Whole Number
ArithmeticA whole number is any non-negative integer: 0, 1, 2, 3, and so on, with no fractions or decimals.
Definition
Whole numbers are the counting numbers plus zero: $0$, $1$, $2$, $3$, $4$, and so on. They never have a decimal point or a fraction part.
Example
You have $0$, $1$, $2$, or $3$ apples. Each of those counts is a whole number. You cannot have $2.5$ apples as a whole number.
Key Insight
Whole numbers are the numbers you use to count things in the real world, plus zero for "none at all."
Definition
The whole numbers are the set $W = \{0, 1, 2, 3, \ldots\}$. They differ from the natural numbers only in including $0$. Whole numbers are closed under addition and multiplication: adding or multiplying two whole numbers always produces another whole number.
Example
$5 + 3 = 8$ (whole number). $4 \times 6 = 24$ (whole number). But $3 - 5 = -2$, which is NOT a whole number, so whole numbers are not closed under subtraction.
Key Insight
Every whole number is an integer, but negative integers like $-1$ and $-2$ are not whole numbers. The whole numbers are a subset of the integers.
Definition
The whole numbers $W = \{0, 1, 2, \ldots\}$ form a commutative monoid under addition (identity $0$) and under multiplication (identity $1$). $W$ is isomorphic to the non-negative part of $\mathbb{Z}$. In the Peano axioms, the natural numbers are often defined to include $0$, making them equivalent to the whole numbers in modern usage.
Example
The well-ordering principle states every non-empty subset of $W$ has a least element. This property drives mathematical induction and the Euclidean algorithm.
Key Insight
The question of whether $0$ belongs to the natural numbers varies by convention. The ISO 80000-2 standard includes $0$ in the naturals, but many textbooks keep a separate "whole number" category to avoid ambiguity.