# associativity of multiplication

It’s important to know the following interpretation of the associative law

 $\displaystyle a\cdot(b\cdot c)=(a\cdot b)\cdot c$ (1)

of multiplication in arithmetics and elementary algebra:

A product ($b\cdot c$) is multiplied by a number ($a$) so that only one ($b$) of the product is multiplied by that number.

This rule is sometimes violated even in high school 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 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

 $\displaystyle a\cdot(b+c)=a\cdot b+a\cdot c,$ (2)

which two different operations, multiplication and addition; both addends must be multiplied separately.

Title associativity of multiplication AssociativityOfMultiplication 2013-03-22 15:09:22 2013-03-22 15:09:22 pahio (2872) pahio (2872) 5 pahio (2872) Application msc 12D99 msc 00A35