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Multiplication

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Multiplication is the process of calculating the result when a number a is added to itself b times. The result of a multiplication is called the product of a and b, and each of the numbers is called a factor of the product ab.

Multiplication is denoted a×b,(a)(b), or simply ab. The symbol '×' is known as the multiplication sign.

The result of multiplying no numbers (empty product) is always 1 (the multiplicative identity, see below). The most common occurences are in exponentiation ([math]\displaystyle{ a^0=1 }[/math]) and factorial numbers (0!=1).

Multiplication properties

Associative property

[math]\displaystyle{ (x*y)z=x(y*z) }[/math]

Commutative property

[math]\displaystyle{ x*y=y*x }[/math]

Distributive property

[math]\displaystyle{ x(y+z)=xy+xz }[/math]

Identity element

[math]\displaystyle{ 1x=x }[/math]

Multiplication by zero

[math]\displaystyle{ 0x=0 }[/math]

Multiplication algorithms

When the product fits in a variable supported by the programming language, or in a register when programming in assembler, the multiplication is trivial. However if the largest available register is n bits wide the factors can only be n/2 bits wide each.

Otherwise there are several algorithms used to calculate products, depending on the size of the factors: