1

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

Properties
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 \(n \geq 1\) and \(a\), so 1 is a sort of identity for all the hyper operators beyond addition. 1 appears frequently as a "default" argument in googological notations, such as BEAF, chained arrow notation, and hyper-E notation.

1 is. It is the only positive integer which is neither prime nor composite.

1 is a, , , etc.

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, obviously, also cannot exist.

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

Googological functions returning 1

 * Rado's Sigma Function: \(\Sigma(1)=1\)
 * 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: \(c(1)=1\)