Properties of Real Numbers

**Properties of Real Numbers**

**Real Numbers **include many sets of numbers: integers, fractions, decimals, rational numbers, and irrational numbers. The one set of numbers that is not in this group is "imaginary numbers."

For all real numbers *x*, *y, *and *z,* the following properties apply*:*

**Closure Properties**

The sum or product of two real numbers is a real number.

**Commutative Properties**

The sum or product of two real numbers is the same regardless of their order.

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**Identity Properties**

The sum of a real number and 0 is that real number.

The product of a real number and 1 is that real number.

**Inverse Properties**

The sum of any real number and its opposite is 0.

The product of any nonzero real number and its reciprocal is 1.

**Associative Properties**

The sum or product of three real numbers is the same no matter which two are added or multiplied first.

**Distributive Property**

When a real number gets multiplied by the sum or difference of real numbers, it gets multiplied by every number within the sum or difference.

*x(y + z) = xy + xz*

*x(y - z) = xy - xz*

**Careful! ** Notice the difference between the associative and commutative properties.

- For the associative properties, the order of the numbers does not change, but their grouping does.

- For the commutative properties, the order of the numbers does change from one side of the equal sign to the other.

**Examples**

1. Simplify the following expression.

Using the associative property, we can simplify it.

2. Simplify the following expression:

First, let us use the commutative property.

Now we'll use the associative property.

3. Use the distributive property to rewrite the following expression:

4. Use the distributive property to factor the following expression:

= 16(3z+ 1)