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Mathematics LibreTexts

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(43)=(54)3 a true statement?

No, because

5(43)=51=4 and (54)3=13=2.

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

Example 2.3.1:

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(43)=5(12)=60 and (54)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.

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

Example 2.3.1:

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÷(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.


2.3: The Associative Property of Addition and Multiplication is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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