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 Depdendent Variable

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 Dependent Variable

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# Properties of Numbers and Definitions

There are several properties that will help you simplify calculations with real numbers. Here is a summary of the properties we will use most often in algebra. In the following examples, a, b, and c represent real numbers.

## Commutative Property

 Commutative Property of Addition Commutative Property of Multiplication When you add two numbers, regardless of the order, the sum is the same. When you multiply two numbers, regardless of the order, the product is the same. 5 + 7 = 7 + 5a + b = b + a 5 Â· 7 = 7 Â· 5 a Â· b = b Â· a

Note

Subtraction and division are not commutative.

For example, 5 -3 =2, but 3 - 5 = 2.

Likewise, 6 Ã·  2 = 3, but 2 Ã· 6 = 1/ 3.

## Associative Property

 Associative Property of Addition Associative Property of Multiplication When you add numbers, regardless of how you group (or associate) them, the sum is the same. When you multiply numbers, regardless of how you group (or associate) them, the product is the same. 2 + (3 + 4) = (2 + 3) + 4a + (b + c) = (a + b) + c 2 Â· (3 Â· 4) = (2 Â· 3) Â· 4 a Â· (b Â· c) = (a Â· b) Â· c

## Distributive Property

This property allows us to convert a product into an equivalent sum.

2 Â· (3 + 4) = 2 Â· 3 + 2 Â· 4

a Â· (b + c) = a Â· b + a Â· c

## Identities

 Additive Identity (Addition Property of 0) Multiplicative Identity (Multiplication Property of 1) The sum of a number and 0 is the number itself. The product of a number and 1 is the number itself. 3 + 0 = 3a + 0 = a 0 is called the additive identity. 3 Â· 1 = 3 a Â· 1 = a 1 is called the multiplicative identity. Multiplication Property of 0The product of a number and 0 is 0. 3 Â· 0 = 0 a Â· 0 = 0

## Inverses

 Additive Inverse (Opposite) Multiplicative Inverse (Reciprocal) Example The sum of a number and its opposite is 0. The product of a number and its reciprocal is 1. The opposite of 7 is -7The opposite of -7 is 7. 7 + (-7) = 0a + (-a) = 0 The reciprocal of 7 is . The reciprocal of is 7.

The opposite of a number is also called the additive inverse of the number.

The reciprocal of a number is also called the multiplicative inverse of the number.

Zero does not have a reciprocal because you may not divide by zero.