Commutative Property
Extra! Extra! Math police give rule on "order" in addition or multiplication!
It's front-page news. The math police have determined that the order of the numbers in an addition or a multiplication problem does not matter, just like it doesn't matter if you get the ticket for speeding first or for driving without a seatbelt first—you've still got two (expensive) tickets. This is what the commutative property is all about.
The commutative property states that the order of addends or factors (fancy words for numbers) in an addition or multiplication problem can change places without affecting the result. For example, 4 + 2 is equal to 2 + 4, just like 5 × 3 is the same as 3 × 5.
That means it doesn't matter if we eat 3 scoops of vanilla then 2 scoops of chocolate, or 2 scoops of chocolate then 3 scoops of vanilla. Either way, we've eaten 5 scoops of ice cream. We do not recommend eating this much ice cream in one sitting. What's the rush?
For more commutative property chatter, visit here.
The Associative Property
The associative property is for the socializers of the math world: multiplication and addition. One minute they're grabbin' pizza with the football team, and the next minute they're buddy-buddy with the drama club adding lines to a play. Regardless of who they're with, they remain their same old people-loving, accepting-of-everyone, bringing-things-together selves.
The associative property states that the addends or factors (again, numbers) of any addition or multiplication problem can change who they associate with (through the use of parentheses) and still maintain their value.
For example, 3 + (2 + 5) is equal to (3 + 2) + 5. Spoiler: both equal 10.
3 + (2 + 5) = 3 + (7) = 10
(3 + 2) + 5 = (5) + 5 = 10
Since (3 × 2) × 4 is equal to 3 × (2 × 4), we know the associative property works for multiplication too.
(3 × 2) × 4 = (6) × 4 = 24
3 × (2 × 4) = 3 × (8) = 24
Just don't try the associative property with subtraction or division. The operations don't associate well. It's probably because they're always borrowing and breaking things apart. Not the best way to make friends.
Case in point: to solve 5 – (3 – 1), we first do what's in the parentheses, then we subtract that from 5.
5 – (3 – 1) = 5 – (2) = 3
If we move the parentheses to the first two numbers, we get (5 – 3) – 1. Try solving that one instead.
(5 – 3) – 1 = (2) – 1 = 1
See? We got two different answers: 3 is definitely not the same as 1. So no moving parentheses around when subtracting. Same goes for dividing.
For more associative property schmoozing, visit here.
Distributive Property
The distributive property is our rule-bender friend in the world of math. He doesn't do anything too outrageous, but he knows how to stretch the rules just enough to be entertaining, without getting into any real trouble.
Thanks to the distributive property, we can ignore the oh-so-precious order of operations (PEMDAS), just for a second. Let's think about 5(3 + 4) as an example. The order of operations would have us add 3 + 4, which is 7, then multiply by 5 because in the order of operations parentheses come before multiplication. We'd end up with 35. The distributive property has another path to get to the answer, though.
The distributive property says we can do our multiplying first. We can multiply the factor that's hangin' out in front of the parentheses by each number inside the parentheses, then find the sum or difference of those products without messing up our answer.
Consider our original example of 5(3 + 4). If we multiply 5 by each number inside the parentheses, we get 5 × 3 = 15 and 5 × 4 = 20. If we add the sums, 15 + 20, we end up with 35, too.
Why would we choose to use the distributive property instead of just doing the stuff inside the parentheses first? In Algebra, we can't always simplify parentheses if there's a variable hiding in there, so we use the distributive property.
For more rule-bending, distributive adventures, visit here.
Additive and Multiplicative Identity
The additive and multiplicative identities are two numbers that could be called the therapists of the math world. They're great listeners and help us see who we really are. They're kind of like mirrors that way.
The additive identity is 0. When we add any number to the additive identity (or 0), we always get that number as the answer. When 5 was experiencing an identity crisis, he added himself to 0 and walked away a better 5, though still a 5. Here are a few more examples of how the additive identity works:
6 + 0 = 6
-9 + 0 = -9
0 + 0.354692354 = 0.354692354
The multiplicative identity is 1 because we can multiply any number by 1 and get that same number as the answer. For example:
3 × 1 = 3
0.25 × 1 = 0.25
0 × 1 = 0
4.2 billion × 1 = 4.2 billion
Great reflective listening, 1 and 0. Where would we be without you?
Additive and Multiplicative Inverse
In our math community of socializers, rule-benders, and therapists, we also have philosophers who have a way of undoing all we've done and bringing us right back to our beginning.
When we add a number and its additive inverse, we get 0 (which is the additive identity).
When we multiply a number and its multiplicative inverse, we always get 1 (which is the multiplicative identity).
Let's subject 3 to our new philosophy of inverses. What can we add to 3 so that the answer is 0? Hopping across 0 on the number line to the opposite of 3, we get -3. And 3 + (-3) = 0. In fact, any number added to its opposite is 0. The additive inverse is the opposite of any given number. Just swap the plus or minus sign.
One more example: What is the additive inverse of 
? We find the opposite, which is 
. We check it by adding them and seeing if we get 0:
Philosophy proved.
Let's try to find the multiplicative inverse. What can we multiply 
by so that the answer is 1? In this case, we think upside down.
Any number multiplied by its reciprocal is 1, so the reciprocal is the multiplicative inverse. Just flip the fraction over.