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The action of repeated [[addition]]s: $a\times b=\underbrace{a+a+\cdots+a}_{\hbox{$b$ times}}$. | The action of repeated [[addition]]s: $a\times b=\underbrace{a+a+\cdots+a}_{\hbox{$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 $\times$ ($\LaTeX$ <tt>times</tt>, 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'': | + | 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 $\times$ ($\hbox{\LaTeX}$ <tt>times</tt>, 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'': |
$$(a+b)(c+d)=ac+ad+bc+bd\,.$$ | $$(a+b)(c+d)=ac+ad+bc+bd\,.$$ | ||
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 $\times$ ($\hbox{\LaTeX}$ 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.