In arithmetic operations it holds the addend, subtrahend, multiplicand, or divisor.
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Multiplication of numbers 0–10. Line labels = multiplicand. X axis = multiplier. Y axis = product.
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The next step is to arrange the multiplier and the multiplicand above the partial products.
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The Multiplier is the number denoting how many times the multiplicand is to be taken.
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Let the task be to multiply a multiplicand of four figures by a multiplier of three.
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Ores are the multiplicand and man the multiplier in the product which represents value or availability.
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That's why the drums are so placed that they engage the multiplicand gears one after the other.
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To multiply by 10, all you have to do is to put a cipher after the multiplicand.
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Therefore zeros in the multiplier would cause a corresponding change of position in the figures of the multiplicand.
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Lay down the multiplicand in the upper row and the multiplier in the lower, with the product in between.
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The most expensive arithmetic operation in ECC is division, which is performed by multiplying the inverse of a multiplicand.
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In short, man is the multiplier and the mineral substance is the multiplicand in the product known as value.
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Suppose for simplicity that the multiplicand has three digits. The leftmost digit is the hundreds-digit, the middle digit is the tens-digit, and the 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.
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This neuron will get a special activation function, and its output is connected as a multiplicand to the output neuron.
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After the multiplicand was multiplied by each digit of the multiplier and the sum accumulated with appropriate shifting, variable Sum contained the product.
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Similarly, when he was instructed to swap the two numbers, the multiplier and the multiplicand, he could not fathom how that could be done.
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In this paper, a novel architecture of Vedic multiplier with 'Urdhava-tiryakbhyam' methodology for 16 bit multiplier and multiplicand is proposed with the use of compressor adders.
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If both first and second number each have only one digit, then their product is given in the multiplication table — thereby making the multiplication algorithm is unnecessary. Then comes the tens-column. The tens- column so far contains only one digit: the tens-digit of the multiplicand (though it might contain a carry digit under the line). Find the product of the multiplier and the tens-digits of the multiplicand.
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The first gear is connected to the cursor by a yoke and can be shifted along the square axle as a function of the multiplicand setting by the operator.
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38x76=2888 al Uqlidis (952 AD)multiplication, a variation of Sunzi multiplication Sunzi Suanjing described in detail the algorithm of multiplication. On the left are the steps to calculate 38×76: #Place the multiplicand on top, the multiplier on bottom. Line up the units place of the multiplier with the highest place of the multiplicand. Leave room in the middle for recording. #Start calculating from the highest place of the multiplicand (in the example, calculate 30×76, and then 8×76). Using the multiplication table 3 times 7 is 21. Place 21 in rods in the middle, with 1 aligned with the tens place of the multiplier (on top of 7). Then, 3 times 6 equals 18, place 18 as it is shown in the image.
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If there is a carry digit (carried over from the tens- column) then write it in the hundreds-column under the line, and the algorithm is finished. When the algorithm finishes, the number under the line is the product of the two numbers. If the multiplicand has a hundreds-digit, find the product of the multiplier and the hundreds-digit of the multiplicand, and to this product add the carry digit if there is one. Then write the resulting sum of the hundreds-column under the line, also in the hundreds column.
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A grid is drawn up, and each cell is split diagonally. The two multiplicands of the product to be calculated are written along the top and right side of the lattice, respectively, with one digit per column across the top for the first multiplicand (the number written left to right), and one digit per row down the right side for the second multiplicand (the number written top-down). Then each cell of the lattice is filled in with product of its column and row digit. As an example, consider the multiplication of 58 with 213.
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Booth's algorithm examines adjacent pairs of bits of the 'N'-bit multiplier Y in signed two's complement representation, including an implicit bit below the least significant bit, y−1 = 0. For each bit yi, for i running from 0 to N − 1, the bits yi and yi−1 are considered. Where these two bits are equal, the product accumulator P is left unchanged. Where yi = 0 and yi−1 = 1, the multiplicand times 2i is added to P; and where yi = 1 and yi−1 = 0, the multiplicand times 2i is subtracted from P. The final value of P is the signed product.
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The promptuary, also known as the 'card abacus' is a calculating machine invented by the 16th-century Scottish mathematician John Napier and described in his book Rabdologiae in which he also described Napier's bones. It is an extension of Napier's Bones, using two sets of rods to achieve multi-digit multiplication without the need to write down intermediate results, although some mental addition is still needed to calculate the result. The rods for the multiplicand are similar to Napier's Bones, with repetitions of the values. The set of rods for the multiplier are shutters or masks for each digit placed over the multiplicand rods.
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When performing any of these multiplication algorithms the following "steps" should be applied. The answer must be found one digit at a time starting at the least significant digit and moving left. The last calculation is on the leading zero of the multiplicand. Each digit has a neighbor, i.e.
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With the 3 in the multiplicand multiplied totally, take the rods off. #Move the multiplier one place to the right. Change 7 to horizontal form, 6 to vertical. #8×7 = 56, place 56 in the second row in the middle, with the units place aligned with the digits multiplied in the multiplier.
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On paper, write down in one column the numbers you get when you repeatedly halve the multiplier, ignoring the remainder; in a column beside it repeatedly double the multiplicand. Cross out each row in which the last digit of the first number is even, and add the remaining numbers in the second column to obtain the product.
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Take 7 out of the multiplier since it has been multiplied. #8×6 = 48, 4 added to the 6 of the last step makes 10, carry 1 over. Take off 8 of the units place in the multiplicand, and take off 6 in the units place of the multiplier. #Sum the 2380 and 508 in the middle, which results in 2888: the product.
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Start with the ones-column. Find the product of the ones-multiplier and the multiplicand and write it down in a row under the line, aligning the digits of the product in the previously-defined columns. If the product has four digits, then the first digit will be the beginning of the thousands-column. Let this product be called the "ones-row".
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By arranging these rulers in the proper order, the user can solve multiplication problems. Consider multiplying 52749 by 4. Five rulers, one for each digit of 52749, are arranged side-by-side, next to the "index" ruler: File:Genaille-Lucas rulers example 1.png The second multiplicand is 4, so we look at the fourth row: File:Genaille-Lucas rulers example 2.
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This is a variation of peasant multiplication. In base 2, long multiplication reduces to a nearly trivial operation. For each '1' bit in the multiplier, shift the multiplicand by an appropriate amount, and then sum the shifted values. In some processors, it is faster to use bit shifts and additions rather than multiplication instructions, especially if the multiplier is small or always same.
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Consider a multiplication where one of the factors has multiple digits, whereas the other factor has only one digit. Write down the multi-digit factor, then write the single-digit factor under the rightmost digit of the multi-digit factor. Draw a horizontal 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.
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The operation of each one of such multiplications was already described in the previous multiplication algorithm, so this algorithm will not describe each one individually, but will only describe how the several multiplications with one-digit multipliers shall be coordinated. The second part will add up all the subproducts of the first part, and the resulting sum will be the product. First part. Let the first factor be called the multiplicand.
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The result is obtained by adding the numbers from the second column for which the corresponding power of two makes up part of the decomposition of the first multiplicand. In the example above, as 25 = 16 + 8 + 1, add the corresponding multiples of 7 to get 25⋅7 = 112 + 56 + 7 = 175. The main advantage of this technique is that it makes use of only addition, subtraction, and multiplication by two.
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The machine had an accumulator and a multiplier register which were arranged to allow double length multiplication. As the multiplicand was repeatedly added to the product, it got longer while the multiplier got shorter so that the product could fill the accumulator and then continue into the multiplier register. Multiplication took up to 640ms for a 32-bit multiplier, which needed 32 drum accesses. Subsequently the design was enhanced with larger capacity drums.
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To perform a single addition or subtraction, the multiplier is simply set at one. To multiply by numbers over 9: #The multiplicand is set into the operand dials. #The first (least significant) digit of the multiplier is set into the multiplier dial as above, and the crank is turned, multiplying the operand by that digit and putting the result in the accumulator. #The input section is shifted one digit to the left with the end crank.
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The inner product on the Leech lattice is defined as 1/8 the sum of the products of respective co-ordinates of the two multiplicand vectors; it is an integer. The square norm of a vector is its inner product with itself, always an even integer. It is common to speak of the type of a Leech lattice vector: half the square norm. Subgroups are often named in reference to the types of relevant fixed points.
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The floating-point register file is read and the data formatted into fraction, exponent, and sign in stage four. If executing add instructions, the adder calculates the exponent difference, and a predictive leading one or zero detector using input operands for normalizing the result is initiated. If executing multiply instructions, a 3 X multiplicand is generated. In stages five and six, alignment or a normalization shift and sticky-bit calculations are performed for adds and subtracts.
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The second Egyptian multiplication and division technique was known from the hieratic Moscow and Rhind Mathematical Papyri written in the seventeenth century B.C. by the scribe Ahmes. Although in ancient Egypt the concept of base 2 did not exist, the algorithm is essentially the same algorithm as long multiplication after the multiplier and multiplicand are converted to binary. The method as interpreted by conversion to binary is therefore still in wide use today as implemented by binary multiplier circuits in modern computer processors.
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The Baby had included two registers, implemented as Williams tubes: the accumulator (A) and the program counter (C). As A and C had already been assigned, the tube holding the two index registers, originally known as B-lines, was given the name B. The contents of the registers could be used to modify program instructions, allowing convenient iteration through an array of numbers stored in memory. The Mark 1 also had a fourth tube, (M), to hold the multiplicand and multiplier for a multiplication operation.
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Thomas started to work on his machine in 1818 while serving in the French Army where he had to do a great deal of calculations. He made use of principles from previous mechanical calculators like the stepped reckoner of Leibniz and Pascal's calculator. He patented it on November 18, 1820. This machine implemented a true multiplication where, by just pulling on a ribbon, the multiplicand entered on the input sliders was multiplied by a one-digit multiplier number and it used the method for subtracting.
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Functional schematic showing the Williams tubes in green. Tube C holds the current instruction and its address; A is the accumulator; M is used to hold the multiplicand and the multiplier for a multiply operation; and B contains the index registers, used to modify instructions. The Baby had been designed by the team of Frederic C. Williams, Tom Kilburn and Geoff Tootill. To develop the Mark 1 they were joined by two research students, D. B. G. Edwards and G. E. Thomas; work began in earnest in August 1948.
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Then, if there is a carry digit (superscripted, under the line and in the tens-column), add it to this product. If the resulting sum is less than ten then write it in the tens- column under the line. If the sum has two digits then write its last digit in the tens-column under the line, and carry its first digit over to the next column: in this case the hundreds column. If the multiplicand does not have a hundreds-digit then if there is no carry digit then the multiplication algorithm has finished.
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Next, the hundreds-column. Find the product of the hundreds-multiplier and the multiplicand and write it down in a row—call it the "hundreds-row"—under the tens-row, but shifted one more column to the left. That is, the ones-digit of the hundreds-row will be in the hundreds-column; the tens-digit of the hundreds-row will be in the thousands- column; the hundreds-digit of the hundreds-row will be in the ten-thousands- column. If the hundreds-row has four digits, then the first digit will be the beginning of the hundred-thousands-column.
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Then the tens-column. Find the product of the tens-multiplier and the multiplicand and write it down in a row—call it the "tens-row"—under the ones-row, but shifted one column to the left. That is, the ones-digit of the tens-row will be in the tens-column of the ones-row; the tens-digit of the tens-row will be under the hundreds-digit of the ones- row; the hundreds-digit of the tens-row will be under the thousands-digit of the ones-row. If the tens-row has four digits, then the first digit will be the beginning of the ten-thousands-column.
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In mathematics, ancient Egyptian multiplication (also known as Egyptian multiplication, Ethiopian multiplication, Russian multiplication, or peasant multiplication), one of two multiplication methods used by scribes, was a systematic method for multiplying two numbers that does not require the multiplication table, only the ability to multiply and divide by 2, and to add. It decomposes one of the multiplicands (preferably the smaller) into a sum of powers of two and creates a table of doublings of the second multiplicand. This method may be called mediation and duplation, where mediation means halving one number and duplation means doubling the other number. It is still used in some areas.
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The Montgomery ladder approach computes the point multiplication in a fixed amount of time. This can be beneficial when timing or power consumption measurements are exposed to an attacker performing a side-channel attack. The algorithm uses the same representation as from double-and-add. R0 ← 0 R1 ← P for i from m downto 0 do if di = 0 then R1 ← point_add(R0, R1) R0 ← point_double(R0) else R0 ← point_add(R0, R1) R1 ← point_double(R1) return R0 This algorithm has in effect the same speed as the double-and-add approach except that it computes the same number of point additions and doubles regardless of the value of the multiplicand d.
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The large rectangle is composed of 20 squares, each having dimensions of 1 by 1. Area of a cloth ; Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction and division. The result of a multiplication operation is called a product. The multiplication of whole numbers may be thought of as a repeated addition; that is, the multiplication of two numbers is equivalent to adding as many copies of one of them, the multiplicand, as the quantity of the other one, the multiplier.
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Public-key cryptography is based on the intractability of certain mathematical problems. Early public-key systems based their security on the assumption that it is difficult to factor a large integer composed of two or more large prime factors. For later elliptic-curve- based protocols, the base assumption is that finding the discrete logarithm of a random elliptic curve element with respect to a publicly known base point is infeasible: this is the "elliptic curve discrete logarithm problem" (ECDLP). The security of elliptic curve cryptography depends on the ability to compute a point multiplication and the inability to compute the multiplicand given the original and product points.
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If a positional numeral system is used, a natural way of multiplying numbers is taught in schools as long multiplication, sometimes called grade-school multiplication, sometimes called Standard Algorithm: multiply the multiplicand by each digit of the multiplier and then add up all the properly shifted results. It requires memorization of the multiplication table for single digits. This is the usual algorithm for multiplying larger numbers by hand in base 10. Computers initially used a very similar shift and add algorithm in base 2, but modern processors have optimized circuitry for fast multiplications using more efficient algorithms, at the price of a more complex hardware realization.
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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 two digits: write this product under the line and in the ones-column. If the product has two digits, then write down only the ones-digit of the product. Write the "carry digit" as a superscript of the yet-unwritten digit in the next column and under the line: in this case the next column is the tens-column, so write the carry digit as the superscript of the yet-unwritten tens-digit of the product (under the line).
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Kochanski multiplication is an algorithm that allows modular arithmetic (multiplication or operations based on it, such as exponentiation) to be performed efficiently when the modulus is large (typically several hundred bits). This has particular application in number theory and in cryptography: for example, in the RSA cryptosystem and Diffie–Hellman key exchange. The most common way of implementing large-integer multiplication in hardware is to express the multiplier in binary and enumerate its bits, one bit at a time, starting with the most significant bit, perform the following operations on an accumulator: #Double the contents of the accumulator (if the accumulator stores numbers in binary, as is usually the case, this is a simple "shift left" that requires no actual computation). #If the current bit of the multiplier is 1, add the multiplicand into the accumulator; if it is 0, do nothing.
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The representations of the multiplicand and product are not specified; typically, these are both also in two's complement representation, like the multiplier, but any number system that supports addition and subtraction will work as well. As stated here, the order of the steps is not determined. Typically, it proceeds from LSB to MSB, starting at i = 0; the multiplication by 2i is then typically replaced by incremental shifting of the P accumulator to the right between steps; low bits can be shifted out, and subsequent additions and subtractions can then be done just on the highest N bits of P. There are many variations and optimizations on these details. The algorithm is often described as converting strings of 1s in the multiplier to a high-order +1 and a low-order −1 at the ends of the string.
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Detail of an arithmometer built before 1851. The one-digit multiplier cursor (ivory top) is the leftmost cursor The arithmometers of this period were four-operation machines; a multiplicand inscribed on the input sliders could be multiplied by a single-digit multiplier by simply pulling on a ribbon (quickly replaced by a crank handle). It was a complicated designScientific American, Volume 5, Number 1, page 92, September 22, 1849 and very few machines were built. Additionally, no machines were built between 1822 and 1844. This hiatus of 22 years coincides almost exactly with the period of time during which the British government financed the design of Charles Babbage's difference engine, which on paper was far more sophisticated than the arithmometer, but wasn’t finished at this time.The British Parliament financed this project from 1822 to 1842 (James Essinger, Jacquard's Web, pages 77 & 102–106, Oxford University Press, 2004).
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Based on the Manchester Mark 1, which was designed at the University of Manchester by Freddie Williams and Tom Kilburn, the machine was built by Ferranti of the United Kingdom. The main improvements over it were in the size of the primary and secondary storage, a faster multiplier, and additional instructions. The Mark 1 used a 20-bit word stored as a single line of dots of electric charges settled on the surface of a Williams tube display, each cathodic tube storing 64 lines of dots. Instructions were stored in a single word, while numbers were stored in two words. The main memory consisted of eight tubes, each storing one such page of 64 words. Other tubes stored the single 80-bit accumulator (A), the 40-bit "multiplicand/quotient register" (MQ) and eight "B-lines", or index registers, which was one of the unique features of the Mark 1 design.
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An associative algebra over K is given by a K-vector space A endowed with a bilinear map A × A → A having two inputs (multiplicator and multiplicand) and one output (product), as well as a morphism K → A identifying the scalar multiples of the multiplicative identity. If the bilinear map A × A → A is reinterpreted as a linear map (i. e., morphism in the category of K-vector spaces) A ⊗ A → A (by the universal property of the tensor product), then we can view an associative algebra over K as a K-vector space A endowed with two morphisms (one of the form A ⊗ A → A and one of the form K → A) satisfying certain conditions that boil down to the algebra axioms. These two morphisms can be dualized using categorial duality by reversing all arrows in the commutative diagrams that describe the algebra axioms; this defines the structure of a coalgebra.
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