associativity of multiplication

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

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

of multiplication in arithmetics and elementary algebra:

A product ($b\mathrm{\cdot }c$) 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

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

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

Title	associativity of multiplication
Canonical name	AssociativityOfMultiplication
Date of creation	2013-03-22 15:09:22
Last modified on	2013-03-22 15:09:22
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	5
Author	pahio (2872)
Entry type	Application
Classification	msc 12D99
Classification	msc 00A35