0

== Number ==
$0$ (zero) is:
:The only number which is both positive and negative
:The only integer which is divisible by $0$
:The only integer which is divisible by all integers
:The identity element for the operation of addition
:The zero element for the operation of multiplication
:The value of $\log_b 1$ for all bases $b$
:The $0$th (and therefore smallest) cardinal number
:The $0$th (and therefore smallest) ordinal number
:The $0$th Fibonacci number
:The $0$th square number:
::$0 = 0^2$
:The $0$th triangular number:
::$0 = \dfrac {0 \paren {0 + 1} } 2$
:The $0$th number to be both square and triangular:
::$0 = 1^0 = \dfrac {0 \times \paren {0 + 1} } 2$
:The $0$th square which has no more than $2$ distinct digits and does not end in $0$:
::$0 = 0^2$
:The $1$st subfactorial:
::$0 = 1! \paren {1 - \dfrac 1 {1!} }$
:The $1$st integer $m$ such that $m! + 1$ (its factorial plus $1$) is prime:
::$0! + 1 = 1 + 1 = 2$
:The $1$st integer such that its double factorial plus $1$ is prime:
::$0!! + 1 = 2$
:The $1$st integer $n$ such that $2^n$ contains no zero in its decimal representation:
::$2^0 = 1$
:The $1$st integer $n$ such that $5^n$ contains no zero in its decimal representation:
::$5^0 = 1$
:The $1$st integer $n$ such that both $2^n$ and $5^n$ have no zeroes:
::$2^0 = 1, 5^0 = 1$
:The $1$st integer which is palindromic in both decimal and binary:
::$0_{10} = 0_2$
:The $1$st integer which is palindromic in both decimal and ternary:
::$0_{10} = 0_3$
:The $1$st number which is (trivially) the sum of the increasing powers of its digits taken in order:
::$0^1 = 0$
:The $1$st integer $m$ such that $m^2 = \dbinom n 0 + \dbinom n 1 + \dbinom n 2 + \dbinom n 3$ for integer $n$:
::$0^2 = \dbinom {-1} 0 + \dbinom {-1} 1 + \dbinom {-1} 2 + \dbinom {-1} 3$
:The $1$st of the $5$ Fibonacci numbers which are also triangular
:The $1$st palindromic triangular number
:The $1$st palindromic integer which is the index of a palindromic triangular number
::$T_0 = 0$
:The $1$st palindromic integer whose square is also palindromic integer
::$0^2 = 0$
:The $1$st Ramanujan-Nagell number:
::$0 = 2^0 - 1 = \dfrac {0 \paren {0 + 1} } 2$
:The $1$st of the $3$ Fibonacci numbers which equals its index:
::$F_0 = 0$
:The $1$st Dudeney number:
::$0^3 = 0$
:The $1$st non-negative integer $n$ such that the Fibonacci number $F_n$ ends in $n$
:The $1$st of the $6$ integers which are the middle term of a sequence of $5$ consecutive integers whose cubes add up to a square:
::$\paren {-2}^3 + \paren {-1}^3 + 0^3 + 1^3 + 2^3 = 0 = 0^2$
:The $1$st of the only $2$ integers which are equal to the sum of the squares of their digits when expressed in base $10$:
::$0 = 0^2$
:The $2$nd integer $n$ after $-1$ such that $\dbinom n 0 + \dbinom n 1 + \dbinom n 2 + \dbinom n 3 = m^2$ for integer $m$:
::$\dbinom 0 0 + \dbinom 0 1 + \dbinom 0 2 + \dbinom 0 3 = 1^2$
== Also see ==
* Definition:Zero
* Zero Divides Zero
* Integer Divides Zero
* General Logarithm of 1 is 0
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== Historical Note ==
== Linguistic Note ==
== Sources ==
* : $0$
Category:Specific Numbers
Category:0