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Mathematics LibreTexts

2.3: The Associative Property of Addition and Multiplication

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    13969

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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 \(\PageIndex{1}\):

    Add text here. For the automatic number to work, you need to add the "AutoNum" template (preferably at the end) to the page.

    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.

    ***************************************************

    Example \(\PageIndex{1}\):

    Add text here. For the automatic number to work, you need to add the "AutoNum" template (preferably at the end) to the page.

    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.