11,328
pages

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.

For real numbers, the product $$x \times y$$ is defined as the product of the equivalence classes of Cauchy sequences of rational numbers. For positive reals $$x$$ and $$y$$, it can be thought as a product of width and height of a rectangle.

Like addition, multiplication is commutative and associative 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 $$c$$. Repeated multiplication is called exponentiation.

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

It is possible to define multiplication for natural numbers in the recursive way:

\begin{eqnarray*} a \times 0 & = & 0 \\ a \times (b+1) & = & (a \times b) + a \end{eqnarray*}

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


## Dependency of properties on arithmetics

When we work in ZFC set theory, properties of multiplication such as commutativity and associativity hold. However, in Robinson arithmetic, the commutativity does not necessarily hold. That is, the statement $$\forall a,b \in \mathbb{N} : a \times b = b \times a$$ is independent of Robinson arithmetic.