
# 2.3: The Associative Property of Addition and Multiplication


(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.