Multiplication

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 ($a^0=1$) and factorial (0!=1).

Multiplication properties

Associative property

$(x\ast y)z=x(y\ast z)$

Commutative property

$x\ast y=y\ast x$

Distributive property

$x(y+z)=xy+xz$

Identity element

$1x=x$

Multiplication by zero

$0x=0$

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:

• Long multiplication: very fast for small factors, but slower than the other methods when the factors are large.
• Karatsuba multiplication: useful for numbers in the range of about 1000-10000 digits.
• Fourier transform multiplication: uses FFT to multiply extremely large numbers faster than the previous methods.
  • Schöhage-Strassen: useful for numbers in the range of about 10000-40000+ digits.
  • Färer: currently of theoretical interest only.