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For example, 4 multiplied by 3, often written as and spoken as "3 times 4", can be calculated by adding 3 copies of 4 together: 3 × 4 = 4 + 4 + 4 = 12. {\displaystyle 3\times 4=4+4+4=12.} Here, 3 (the multiplier ) and 4 (the multiplicand ) are the factors , and 12 is the product .
Adding 4 hours to 9 o'clock gives 1 o'clock, since 13 is congruent to 1 modulo 12. In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones ...
n. In modular arithmetic, the integers coprime (relatively prime) to n from the set of n non-negative integers form a group under multiplication modulo n, called the multiplicative group of integers modulo n. Equivalently, the elements of this group can be thought of as the congruence classes, also known as residues modulo n, that are coprime to n.
For example, to multiply 7 and 15 modulo 17 in Montgomery form, again with R = 100, compute the product of 3 and 4 to get 12 as above. The extended Euclidean algorithm implies that 8⋅100 − 47⋅17 = 1, so R′ = 8. Multiply 12 by 8 to get 96 and reduce modulo 17 to get 11. This is the Montgomery form of 3, as expected.
The Karatsuba algorithm is a fast multiplication algorithm. It was discovered by Anatoly Karatsuba in 1960 and published in 1962. [1][2][3] It is a divide-and-conquer algorithm that reduces the multiplication of two n -digit numbers to three multiplications of n /2-digit numbers and, by repeating this reduction, to at most single-digit ...
A multiplication algorithm is an algorithm (or method) to multiply two numbers. Depending on the size of the numbers, different algorithms are more efficient than others. Numerous algorithms are known and there has been much research into the topic. The oldest and simplest method, known since antiquity as long multiplication or grade-school ...
Order of operations. In mathematics and computer programming, the order of operations is a collection of rules that reflect conventions about which operations to perform first in order to evaluate a given mathematical expression. These rules are formalized with a ranking of the operations. The rank of an operation is called its precedence, and ...
Chisanbop or chisenbop (from Korean chi (ji) finger + sanpŏp (sanbeop) calculation [1] 지산법/指算法), sometimes called Fingermath, [2] is a finger counting method used to perform basic mathematical operations. According to The Complete Book of Chisanbop[3] by Hang Young Pai, chisanbop was created in the 1940s in Korea by Sung Jin Pai ...