* Site-Outage Notice: Our engineering elves will be tweaking the Shmoop site from Monday, December 22 10:00 PM PST to Tuesday, December 23 5:00 AM PST. The site will be unavailable during this time.
Dismiss

# Properties of Addition

No, additions are not incredibly wealthy mathematical operations that own multiple houses in multiple states. Although, if they were, they'd 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.

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

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

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. Example: 4 + 0 = 4

That is, 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 zero again. Who knows why you went all the way back to zero. 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.