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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 commutative and associative: $$a \times b = b \times a$$ and $$(a \times b) \times c = a \times (b \times c)$$. Repeated multiplication is called exponentiation.

In googology, it is the second hyper operator

## Other properties Edit

• $$0 \times n = 0$$
• $$1 \times n = n$$
• $$(-a) \times (-b) = a \times b$$
• $$(-a) \times b = a \times (-b) = -(a \times b)$$

## Turing machine code Edit

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