# Properties of Addition

No, addition isn't some incredibly wealthy mathematical operation that owns multiple houses in multiple states. Although if it were, it would probably be building additions onto each of them.

If we add two real numbers, we get the same answer no matter which number comes first. This means that addition of real numbers is **commutative**: the numbers can commute (travel past each other with a cup of coffee while listening to their favorite morning radio station) without changing the answer. Wait, no, that's not right. "Commute" just means "switch places" when we're talking about math.

To write "addition of real numbers is commutative" in symbols—because goodness knows we love us some symbols—we say that for real numbers *a* and *b*:

*a *+ *b = b* + *a*

Both terms can swap spots without changing the total sum.

Addition of real numbers is also **associative**: we don't care how the numbers associate with each other. As long as there isn't bloodshed. When adding *more* than two numbers, you'll get the same answer no matter which end you start adding from. If you have three friends, one of whom tells you 2 jokes, one of whom tells you 4 jokes, and one of whom tells you 7 jokes, you'll end up having heard 13 jokes. Regardless of the order you hear them in, you'll be the one who has the last laugh.

### Sample Problem

1 + 2 + 3 = ?

If we first add up 1 and 2, we get:

(1 + 2) + 3 = 3 + 3 = 6

If we first add up 2 and 3, we get:

1 + (2 + 3) = 1 + 5 = 6

Same deal either way. To write "addition of real numbers is associative" in symbols, we say that for real numbers *a*, *b*, and *c*:

(*a* + *b*) + *c = a *+ (*b *+ *c*)

Addition has an **additive identity**: the number 0. If you add 0 to any number, that number keeps its identity. For example, 4 + 0 = 4.

In other words, if you take 4 steps, and then 0 steps, you've still taken only 4 steps overall. By the way, you look pretty silly trying to take 0 steps.

If you walk a certain distance on the number line and then walk the same distance in the opposite direction, you'll end up back where you started. This means that if you add a real number *n* and its negative -*n*, you end up back at 0 again. Who knows why you went all the way back to 0? Maybe you forgot your keys.

*n *+ (*-n*) = 0

(*-n*) + *n* = 0

2 + (-2) = 0

(-8.7) + 8.7 = 0

If *n* is a real number (positive or negative or zero), we call -*n* the **additive inverse** of *n*. "Additive inverse of *n*" is just a fancy way of saying "the number we add to *n* to get back to zero." Remember that -*n* is only negative if *n* is positive. For example if *n *= -3, then *n *is negative and *-n* is positive. It's a tad confusing, but if you get confused, just ask Tad.