Properties of addition

In this article, we'll learn the three main properties of addition. Here's a quick summary of these properties:

Commutative property of addition: Changing the order of addends does not change the sum. For example, $4+2=2+4$‍.

Associative property of addition: Changing the grouping of addends does not change the sum. For example, $(2+3)+4=2+(3+4)$‍.

Identity property of addition: The sum of $0$‍ and any number is that number. For example, $0+4=4$‍.

The commutative property of addition says that changing the order of addends does not change the sum. Here's an example:

Notice how both sums are $6$‍ even though the ordering is reversed.

Here's another example with more addends:

The associative property of addition says that changing the grouping of the addends does not change the sum. Here's an example:

Remember that parentheses tell us to do something first. So here's how we evaluate the left-hand side:

And here's how we evaluate the right-hand side:

Notice that both sides sum to $9$‍ even though we added the $2$‍ and the $3$‍ first on the left-hand side, and we added the $3$‍ and the $4$‍ first on the right-hand side.

The identity property of addition says that the sum of $0$‍ and any number is that number. Here's an example:

This is true because the definition of $0$‍ is "no quantity", so when we add $0$‍ to $4$‍, the quantity of $4$‍ doesn't change!

The commutative property of addition tells us that it doesn't matter if the $0$‍ comes before or after the number. Here's an example of the identity property of addition with the $0$‍ after the number: