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.

Like addition, multiplication is and  on \(\mathbb{N}\) and \(\mathbb{R}\): \(a \times b = b \times a\) and \((a \times b) \times c = a \times (b \times c)\). Repeated 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