AssociativityOfMultiplication 2005-03-24 17:40:42 2005-03-25 14:51:24 Application associative % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here It's important to know the following interpretation of the associative law \begin{align} a\cdot(b\cdot c) = (a\cdot b)\cdot c \end{align} of multiplication in arithmetics and elementary algebra: {\em A product ($b\cdot c$) is multiplied by a number ($a$) so that only one \PMlinkescapetext{factor} ($b$) of the product is multiplied by that number.} This rule is sometimes violated even in high school \PMlinkescapetext{level. \,A pupil may calculate} e.g. like $$10 \cdot 2.5 \cdot 0.3 = 25 \cdot 3 = 75,$$ which is wrong. \,Or when solving an equation like $$x\cdot\frac{2x-1}{3} = 1$$ one would like to multiply both sides by 3 for removing the denominator, getting perhaps $$3x(2x-1) = 3;$$ then the both \PMlinkescapetext{factors} of left side have incorrectly been multiplied by 3. The reason of such mistakes is very likely that one confuses the associative law with the distributive law; \,cf. (1) with this latter \begin{align} a\cdot(b+c) = a\cdot b+a\cdot c, \end{align} which \PMlinkescapetext{contains} two different operations, multiplication and addition; both {\em addends must be multiplied separately}.