1

1

one

This is a number SO AMAZINGLY LARGE ... THAT it's not small! In fact, it's the smallest number with this property. We say things are "small", when they are smaller than us (like atoms), and things are "large" when they are larger than us (like galaxies). A person our own height however is neither small nor large relative to us, merely the same. Thus in this sense we can say that one is neither large nor small. If we define large numbers as being numbers larger than 1, then a very small large number is one that is very close, but greater than 1. It follows then that 1 is an infinitely small large number, being the limit of the small large numbers.

One is important in googology because it is often the first member of a sequence. It is the first of the sequence of Positive Integers (pintegers as I like to call them). It is also the first square number, the first cubic number, the first quartic number, the first quintic number etc. the first triangular number, the first tetrahedral number, the first pentachoral number, the first hexapetal number etc. If N is a whole number, then 1^N, 1^^N, 1^^^N, 1^^^^N, etc. all equal 1! Using Steinhaus-Moser polygon notation we find that 1 in a triangle is 1. So is 1 in a square, 1 in a pentagon, 1 in a hexagon, etc. The empty array in Jonathan Bower's array notation, " < > ", is equal to 1. One is also the smallest Busy Beaver Number, since BB(1) = 1. One is also the first Harmonic Number.

One is often the smallest value an argument may have in a large number function. The default value of an entry in array notation is 1. The smallest input in Conway Chain arrows is also 1. For my Hyper-E and Cascading-E Notations, the smallest value for any argument is 1.

One also has many unique properties. It is the only positive integer that is not a prime or composite number. It is the only positive real number whose every integer power is equal to itself. In fact it's the only positive real number for which all real powers equals itself!

One seems to be a natural place to begin a large number list, especially one which deals exclusively with the positive integers, because it represents the very first step towards infinity and away from zero. If we define googolism's as any positive integer given a name, then One can be thought of as the small "googolism" since it's the smallest positive integer with a name :)

...but now that I've said what needs to be said about one and it's importance in googology, let move on to some much much larger googolisms.

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