder"". See Euclidean division for a proof of this result and division algorithm for algorithms describing how to calculate the remainder. The remainder, as defined above, is called the ""least positive remainder"" or simply the ""remainder"". The integer ""a"" is either a multiple of ""d"" or lies in the interval between consecutive multiples of ""d"", namely, ""q⋅d"" and (""q"" + 1)""d"" (for positive ""q""). At times it is convenient to carry out the division so that ""a"" is as close as possible to an integral multiple of ""d"", that is, we can write In this case, ""s"" is called"
"the numerator and denominator are both multiplied by 2, the result is formula_59, which has the same value (0.5) as formula_1. To picture this visually, imagine cutting a cake into four pieces; two of the pieces together (formula_59) make up half the cake (formula_1). Dividing the numerator and denominator of a fraction by the same non-zero number will also yield an equivalent fraction. This is called reducing or simplifying the fraction. A simple fraction in which the numerator and denominator are coprime (that is, the only positive integer that goes into both the numerator and denominator evenly is 1) is"
"exactly is formula_95, which has slightly more than 11 billion digits. Partition function (number theory) In number theory, the partition function formula_1 represents the number of possible partitions of a non-negative integer formula_2. For instance, formula_3 because the integer formula_4 has the five partitions formula_5, formula_6, formula_7, formula_8, and formula_4. No closed-form expression for the partition function is known, but it has both asymptotic expansions that accurately approximate it and recurrence relations by which it can be calculated exactly. It grows as an exponential function of the square root of its argument. The multiplicative inverse of its generating function is"
"then 12 divided by 3 equals 4. Multiplication by 3, followed by division by 3, yields the original number (since the division of a number other than 0 by itself equals 1). Multiplication is also defined for other types of numbers, such as complex numbers, and more abstract constructs, like matrices. For some of these more abstract constructs, the order in which the operands are multiplied together matters. A listing of the many different kinds of products that are used in mathematics is given in the product (mathematics) page. In arithmetic, multiplication is often written using the sign ""×"" between"
"A positive divisor of formula_1 which is different from formula_1 is called a proper divisor or an aliquot part of formula_1. A number that does not evenly divide formula_1 but leaves a remainder is called an aliquant part of formula_1. An integer formula_63 whose only proper divisor is 1 is called a prime number. Equivalently, a prime number is a positive integer that has exactly two positive factors: 1 and itself. Any positive divisor of formula_1 is a product of prime divisors of formula_1 raised to some power. This is a consequence of the fundamental theorem of arithmetic. A number"
"line under the single-digit factor. Henceforth, the multi-digit factor will be called the multiplicand, and the single-digit factor will be called the multiplier. Suppose for simplicity that the multiplicand has three digits. The first digit is the hundreds-digit, the middle digit is the tens-digit, and the last, rightmost, digit is the ones-digit. The multiplier only has a ones-digit. The ones-digits of the multiplicand and multiplier form a column: the ones-column. Start with the ones-column: the ones-column should contain a pair of digits: the ones-digit of the multiplicand and, under it, the ones-digit of the multiplier. Find the product of these"
"Quotition and partition In arithmetic, quotition and partition are two ways of viewing fractions and division. In quotition division one asks how many parts there are; in partition division one asks what the size of each part is. For example, the expression can be construed in either of two ways: It is a fact of elementary theoretical mathematics that the numerical answer is always the same either way: 6 ÷ 2 = 3. This is essentially equivalent to the commutativity of multiplication. Division involves thinking about a whole in terms of its parts. One frequent division notion, a natural number"
"the Euclidean division of the integers. It results in two polynomials, a ""quotient"" and a ""remainder"" that are characterized by the following property of the polynomials: given two polynomials and such that , there exists a unique pair of polynomials, , the quotient, and , the remainder, such that and (here the polynomial zero is supposed to have a negative degree). By hand as well as with a computer, this division can be computed by the polynomial long division algorithm. All polynomials with coefficients in a unique factorization domain (for example, the integers or a field) also have a factored"
"Division of any number by zero in most mathematical systems is undefined because zero multiplied by any finite number always results in a product of zero. Entry of such an expression into most calculators produces an error message. However, in certain higher level mathematics division by zero is possible by the zero ring and algebras such as wheels. In these algebras, the meaning of division is different from traditional definitions. Division (mathematics) Division is one of the four basic operations of arithmetic, the others being addition, subtraction, and multiplication. At an elementary level the division of two natural numbers is"
"addition of the number 1 is the most basic form of counting, the result of adding is usually called the successor of the original number. Addition is commutative and associative, so the order in which finitely many terms are added does not matter. The identity element for a binary operation is the number that, when combined with any number, yields the same number as result. According to the rules of addition, adding to any number yields that same number, so is the additive identity. The ""inverse of a number"" with respect to a binary operation is the number that, when"
"1 is its own factorial, its own square and square root, its own cube and cube root, and so on. One is also the result of the empty product, as any number multiplied by one is itself. It is also the only natural number that is neither composite nor prime with respect to division, but instead considered a unit (meaning of ring theory). The glyph used today in the Western world to represent the number 1, a vertical line, often with a serif at the top and sometimes a short horizontal line at the bottom, traces its roots back to"
"Modulo operation In computing, the modulo operation finds the remainder after division of one number by another (sometimes called ""modulus""). Given two positive numbers, (the dividend) and (the divisor), modulo (abbreviated as ) is the remainder of the Euclidean division of by . For example, the expression ""5 mod 2"" would evaluate to 1 because 5 divided by 2 leaves a quotient of 2 and a remainder of 1, while ""9 mod 3"" would evaluate to 0 because the division of 9 by 3 has a quotient of 3 and leaves a remainder of 0; there is nothing to subtract"
"Continued fraction }</math> In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on. In a finite continued fraction (or terminated continued fraction), the iteration/recursion is terminated after finitely many steps by using an integer in lieu of another continued fraction. In contrast, an infinite continued fraction is an infinite expression. In either case, all integers in the sequence, other than the first,"
"located between the first and second lines. The answer is In mathematics, especially in elementary arithmetic, division is an arithmetic operation which 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. Division by zero (i.e. where the divisor is zero) is not defined. Division is most often shown by placing the ""dividend"" over the ""divisor"" with a horizontal line, also called a vinculum, between"
"Aliquot sequence In mathematics, an aliquot sequence is a sequence of positive integers in which each term is the sum of the proper divisors of the previous term. If the sequence reaches the number 1, it ends, since the sum of the proper divisors of 1 is 0. The aliquot sequence starting with a positive integer ""k"" can be defined formally in terms of the sum-of-divisors function σ or the aliquot sum function ""s"" in the following way: and ""s""(0) is undefined. For example, the aliquot sequence of 10 is 10, 8, 7, 1, 0 because: Many aliquot sequences terminate"
"Stirling transform In combinatorial mathematics, the Stirling transform of a sequence { ""a"" : ""n"" = 1, 2, 3, ... } of numbers is the sequence { ""b"" : ""n"" = 1, 2, 3, ... } given by where formula_2 is the Stirling number of the second kind, also denoted ""S""(""n"",""k"") (with a capital ""S""), which is the number of partitions of a set of size ""n"" into ""k"" parts. The inverse transform is where ""s""(""n"",""k"") (with a lower-case ""s"") is a Stirling number of the first kind. Berstein and Sloane (cited below) state ""If ""a"" is the number of"
"Fraction (mathematics) A fraction (from Latin """", ""broken"") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A ""common"", ""vulgar"", or ""simple"" fraction (examples: formula_1 and 17/3) consists of an integer numerator displayed above a line (or before a slash), and a non-zero integer denominator, displayed below (or after) that line. Numerators and denominators are also used in fractions that are not ""common"", including compound fractions, complex fractions, and mixed numerals. We begin"