1

1 (one) is a positive integer following 0 and preceding 2. Its ordinal form is written "1st" or "first".

Properties

 * 1 is the first natural number. 1 is, and it's the only natural number that is neither prime nor composite.
 * 1 is a, , , etc. It's also a , cubic number, etc.
 * 1 is the multiplicative identity, meaning that \(a = a \times 1\) for all \(a\). In fact, \(a \underbrace{\uparrow\uparrow\ldots\uparrow\uparrow}_n 1 = a\) (arrow notation) for all \(n,a \geq 0\), so 1 is a sort of identity for all the hyper operators beyond addition. Furthermore, for all \(n,a > 0\), \(a \underbrace{\uparrow\uparrow\ldots\uparrow\uparrow}_n 0 = 1\) and \(1 \uparrow\uparrow\ldots\uparrow\uparrow n\). 1 appears frequently as a "default" argument in googological notations, such as BEAF, chained arrow notation, and hyper-E notation.
 * All natural numbers are just strings of 1's added together. For example, 1,000,000 is a string of 1 million 1's added together.
 * 1 is the only odd.
 * 1 is the only known odd.
 * The base one numbering system is called, as demonstrated by the video below:



In googology
Sbiis Saibian argued that all numbers larger than 1 should be called "large numbers," because the reciprocals of large numbers are small. The large numbers and small numbers are "mirrored" about 1, so it makes sense to say that 1 is the threshold of largeness. A number like 1 + 1/googol could be called a "very small large number." The smallest large number is a logical impossibility, of course.

1 was also the first number by Adam Elga in the Big Number Duel.

1 can be named garone, fzone, fugaone, megafugaone, or boogaone with the gar-, fz-, fuga-, megafuga-, and booga- prefixes respectively.

Googological functions returning 1

 * Rado's Sigma Function: \(\Sigma(1)=1\)
 * Maximum shifts function: \(S(1)=1\)
 * Xi function: \(\Xi(1)=1\)
 * Goodstein function: \(G(1)=1\)
 * Weak Goodstein function: \(g(1)=1\)
 * Kirby-Paris hydra: \(\text{Hydra}(1)=1\)
 * Buchholz hydra: \(\text{BH}(2)=1\)
 * TREE function: \(\text{TREE}(1)=1\)
 * Weak tree function: \(\text{tree}(0)=1\)
 * Fusible numbers: \(m_1(0) = 1\)
 * Exploding Tree function: \(E(1)=1\)
 * Latin square: \(L(1)=1\)
 * Gijswijt's sequence: \(c(1)=1\)
 * Tetration: \(1 \uparrow\uparrow n = 1\)

As a cash denomination
Some currencies, such as the and the, have banknotes with this number in the denomination. Likewise, some currencies, such as the, the and the , have coins with this number in the denomination.