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. 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 \(n,a > 0\), \(a \underbrace{\uparrow\uparrow\ldots\uparrow\uparrow}_n 0 = 1\). 1 appears frequently as a "default" argument in googological notations, such as BEAF, chained arrow notation, and hyper-E notation.

By definition all natural numbers are just strings of 1's added together. For example, 1000000 is a string of 1 million 1's added together.

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.

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: \(\Sigma(1)=1\)
 * 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: \(c(1)=1\)

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

Some currencies, such as the and the, have coins with this number in the denomination.