multiplication

Multiplication is a mathematical operation in which two or more numbers are added up to themselves by a factor of other numbers. For example, $2\times 3=2+2+2=3+3=6$. The numbers may be real, imaginary or complex, they may be integers or fractions. Among real numbers, if an odd number of multiplicands are negative, the overall result is negative; if an even number of multiplicands are negative, the overall result is positive. Two examples: $(-3)\times (-5)=15$; $(-2)\times (-3)\times (-5)=(-30)$.

The usual operator is the cross with its four arms of equal length pointing northeast, northwest, southeast and southwest: $\times$. Other options are the central dot $\cdot$ and the tacit multiplication operator. In many computer programming languages the asterisk is often used as it is almost always available on the keyboard (Shift-8 in most American layouts, as well as dedicated key if the keyboard has a numeric keypad), and this is the operator likely to be used in a computer implementation of a reverse Polish notation calculator. In Mathematica, the space can sometimes function as a multiplication operator, but more experienced users warn novices not to rely on this feature.

Just as with addition, multiplication is commutative: $xyz=xzy=yxz$, etc.

The iterative operator is the Greek capital letter pi:

$$\prod _{i=1}^n{a}_i,$$

which is a compact way of writing $a_1\times a_2\times \mathrm{\dots }\times a_n$.

Multiplication of complex numbers is helped by the following identity: $(a+bi)\times (x+yi)=(ax-by)+(ay+bx)i$. To give three examples: $(17+29i)(11+38i)=-915+965i$ (the result has both real and imaginary parts), $(1+2i)(1-2i)=5$ (the result is a real prime) and $(4+7i)(7+4i)=65i$ (the result has only an imaginary part).