The set of positive integers and zero are collectively called non-negative integers.

The set of integers denoted by **Z** consists of,

**Z** = {….,-3,-2,-1,0,1,2,3,…..} from which on removing the negative numbers we obtain the set of non-negative integers.

**Non- negative integers and whole numbers**:

The collection of all positive integers like 1, 2, 3,…. is known as the set of natural numbers. It is denoted by the letter **N.**

The set of natural numbers along with 0 is called the set of whole numbers. It is denoted by the letter **W**. So we see that the set of non-negative integers is the same as the set of whole numbers.

Therefore, Non-negative integers = **W** = **N** U {0}

** Examples**:

- -3 is not a non-negative integer (It is in fact negative although it is an integer).
- 3.5 is not a non-negative integer (It is positive but not an integer since it is a non-trivial decimal fraction).
- -3.5 is not a non-negative integer (It is neither positive nor an integer).
- 5 and 0 are examples of non-negative integers.

**Why are non-negative integers so called?**

Non-negative integers are so called because they are obtained by removing the negative numbers from the set of integers.

**Is 0 a positive integer?**

The number 0 is considered to be neither a positive nor a negative integer. It belongs to the set of non-negative integers.

**What is the largest non-negative integer?**

The numbers which are to the right of the number line are considered larger than those who are to its left. There is no largest non-negative integer because we can always find a greater number by adding 1 to it. If we claim that **X** is the largest then we get a contradiction because **X+1** is clearly greater than **X. **This proves that there is no largest non-negative integer.

**Addition of Non-negative integers**:

- The addition of non-negative integers is commutative, that is, a+b=b+a.
- The addition of non-negative integers is associative, that is, a+(b+c)=(a+b)+c.
- Adding 0 does not change the value of a given integer, that is, a+0=a.

These three properties imply that the set of non-negative integers forms what is called an abelian monoid.

**Multiplication of Non-negative integers**:

- The product of non-negative integers is commutative, that is, a*b=b*a.
- The product of non-negative integers is associative, that is, a*(b*c)=(a*b)*c.
- Taking product by 0 gives 0, that is, a*0=0.