The primary properties of integers include the following:

### 1. Closure Property

The set of integers is closed under addition and multiplication, meaning that the sum or product of any two integers is also an integer.

### 2. Associative Property

In integer addition and multiplication, the way in which numbers are grouped does not change the result. Here are two examples.

*(a + b) + c = a + (b + c)*

and

*(a x b) x c = a x (b x c)*

### 3. Commutative Property

The order of integers in addition and multiplication does not affect the result. Here are two examples.

and

### 4. Distributive Property

Multiplication distributes over addition for integers, meaning:

*a x (b + c) = (a x b) + (a x c)*

### 5. Additive Inverse Property

Every integer *a* has an additive inverse *–a* such that:

### 6. Multiplicative Inverse Property

Every nonzero integer *a* has a multiplicative inverse *1/a*, but since *1/a* is typically not an integer, this property mostly applies to rational numbers.

### 7. Identity Property

The identity element for addition is 0, since *a + 0 = a*. The identity element for multiplication is 1, since *a x 1 = a*.

*We created this article in conjunction with AI technology, then made sure it was fact-checked and edited by a HowStuffWorks editor.*