2.3: The Associative Property of Addition and Multiplication

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Is addition associative?

(0,5)(5,-3) (10,0)\(5\ +\ \underbrace{( {\boldmath \ 4}\ +\ 3)}\ =\underbrace{(5\ +\ {\boldmath 4}\ )}\ +\ 3\\) (8,-2)

\(4\) is associated (grouped) with \(3\)

(18,-2)

\(4\) is associated (grouped) with \(5\)

(13.4,0.2) (17.4,0.2)

\(5+(4+3)=5+7=12\) and \((5+4)+3=9+3=12\) also.

The picture illustrates the Associative property of addition.

In general \[\fbox{\Large \boldmath a+(b+c)=(a+b)+c}\]
where \(a\), \(b\), and \(c\) are any real numbers.

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Is subtraction associative?

Is \(5-(4-3)=(5-4)-3\) a true statement?

No, because

\(5-(4-3)=5-1=4\) and \((5-4)-3=1-3=-2\).

The general property (using \(a\), \(b\) and \(c\)) does not exist.

Example \(\PageIndex{1}\):

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Is multiplication associative?

(0,5)(7,-5) (10,0)\(5\ \cdot\ \underbrace{( {\boldmath \ 4}\ \cdot\ 3)}\ =\underbrace{(5\ \cdot\ {\boldmath 4}\ )}\ \cdot\ 3\\) (8,-4)

\(4\) is associated (grouped) with \(3\)

(18,-4)

\(4\) is associated (grouped) with \(5\)

(12.8,0.2) (16.5,0.2)

\(5(4\cdot 3)=5(12)=60\) and \((5\cdot 4)\cdot 3=(20)3=60\) also.

The picture illustrates the Associative Property of Multiplication.

In general

\[\boxed{\Large \boldmath a(bc)=(ab)c}\]

where \(a\), \(b\), and \(c\) are any real numbers.

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Example \(\PageIndex{1}\):

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Is division associative?

Is \(16\div (4\div 2)=(16\div 4)\div 2\) a true statement?

No, because
\(16\div (4\div 2)=16\div 2=16\div 2=8\)
and
\((16\div 4)\div 2=(4)\div 2=2\).
Division is not associative, the general property (using \(a\), \(b\), and \(c\)) does not exist.