Associative Property of Addition

Intuitively, we understand, that it is correct.

Indeed, suppose your friend Ann has 3 apples, Bob has 5 apples and Cliff has 4 apples.

Bob and Cliff give you their apples, then Ann gives you her apples: $$${3}+{\left({5}+{4}\right)}={3}+{9}={12}$$$ $$$\left({a}+{\left({b}+{c}\right)}\right)$$$.

But, what if Ann and Bob gave you their apples, and then Cliff gave you his apples?

You will get the same number of apples: $$${\left({3}+{5}\right)}+{4}={8}+{4}={12}$$$ $$$\left({\left({a}+{b}\right)}+{c}\right)$$$.

So, regardless of order, you will get all 12 apples of your friends.

However, associative property of addition works for negative numbers (in fact, for real numbers) as well.

Example 1. $$${5}+{\left({3}+{\left(-{4}\right)}\right)}={\left({5}+{3}\right)}+{\left(-{4}\right)}={4}$$$.

Example 2. $$${\left(-{5.89}\right)}+{\left({2.51}+{\left(-{3.4}\right)}\right)}={\left({\left(-{5.89}\right)}+{2.51}\right)}+{\left(-{3.4}\right)}=-{6.78}$$$.

Example 3. $$${\left(\frac{{5}}{{8}}+{3}\right)}+\sqrt{{{2}}}=\frac{{5}}{{8}}+{\left({3}+\sqrt{{{2}}}\right)}$$$.