Suppose I have 7 frogs in an enclosure.
I buy 3 more, and put them in with the others.
The equation for this is simple to write down. It is:
$$7 + 3 = 10$$
I have 10 frogs.
On the other hand, suppose that I start with 3 frogs, and buy 7 more.
This can be written as:
$$3 + 7 = 10$$
It seems that we ended up with the same amount!
We can write
$$7 + 3 = 3 + 7$$
since these quantities are equal to each other.
If we think about it, it's not hard to realize that this work with any two numbers.
Let's choose two numbers! Then we can say that
$$\text{The first number I chose} + \text{The second number I chose} = \text{The second number I chose} + \text{The first number I chose}$$
It doesn't matter which one comes first! These two ways of adding are equal to each other!
Remember what we talked about in Letters in math?
We abbreviate to avoid long lines like that above!
Call the first number we chose \(a\), and the second number we chose \(b\).
Then we can write
$$a + b = b + a$$
Switcheroo!
Next, suppose that I have a bookshelf with 2 books on it.
I have a set of 4 and 12 books that I want to put on the shelf.
We can write
$$2 + 4 + 12$$
to denote this operation.
In what order do we place the books on the shelf? Does it even matter
First, take the first set of 4 books and put them on the bookshelf next to the 2 already on the shelf.
After that, put the remaining 12 books on the shelf too. This is one possibility.
Another way of doing it would be the combine the given 4 and 12 books into a larger pile, and place them on the bookshelf after that.
Let's examine these two cases.
In the first case, we said that we first add the 4 books not yet on the bookshelf to the 2 already on the shelf, and add the remaining 12 after that.
In written language, we use parentheses to mark the operation that comes first. Like so:
$$(2 + 4) + 12$$
In the second case, we combine the sets of 4 and 12 books and put them on the shelf next to the 2 already in its place.
This can be written down like this:
$$2 + (4 + 12)$$
Now, let's perform these operation!
As for the first one, we do 2 + 4 first, which gives 6. All we have left is to add the remaining 12 books the the 6 already on the shelf, which gives 18 books.
In the second case, we perform 4 + 12 first, since they are in parentheses. That gives 16. We place these 16 books next to the 2 already on the shelf, which can be written down as 2 + 16. This is equal to 18 of course.
We now see that these two ways of adding three number are equal! Very nice!
In fact, it's pretty easy to see, that this happens with any three numbers!
Using letters, this can be written down as:
$$(a + b) + c = a + (b + c)$$