9

Definition
$9$ (Nine) is:


 * The third square number after $1$ and $4$:
 * $9 = 3^2$
 * and therefore from Sum of Consecutive Triangular Numbers is Square, the sum of $2$ consecutive triangular numbers:
 * $9 = 3 + 6$


 * In ternary:
 * $100_3 = 9_{10}$


 * The first odd prime power:
 * $9 = 3^2$


 * The fourth lucky number, and the second lucky square after $1$:
 * $1, 3, 7, 9, \ldots$


 * The sum of the first $3$ factorials:
 * $9 = 1! + 2! + 3!$


 * The $4$th subfactorial after $0, 1, 2, \ldots$
 * $9 = 4! \left({\dfrac 1 - \dfrac 1 {1!} + \dfrac 1 {2!} - \dfrac 1 {3!} + \dfrac 1 {4!} }\right)$

Also see

 * 9 is Only Square which is Sum of 2 Consecutive Cubes
 * Nine Regular Polyhedra
 * Dissection of Rectangle into 9 Distinct Integral Squares