# Algebra: Properties of the Real Numbers

## Properties of the Real Numbers

This algebra video tutorial introduces some of the fundamental math concepts in algebra. Although seemingly trivial, these properties form the basis of much of the material covered in early algebra courses, and a full understanding of them is essential to a proper understanding of algebra.

The properties of the real numbers are closure, inverse, identity, associative, commutative, and distributive. The first three properties were introduced in lesson 2. For completeness, we give the formal definitions here.

For closure under addition, we say that for any real numbers **x** and **y**, **(x + y) ** is a real number. For multiplication, we have that for any real numbers **x** and **y**, **(x * y)** is also a real number.

For the additive inverse property, we have that for any real numbers **x**, there exists a real number **y** such that **x + y = 0**. The additive inverse of **x** is **–x**. For the multiplicative inverse property, we have that for any real numbers **x**, there exists a real number **y** such that **x * y = 1**. The multiplicative inverse of **x** is the reciprocal **1/x**.

The identity property for addition states there is a real number that does not affect the any other real number when it is added to it, namely **0**. For multiplication, we have a real number that does not affect any other real number when it is multiplied by it; that number is **1**.

The associative property tells us that if we add or multiply three real numbers together with two operations, it does not matter which of the two operations we do first.

The commutative property tells us that if we are adding or multiplying two real numbers, we can flip the sides of the operands and we will get the same result.

The distributive property tells us that a real number that is multiplied by a sum of two real numbers is equal to the sum of the first number times the sum of each of the real numbers in the sum. There is a left-distributive property and a right-distributive property. However, the commutative property makes these logically equivalent, as we showed in the video.