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Teachers & SchoolsStudy Guide

Long ago, and in a guide far, far away, we learned the properties of numbers: commutative, associative, distributive, inverse, and identity. These properties also apply to adding and multiplying with variables, and they even have the same names.

This table provides a quick review.

The **Commutative Property** states that we can add or multiply numbers in any order. For example: 7*xy* is equal to *y*7*x* and *yx*7. In Algebra we almost always put the coefficient in front of the variables, but just for consistency, not because it *needs* to be that way mathematically.

The **Associative Property** allows us to move the parentheses to a different pair of numbers as long as everything is being multiplied or everything is being added. For example: 7 + (*x* + 10) is the same thing as (7 + *x*) + 10.

Notice that none of these examples have subtraction or division in them. The properties above do *NOT work with subtraction and division.*

The** Identity Property**: The identity for addition, or the additive identity, is 0. This is the number that we can add anything to and it won't change. For example: *x* + 0 = *x*.

The multiplicative identity is 1. This is the number that we can multiply anything by and it won't change. For example: (*x*)(1) = *x*.

The **Inverse Property** states that a number added to or multiplied by its inverse equals the identity. This works for variables too. When we add a variable to the same variable with the opposite sign, we get zero (the additive identity).

**Examples:**

*x* + (-*x*) = 0

-4*xy* + 4*xy* = 0

When we multiply a variable by its reciprocal, we get 1 (the multiplicative identity). Remember the reciprocal of a number (or variable) has the same sign. We don't change the sign; just flip the fraction.

**Examples:**

Note: both of these are true only when *x* does *not* equal zero, because we can't have a zero in the denominator.