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]a^0=1[/math]) and factorial numbers (0!=1).
Multiplication by zero
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.