Natural Number
ArithmeticA natural number is a positive counting number (1, 2, 3, ...) used to count and order objects.
Definition
Natural numbers are the numbers you use when counting: $1$, $2$, $3$, $4$, $5$, and so on. They start at $1$ and keep going forever.
Example
Counting students in a class: $1, 2, 3, \ldots, 28$. Every count is a natural number.
Key Insight
They are called "natural" because counting is the most natural thing humans do with numbers. They come before all the other kinds of numbers we invent.
Definition
The natural numbers $\mathbb{N} = \{1, 2, 3, \ldots\}$ are the positive integers. They are closed under addition and multiplication. Many textbooks use $\mathbb{N}$ to mean $\{0, 1, 2, \ldots\}$; when $0$ is excluded the set is sometimes written $\mathbb{N}^+$ or $\mathbb{Z}^+$.
Example
The sum and product of any two natural numbers is always a natural number: $7 + 12 = 19$, $4 \times 9 = 36$. Subtraction and division can leave the set: $3 - 5 = -2$, $7 / 2 = 3.5$.
Key Insight
By the Fundamental Theorem of Arithmetic, every natural number greater than $1$ is either prime or can be written as a unique product of primes, making the naturals the home of all number theory.
Definition
The natural numbers can be constructed from first principles using the Peano axioms: there is a first element ($1$ or $0$), and every element has a unique successor. Addition and multiplication are defined recursively from the successor function. The resulting structure is the initial object in the category of semirings.
Example
Proof by strong induction lives entirely in $\mathbb{N}$: to prove $P(n)$ for all $n$ in $\mathbb{N}$, show $P(1)$ holds, then show that if $P(k)$ holds for all $k < n$, it holds for $n$.
Key Insight
Kronecker's famous quote: "God made the integers; all else is the work of man." The naturals are the foundation on which all other number systems ($\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$) are constructed.