2.3: The Associative Property of Addition and Multiplication
( \newcommand{\kernel}{\mathrm{null}\,}\)
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 \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 2.3.1:
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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⋅3)=5(12)=60 and (5⋅4)⋅3=(20)3=60 also.
The picture illustrates the Associative Property of Multiplication.
In general
\boldmatha(bc)=(ab)c
where a, b, and c are any real numbers.
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Example 2.3.1:
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Is division associative?
Is 16÷(4÷2)=(16÷4)÷2 a true statement?
No, because
16÷(4÷2)=16÷2=16÷2=8
and
(16÷4)÷2=(4)÷2=2.
Division is not associative, the general property (using a, b, and c) does not exist.