# 0

## Contents

## Number

$0$ (**zero**) is:

- 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 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 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 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 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

### Previous in sequence: $-1$

### Next in sequence: $1$

*Next*: Subfactorial*Next*: Sequence of Integers whose Factorial plus 1 is Prime*Next*: Prime Values of Double Factorial plus 1*Next*: Fibonacci Number*Next*: Powers of 2 with no Zero in Decimal Representation*Next*: Powers of 5 with no Zero in Decimal Representation*Next*: Powers of 2 and 5 without Zeroes*Next*: Palindromes in Base 10 and Base 3*Next*: Numbers which are Sum of Increasing Powers of Digits*Next*: Square Formed from Sum of 4 Consecutive Binomial Coefficients*Next*: Triangular Number*Next*: Palindromes in Base 10 and Base 2*Next*: Palindromes in Base 10 and Base 3*Next*: Triangular Fibonacci Numbers*Next*: Palindromic Triangular Numbers*Next*: Palindromic Indices of Palindromic Triangular Numbers*Next*: Ramanujan-Nagell Number*Next*: Fibonacci Numbers which equal their Index*Next*: Dudeney Number*Next*: Sequence of Fibonacci Numbers ending in Index*Next*: Numbers Equal to Sum of Squares of Digits

## 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**.

## Sources

- 1983: François Le Lionnais and Jean Brette:
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