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### Karatsuba Multiplication

It is possible to perform multiplication of large numbers in (many) fewer operations than the usual brute-force technique of "long multiplication." As discovered by Karatsuba (Karatsuba and Ofman 1962), multiplication of two -digit numbers can be done with a bit complexity of less than using identities of the form(1)Proceeding recursively then gives bit complexity , where (Borwein et al. 1989). The best known bound is steps for (Schönhage and Strassen 1971, Knuth 1998). However, this algorithm is difficult to implement, but a procedure based on the fast Fourier transform is straightforward to implement and gives bit complexity (Brigham 1974, Borodin and Munro 1975, Borwein et al. 1989, Knuth 1998).As a concrete example, consider multiplication of two numbers each just two "digits" long in base ,(2)(3)then their product is(4)(5)(6)Instead of evaluating products of individual digits, now write(7)(8)(9)The..

### Trial Division

A brute-force method of finding a divisor of an integer by simply plugging in one or a set of integers and seeing if they divide . Repeated application of trial division to obtain the complete prime factorization of a number is called direct search factorization. An individual integer being tested is called a trial divisor.

### Integer Division

Integer division is division in which the fractional part (remainder) is discarded is called integer division and is sometimes denoted . Integer division can be defined as , where "/" denotes normal division and is the floor function. For example,soInteger division is implemented in the Wolfram Language as Quotient[a, b].

### Times

The operation of multiplication, i.e., times . Various notations are , , , , and . The "multiplication sign" is based on Saint Andrew's cross (Bergamini 1969). Floating-point multiplication is sometimes denoted .

### Infinite Product

A product involving an infinite number of terms. Such products can converge. In fact, for positive , the product converges to a nonzero number iff converges.Infinite products can be used to define the cosine(1)gamma function(2)sine, and sinc function.They also appear in polygon circumscribing,(3)An interesting infinite product formula due to Euler which relates and the th prime is(4)(5)(Blatner 1997). Knar's formula gives a functional equation for the gamma function in terms of the infinite product(6)A regularized product identity is given by(7)(Muñoz Garcia and Pérez-Marco 2003, 2008).Mellin's formula states(8)where is the digamma function and is the gamma function.The following class of products(9)(10)(11)(12)(13)(Borwein et al. 2004, pp. 4-6), where is the gamma function, the first of which is given in Borwein and Corless (1999), can be done analytically. In particular, for ,(14)where (Borwein et..

### Pippenger Product

The Pippenger product is an unexpected Wallis-like formula for given by(1)(OEIS A084148 and A084149; Pippenger 1980). Here, the th term for is given by(2)(3)where is a double factorial and is the gamma function.

### Multiplication Table

A multiplication table is an array showing the result of applying a binary operator to elements of a given set . For example, the following table is the multiplication table for ordinary multiplication. 12345678910112345678910224681012141618203369121518212427304481216202428323640551015202530354045506612182430364248546077142128354249566370881624324048566472809918273645546372819010102030405060708090100The results of any binary mathematical operation can be written as a multiplication table. For example, groups have multiplication tables, where the group operation is understood as multiplication. However, different labelings and orderings of a multiplication table may describe the same abstract group. For example, the multiplication table for the cyclic group C4 may be written in three equivalent ways--denoted here by , , and --by permuting the symbols used for the group elements (Cotton 1990, p. 11).The..

### Russian Multiplication

Also called "Ethiopian multiplication." To multiply two numbers and , write and in two columns. Under , write , where is the floor function, and under , write . Continue until . Then cross out any entries in the column which are opposite an even number in the column and add the column. The result is the desired product. For example, for Russian multiplication works because it implements binarymultiplication: 1. If , accumulate . 2. Right-shift one bit. 3. If , exit. 4. Left-shift one bit. 5. Loop.

### Multiplication

In simple algebra, multiplication is the process of calculating the result when a number is taken times. The result of a multiplication is called the product of and , and each of the numbers and is called a factor of the product . Multiplication is denoted , , , or simply . The symbol is known as the multiplication sign. Normal multiplication is associative, commutative, and distributive.More generally, multiplication can also be defined for other mathematical objects such as groups, matrices, sets, and tensors.Karatsuba and Ofman (1962) discovered that multiplication of two digit numbers can be done with a bit complexity of less than using an algorithm now known as Karatsuba multiplication.Eddy Grant's pop song "Electric Avenue" (Electric Avenue, 2001) includes the commentary: "Who is to blame in one country; Never can get to the one; Dealin' in multiplication; And they still can't feed everyone, oh no."..

### Division Lemma

When is divisible by a number that is relatively prime to , then must be divisible by .

### Division by Zero

Division by zero is the operation of taking the quotient of any number and 0, i.e., . The uniqueness of division breaks down when dividing by zero, since the product is the same for any , so cannot be recovered by inverting the process of multiplication. 0 is the only number with this property and, as a result, division by zero is undefined for real numbers and can produce a fatal condition called a "division by zero error" in computer programs.To the persistent but misguided reader who insists on asking "What happens if I do divide by zero," Derbyshire (2004, p. 36) provides the slightly flippant but firm and concise response, "You can't. It is against the rules." Even in fields other than the real numbers, division by zero is never allowed (Derbyshire 2004, p. 266).There are, however, contexts in which division by zero can be considered as defined. For example, division by zero for in the extended complex..

### Division

Taking the ratio of two numbers and , also written . Here, is called the dividend, is called the divisor, and is called a quotient. The symbol "/" is called a solidus (sometimes, the "diagonal"), and the symbol "" is called the obelus. If left unevaluated, is called a fraction, with known as the numerator and known as the denominator.Division in which the fractional (remainder) is discarded is called integer division, and is sometimes denoted using a backslash, .Division is the inverse operation of multiplication,so that ifthen can be recovered asas long as . In general, division by zero is not defined since the ability to "invert" to recover breaks down if (in which case is always 0, independent of ).Cutting or separating an object into two or more parts is also called division...

### Remainder

In general, a remainder is a quantity "left over" after performing a particular algorithm. The term is most commonly used to refer to the number left over when two integers are divided by each other in integer division. For example, , with a remainder of 6. Of course in real division, there is no such thing as a remainder since, for example, .The term remainder is also sometimes applied to the residueof a congruence.

### Divide

To divide is to perform the operation of division, i.e., to see how many times a divisor goes into another number . divided by is written or . The result need not be an integer, but if it is, some additional terminology is used. is read " divides " and means that is a divisor of . In this case, is said to be divisible by . Clearly, and . By convention, for every except 0 (Hardy and Wright 1979, p. 1). The "divisibility" relation satisfies(1)(2)(3)where the symbol means implies. is read " does not divide " and means that is not a divisor of . means divides exactly. If and are relatively prime, the notation or sometimes is used.

### Long Division

Long division is an algorithm for dividing two numbers, obtaining the quotient one digit at a time. The example above shows how the division of 123456/17 is performed to obtain the result 7262.11....The term "long division" is also used to refer to the method of dividing one polynomial by another, as illustrated above. This example illustrates the resultThe symbol separating the dividend from the divisor seems to have no established name, so can be simply referred to as the long division symbol (or sometimes the division bracket).The chorus of the song "Singular Girl" by Rhett Miller (The Believer, 2006) contains the slightly cryptic line "Talking to you girl is like long division, yeah." Coincidentally, Long Division (1995) is also the name of the second album by the band Low...

### Lattice Method

The lattice method is an alternative to long multiplication for numbers. In this approach, a lattice is first constructed, sized to fit the numbers being multiplied. If we are multiplying an -digit number by an -digit number, the size of the lattice is . The multiplicand is placed along the top of the lattice so that each digit is the header for one column of cells (the most significant digit is put at the left). The multiplier is placed along the right side of the lattice so that each digit is a (trailing) header for one row of cells (the most significant digit is put at the top). Illustrated above is the lattice configuration for computing .Before the actual multiplication can begin, lines must be drawn for every diagonal path in the lattice from upper right to lower left to bisect each cell. There will be 5 diagonals for our lattice array.Now we calculate a product for each cell by multiplying the digit at the top of the column and the digit at the right of the..

### Vector Division

In general, there is no unique matrix solution to the matrix equationEven in the case of parallel to , there are still multiple matrices that perform this transformation. For example, given , all the following matrices satisfy the above equation:Therefore, vector division cannot be uniquely defined in terms of matrices.However, if the vectors are represented by complex numbers or quaternions, vector division can be uniquely defined using the usual rules of complex division and quaternion algebra, respectively.

### Complex Division

The division of two complex numbers can be accomplished by multiplying the numerator and denominator by the complex conjugate of the denominator, for example, with and , is given by(1)(2)(3)(4)(5)where denotes the complex conjugate. In component notation with ,(6)

### Reciprocal

The reciprocal of a real or complex number is its multiplicative inverse , i.e., to the power . The reciprocal of zero is undefined. A plot of the reciprocal of a real number is plotted above for .Two numbers are reciprocals if and only if their product is 1. To put it another way, a number and its reciprocal are inversely related. Therefore, the larger a (positive) number, the smaller its reciprocal.The reciprocal of a complex number is given byPlots of the reciprocal in the complex plane are given above.Given a geometric figure consisting of an assemblage of points, the polars with respect to an inversion circle constitute another figure. These figures are said to be reciprocal with respect to each other. Then there exists a duality principle which states that theorems for the original figure can be immediately applied to the reciprocal figure after suitable modification (Lachlan 1893)...

### Long Multiplication

Long multiplication is the method of multiplication that is commonly taught to elementary school students throughout the world. It can be used on two numbers of arbitrarily large size or number of decimal digits. The numbers to be multiplied are placed vertically over one another with their least significant digits aligned. The top number is named the multiplicand and the lower number is the multiplier. The result of the multiplication is the product.For example, we can multiply . The number with more digits is usually selected as the multiplicand: The long multiplication algorithm starts with multiplying the multiplicand by the least significant digit of the multiplier to produce a partial product, then continuing this process for all higher order digits in the multiplier. Each partial product is right-aligned with the corresponding digit in the multiplier. The partial products are then summed: Implicit in using this method is the following..

### Multiple

A multiple of a number is any quantity with an integer. If and are integers, then is called a factor of .

### Steffi Problem

A homework problem proposed in Steffi's math class in January 2003 asked students to prove that no ratio of two unequal numbers obtained by permuting all the digits 1, 2, ..., 7 results in an integer. If such a ratio existed, then some permutation of 1234567 would have to be divisible by . can immediately be restricted to , since a ratio of two permutations of the first seven digits must be less than , and the permutations were stated to be unequal, so . The case can be eliminated by the divisibility test for 3, which says that a number is divisible by 3 iff the sum of its digits is divisible by 3. Since the sum of the digits 1 to 7 is 28, which is not divisible by 3, there is no permutation of these digits that is divisible by 3. This also eliminates as a possibility, since a number must be divisible by 3 to be divisible by 6.This leaves only the cases , 4, and 5 to consider. The case can be eliminated by noting that in order to be divisible by 5, the last digits of the numerator..

### Quotient

The term "quotient" is most commonly used to refer to the ratio of two quantities and , where .Less commonly, the term quotient is also used to mean the integer part of such a ratio. In the Wolfram Language, the command Quotient[r, s] is defined in this latter sense, returningwhere is the floor function. This is sometimes called integer division.Since usage concerning fractional part/value and integer part/value can be confusing, the following table gives a summary of names and notations used. Here, S&O indicates Spanier and Oldham (1987).notationnameS&OGraham et al. Wolfram Languageceiling function--ceiling, least integerCeiling[x]congruence----Mod[m, n]floor functionfloor, greatest integer, integer partFloor[x]fractional valuefractional part or SawtoothWave[x]fractional partno nameFractionalPart[x]integer partno nameIntegerPart[x]nearest integer function----Round[x]quotient----Quotient[m,..

### Product

The term "product" refers to the result of one or more multiplications. For example, the mathematical statement would be read " times equals ," where is the product.More generally, it is possible to take the product of many different kinds of mathematical objects, including those that are not numbers. For example, the product of two sets is given by the Cartesian product. In topology, the product of spaces can be defined by using the product topology. The product of two groups, vector spaces, or modules is given by the direct product. In category theory, the product of objects is given using the category product.The product symbol is defined by(1)Useful product identities include(2)(3)

### Polynomial Remainder

The remainder obtained when dividing a polynomial by another polynomial . The polynomial remainder is implemented in the Wolfram Language as PolynomialRemainder[p, q, x], and is related to the polynomial quotient byFor example, the polynomial remainder of and is , corresponding to polynomial quotient .

### Polynomial Quotient

The quotient of two polynomials and , discarding any polynomial remainder. Polynomial quotients are implemented in the Wolfram Language as PolynomialQuotient[p, q, x], and are related to the polynomial remainder byFor example, the polynomial quotient of and is , leaving remainder .

### Synthetic Division

Synthetic division is a shortcut method for dividing two polynomials which can be used in place of the standard long division algorithm. This method reduces the dividend and divisor polynomials into a set of numeric values. After these values are processed, the resulting set of numeric outputs is used to construct the polynomial quotient and the polynomial remainder.For an example of synthetic division, consider dividing by . First, if a power of is missing from either polynomial, a term with that power and a zero coefficient must be inserted into the correct position in the respective polynomial. In this case the term is missing from the dividend while the term is missing from the divisor; therefore, is added between the quintic and the cubic terms of the dividend while is added between the cubic and the linear terms of the divisor:(1)and(2)respectively.Next, all the variables and their exponents () are removed from the dividend, leaving instead..

### Ruffini's Rule

Ruffini's rule a shortcut method for dividing a polynomial by a linear factor of the form which can be used in place of the standard long division algorithm. This method reduces the polynomial and the linear factor into a set of numeric values. After these values are processed, the resulting set of numeric outputs is used to construct the polynomial quotient and the polynomial remainder.Note that Ruffini's rule is a special case of the more generalized notion of synthetic division in which the divisor polynomial is a monic linear polynomial. Confusingly, Ruffini's rule is sometimes referred to as synthetic division, thus leading to the common misconception that the scope of synthetic division is significantly smaller than that of the long division algorithm.For an example of Ruffini's rule, consider divided by . First, if a power of is missing from the dividend, a term with that power and a zero coefficient must be inserted into the correct position..

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