# Multiplication

The action of repeated additions: $a\times b=\underbrace{a+a+\cdots+a}_{\hbox{$b$times}}$.

By looking at it from a geometric point of view, it is not difficult—although not trivial either—to see that $a\times b=b\times a$. In typesetting, we do not use $.$ or $\cdot$ and usually omit the multiplication sign $\times$ ($\mathrm{\LaTeX}$ times, not x (ex)), so that we write $ab$. The most important rule of multiplication is how it combines with addition, with the property of distributivity:

$$(a+b)(c+d)=ac+ad+bc+bd\,.$$

An interesting problem regards the complexity, or cost of multiplication. In 1960, Karatsuba improved on the "school method" by finding a clever way to write $(a+b)(c+d)$ as a sum of three products, rather than four.