Multiplication

Multiplication is an elementary binary operation, written \(ab\), \(a \times b\), or \(a \cdot b\) (pronounced "\(a\) times \(b\)"). It is defined as repeated addition: \(a \times b = \underbrace{a + a + \cdots + a + a}_b\). The result of a multiplication problem is called the product.

For example, the product \(3 × 4\) is equal to the sum of four 3s: \(3 + 3 + 3 + 3\).

Like addition, multiplication is and : \(a \times b = b \times a\) and \((a \times b) \times c = a \times (b \times c)\). Repeated multiplication is called exponentiation.

In the fast-growing hierarchy, multiplication grows faster than \(f_1(n)\).

It is the second hyper operator.

Turing machine code
; We remove 1 from a 0 1 _ r 1 1 1 1 r 1 1 _ _ r 2 ; Now we copy b to third string, which at the beginning is empty ; At the same moment, we check if remains of b is one or more 2 _ _ r 2 2 1 _ r 3 3 1 1 r 4 3 _ _ r 9 ; If it is, we must restore b 4 1 1 r 4 4 _ _ r 5 5 1 1 r 5 5 _ 1 l 6 6 1 1 l 6 6 _ _ l 7 7 _ _ l 7 7 1 1 l 8 8 1 1 l 8 8 _ _ r 2 ; Number of blanks between 1st and 3rd string is always b+2 ; We skip first blank and write b+1 ones on tape 9 1 1 r 9 9 _ 1 l 10 10 1 1 l 10 10 _ _ l 11 11 _ 1 l 11 11 1 1 r 12 ; We remove one unneeded 1 12 1 _ l 13 ; We check if remains of a is 1 or not 13 1 1 l 14 14 1 1 l 15 14 _ _ r 16 ; If it is, we are nearly at the end 15 1 1 l 15 15 _ _ r 0 ; Otherwise, we repeat from beginning ; Now tape is 1 (b 1's) (a*(b-1) 1's) ; If we delete leftmost 1 and connect 2nd and 3rd string, we get a*b 16 1 _ r 16 16 _ _ r 17 17 _ _ r 17 17 1 _ r 18 18 1 1 r 18 18 _ 1 l halt

Given a number in specified base, and base which need to convert that number.