![]() ![]() It was introduced to Europe in 1202 in Fibonacci's Liber Abaci. It requires the preparation of a lattice (a grid drawn on paper) which guides the calculation and separates all the multiplications from the additions. Lattice, or sieve, multiplication is algorithmically equivalent to long multiplication. Finally, sum along the diagonal tracts and carry as needed to get the answer Then, fill in the boxes with tens digits in the top triangles and units digits on the bottom. Main article: Lattice multiplication First, set up the grid by marking its rows and columns with the numbers to be multiplied. For example, there are several ways to multiply by 10 using only bit-shift and addition. Multiplication by a constant and division by a constant can be implemented using a sequence of shifts and adds or subtracts. On currently available processors, a bit-wise shift instruction is faster than a multiply instruction and can be used to multiply (shift left) and divide (shift right) by powers of two. Most currently available microprocessors implement this or other similar algorithms (such as Booth encoding) for various integer and floating-point sizes in hardware multipliers or in microcode. In base two, long multiplication is sometimes called "shift and add", because the algorithm simplifies and just consists of shifting left (multiplying by powers of two) and adding. ![]() Ĭomputers initially used a very similar algorithm to long multiplication 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. Richard Brent used this approach in his Fortran package, MP. When the number becomes too large, we add part of it to the result, or we carry and map the remaining part back to a number that is less than b. Several additions can then be performed before an overflow occurs. A typical solution is to represent the number in a small base, b, such that, for example, 8 b is a representable machine integer. When implemented in software, long multiplication algorithms must deal with overflow during additions, which can be expensive. More formally, multiplying two n-digit numbers using long multiplication requires Θ( n 2) single-digit operations (additions and multiplications). To multiply two numbers with n digits using this method, one needs about n 2 operations. In arbitrary-precision arithmetic, it is common to use long multiplication with the base set to 2 w, where w is the number of bits in a word, for multiplying relatively small numbers. Some chips implement long multiplication, in hardware or in microcode, for various integer and floating-point word sizes. Multiply ( a, b, base ) // Operands containing rightmost digits at index 1 product = // Allocate space for result for b_i = 1 to q // for all digits in b carry = 0 for a_i = 1 to p // for all digits in a product += carry + a * b carry = product / base product = product mod base product = carry // last digit comes from final carry return product Usage in computers Note that the '+=' operator is used to denote sum to existing value and store operation (akin to languages such as Java and C) for compactness. It keeps only one row to maintain the sum which finally becomes the result. In some countries such as Germany, the above multiplication is depicted similarly but with the original product kept horizontal and computation starting with the first digit of the multiplier: 23958233 īelow pseudocode describes the process of above multiplication. This example uses long multiplication to multiply 23,958,233 (multiplicand) by 5,830 (multiplier) and arrives at 139,676,498,390 for the result (product). A person doing long multiplication on paper will write down all the products and then add them together an abacus-user will sum the products as soon as each one is computed. This is the usual algorithm for multiplying larger numbers by hand in base 10. It requires memorization of the multiplication table for single digits. ![]() Multiply the multiplicand by each digit of the multiplier and then add up all the properly shifted results. If a positional numeral system is used, a natural way of multiplying numbers is taught in schoolsĪs long multiplication, sometimes called grade-school multiplication, sometimes called the Standard Algorithm: Efficient multiplication algorithms have existed since the advent of the decimal system. Depending on the size of the numbers, different algorithms are more efficient than others. ( August 2022) ( Learn how and when to remove this template message)Ī multiplication algorithm is an algorithm (or method) to multiply two numbers. Use the lead layout guide to ensure the section follows Wikipedia's norms and is inclusive of all essential details. The lead section of this article may need to be rewritten. ![]()
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