What is a natural number? A whole number? An integer?

I've discovered:

- Natural number: positive integer, nonnegative integer
- Whole number: positive integer, nonnegative integer, integer

For counting, we always use positive integers: 1, 2, 3, 4, 5, ...

But for set theory, we also need zero, giving the nonnegative integers: 0, 1, 2, 3, 4, 5, ...

One also needs zero to get addition and multiplication in a straightforward way from Peano's axioms of arithmetic. Those axioms define natural numbers from some initial number and a successor for each number.

As to the word "integer", it comes from Latin "integer" and I find

integer - Wiktionary - Latin adjective

- complete, whole, intact
- uninjured, sound, healthy

Charlton T. Lewis, Charles Short, A Latin Dictionary, intĕger - numerous meanings, including "Undiminished, whole, entire, complete, perfect" - also has lots of usage examples

So "integer" is from a word that means "whole".