Multiplication

Multiplication is an elementary binary operation, written \(ab\), \(a \times b\), \(a(b)\) or \(a \cdot b\) (pronounced "\(a\) times \(b\)"). For natural numbers, it is defined as repeated addition:

\[a \times b = \underbrace{a + a + \cdots + a + a}_b.\]

For example, \(3 \times 4 = 3 + 3 + 3 + 3 = 12\). The result of a multiplication problem is called the product.

In certain arithmetics like Peano arithmetic, multiplication is and  on \(\mathbb{N}\) and \(\mathbb{R}\): \(a \times b = b \times a\) and \((a \times b) \times c = a \times (b \times c)\) for all natural and real values of \(a,b\) and \(c\). This applies in certain aRepeated multiplication is called exponentiation.

However, multiplication is not commutative on ordinals. For any limit ordinal \(\alpha\), \(2 \times \alpha = \alpha \neq \alpha \times 2\).

In googology, it is the second hyper operator.

Other properties

 * \(0 \times n = 0\)
 * \(1 \times n = n\)
 * \((-a) \times (-b) = a \times b\)
 * \((-a) \times b = a \times (-b) = -(a \times b)\)

Turing machine code
Given input of form (string of a 1's) (string of b 1's) it outputs string of a*b 1's

0 1 _ r 1 0 _ _ r 9 1 1 1 r 1 1 _ _ r 2 2 1 _ r 3 2 _ _ l 7 3 1 1 r 3 3 _ _ r 4 4 1 1 r 4 4 _ 1 l 5 5 1 1 l 5 5 _ _ l 6 6 1 1 l 6 6 _ 1 r 2 7 1 1 l 7 7 _ _ l 8 8 1 1 l 8 8 _ _ r 0 9 1 _ r 9 9 _ _ r halt