# 0

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## 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 triangular number:
$0 = \dfrac {0 \paren {0 + 1} } 2$

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 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

The $1$st integer equal to the sum of the digits of its cube:
$0^3 = 0$

The $1$st non-negative integer $n$ such that the Fibonacci number $F_n$ ends in $n$

The $1$st of the 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 $m$:
$\dbinom 0 0 + \dbinom 0 1 + \dbinom 0 2 + \dbinom 0 3 = 1^2$

## Historical Note

The Babylonians from the $2$nd century BCE used a number base system of arithmetic, with a placeholder to indicate that a particular place within a number was empty, but its use was inconsistent. However, they had no actual recognition of zero as a mathematical concept in its own right.

The Ancient Greeks had no conception of zero as a number.

The concept of zero was invented by the mathematicians of India. The Bakhshali Manuscript from the $3$rd century CE contains the first reference to it.

However, even then there were reservations about its existence, and misunderstanding about how it behaved.

In Ganita Sara Samgraha of Mahaviracharya, c. $850$ CE appears:

A number multiplied by zero is zero and that number remains unchanged which is divided by, added to or diminished by zero.

It was not until the propagation of Arabic numbers, where its use as a placeholder made it important, that it became commonplace.

## Linguistic Note

The Sanskrit word used by the early Indian mathematicians for zero was sunya, which means empty, or blank.

In Arabic this was translated as sifr.

This was translated via the Latin zephirum into various European languages as zero, cifre, cifra, and into English as zero and cipher.