PlanetMath: associativity of multiplication

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associativity of multiplication

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

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

(1)

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

This rule is sometimes violated even in high school level. A pupil may calculate e.g. like

$\displaystyle 10 \cdot 2.5 \cdot 0.3 = 25 \cdot 3 = 75,$

which is wrong. Or when solving an equation like

$\displaystyle x\cdot\frac{2x-1}{3} = 1$

one would like to multiply both sides by 3 for removing the denominator, getting perhaps

$\displaystyle 3x(2x-1) = 3;$

then the both 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

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

(2)

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

"associativity of multiplication" is owned by pahio.

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Cross-references: addition, operations, distributive law, denominator, sides, equation, even, product, algebra, arithmetics, multiplication, associative, interpretation
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This is version 2 of associativity of multiplication, born on 2005-03-24, modified 2005-03-25.
Object id is 6904, canonical name is AssociativityOfMultiplication.
Accessed 1799 times total.

Classification:

AMS MSC:	00A35 (General :: General and miscellaneous specific topics :: Methodology of mathematics, didactics)
	12D99 (Field theory and polynomials :: Real and complex fields :: Miscellaneous)

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