Positive Integer — Definition, Formula & Examples

A positive integer is any whole number greater than zero. The positive integers are 1, 2, 3, 4, 5, and so on, continuing forever.

The set of positive integers is $\{1, 2, 3, 4, \ldots\}$ , which equals the set of natural numbers (excluding zero). Every positive integer $n$ satisfies $n \in \mathbb{Z}$ and $n > 0$ .

How It Works

To determine whether a number is a positive integer, check two things: it must be a whole number (no fractions or decimals), and it must be greater than zero. For example, 7 passes both checks, but

-3

fails because it is less than zero, and 4.5 fails because it is not a whole number. Zero itself is an integer but not a positive integer.

Worked Example

Problem: From the list {−2, 0, 3, 4.7, 8, ½}, identify all the positive integers.

Check each number: Go through the list one by one. A positive integer must be a whole number AND greater than zero.

Eliminate non-qualifying numbers:

-2

is negative, so it fails.

0

is not greater than zero, so it fails.

4.7

has a decimal, so it is not a whole number.

\frac{1}{2}

is a fraction, not a whole number.

Identify the positive integers: Only 3 and 8 are whole numbers greater than zero.

Answer: The positive integers in the list are 3 and 8.

Why It Matters

Positive integers are the counting numbers you use when tallying objects, numbering items in a list, or describing quantities like "5 apples." They form the foundation for topics like factors, multiples, and prime numbers, all of which are defined only for positive integers.

Common Mistakes

Mistake: Including zero as a positive integer.

Correction: Zero is neither positive nor negative. The positive integers start at 1. Zero is classified as nonnegative, but not positive.