The commutative property of addition says that we can add numbers in any order. The commutative property of multiplication is very similar. It says that we can multiply numbers in any order we want without changing the result.
5 + 4 = 4 + 5
3 x 8 x 5 = 5 x 3 x 8
Both addition and multiplication can actually be done with two numbers at a time. So if there are more numbers in the expression, how do we decide which two to "associate" first? The associative property of addition tells us that we can group numbers in a sum in any way we want and still get the same answer. The associative property of multiplication tells us that we can group numbers in a product in any way we want and still get the same answer.
(4 + 2) + 7 = 4 + (2 + 7)
2(3) = 3(2)
The distributive property comes into play when an expression involves both addition and multiplication. A longer name for it is, "the distributive property of multiplication over addition." It tells us that if a term is multiplied by terms in parenthesis, we need to "distribute" the multiplication over all the terms inside.
2(5 + 8) = (2 x 5) + (2 x 8)
Even though order of operations says that you must add the terms inside the parenthesis first, the distributive property allows you to simplify the expression by multiplying every term inside the parenthesis by the multiplier. This simplifies the expression.
The identity property for addition tells us that zero added to any number is the number itself. Zero is called the "additive identity." The identity property for multiplication tells us that the number 1 multiplied times any number gives the number itself. The number 1 is called the "multiplicative identity."
5y + 0 = 5y
2c × 1 = 2c
The inverse of something is that thing turned inside out or upside down. The inverse of an operation undoes the operation: division undoes multiplication.
A number's additive inverse is another number that you can add to the original number to get the additive identity. For example, the additive inverse of 67 is -67, because 67 + -67 = 0, the Additive Identity.
Similarly, if the product of two numbers is the multiplicative identity, the numbers are multiplicative inverses. Since 6 x 1/6 = 1 (the Multiplicative Identity), the multiplicative inverse of 6 is 1/6.
NOTE: Zero does not have a multiplicative inverse, since no matter what you multiply it by, the answer is always 0, not 1.
Basic Properties of Equality and Inequality
The basic properties can be confusing is you do not try to break them down to understand what they each mean. What follows is a simple explanation of each property that breaks it down in terms that are easier to follow.
If x = y, then x may be replaced by y in any equation or expression. These are all so basic that it can be hard to see why we bother sayingthem! What mathematicians try to do is to boil down our reasoning to the simplest clear statements we can make, so that we can prove everything we say on the basis of those "axioms". Let's look at what they mean.
Suppose I know that "big" is just another word for "large". That means that anywhere I read the word "large", I can replace it with theword "big". That's obvious. (Though in language, sometimes things get a lot more tricky than in math, since words can have more than one meaning!)
The same thing works in math. If I know that two variables both refer to the same number (that's what "equal" means), then anything I say about one of them is also true of the other.
If I know a = b and you give me the expression 3a^2 – 5 (that is, 3 times the square of a, minus 5), then I know I can replace "a" with "b" in that expression, and 3b^2 – 5 will always have exactly the same value. If, for example, a and b are both equal to 4, then regardless of whether I call it a or b, I get the same answer (43) when I evaluate the expression.