2.3: The Associative Property of Addition and Multiplication

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 $\PageIndex{1}$:

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.