Nonnegative Integer — Definition, Formula & Examples

A nonnegative integer is any integer that is greater than or equal to zero. The set includes 0, 1, 2, 3, 4, and so on — every whole number that is not negative.

The set of nonnegative integers is $\{x \in \mathbb{Z} \mid x \geq 0\}$ , which equals $\{0, 1, 2, 3, \dots\}$ . It is the union of zero and the positive integers.

Key Formula

\mathbb{Z}_{\geq 0} = \{0, 1, 2, 3, \dots\}

Where:

$\mathbb{Z}_{\geq 0}$ = Standard notation for the set of nonnegative integers

Worked Example

Problem: From the list −3, 0, 4.5, 7, −1, 2, identify all the nonnegative integers.

Check each value: A nonnegative integer must be (1) an integer (no decimals or fractions) and (2) greater than or equal to zero.

Eliminate non-integers: 4.5 is not an integer, so it is excluded.

Eliminate negatives: −3 and −1 are negative, so they are excluded.

Answer: The nonnegative integers in the list are 0, 7, and 2.

Why It Matters

Many formulas and problems restrict values to nonnegative integers. For example, you cannot have a negative number of people, coins, or sides of a polygon. Recognizing this constraint helps you set up correct domains in algebra and combinatorics.

Common Mistakes

Mistake: Thinking nonnegative integers and positive integers are the same thing.

Correction: Nonnegative integers include zero. Positive integers start at 1. Zero is nonnegative but not positive.