2
Two
2
The number two is the number defined as S(1) where S(x) is the successor function, aka the function equal to x+1. The successor function is the base of almost all numerical functions. Since two is the smallest positive integer defined from a previous number, it is another especially unique number.
Here are some properties of two: 2 is the one and only even prime number.* It's also the only prime that is one less than another prime. 2+2, 2*2, 2^2, 2^^2, 2^^^2, and 2^^^... ... ...^^^2 with as many ^s as we please all turn into 4. There’s zillions more as well, waaaaaay too many to list, most of which are for trivial reasons - perhaps its most interesting is that it's the third member of the Fibonacci sequence.**
Two is also notable in googology for creating degenerate arrays in Bowers' notation. Since array notation is an extension of up-arrow notation, a 4 or more entry array that starts with a two would simply turn into 2^^^^2 with as many or few ups as we please, which decomposes to 4. For example, take the array of tens that represents the mind-boggling gongulus, or even the more mind-blowing golapulus. Replace the first ten with a 3, the second and last tens with a 2, and all the others with 1, and the number isn’t too much less. But replace the first number with 2, and the array degenerates to four.
* 2018 EDIT: Upon further reflection, I've come to no longer understand why (or rather, no longer agree with) 2 being the only even prime number is considered a special, mind-blowing property. I get that even and odd numbers make a nice duality (more on duality later), but 2 being the only prime number divisible by 2 (that's what "even" means) seems like kind of a tautology?
** 2018 EDIT: I'm almost certain I stole that joke from Robert Munafo.
Because of two's role as a number, unlike with one of something, when we see two of something we're instantly reminded of the number two.
In other words:
when you see one giraffe you think "a GIRAFFE"
when you see two giraffes you think "TWO giraffes"
The number 2 is so small that even animals can perceive it immediately. Two of something is an image that rests clearly in the mind and can always be recognized immediately with our primary number sense. We don’t need to count two of something; we can just immediately recognize its “twoness”.
Two's importance is also reflected in the linguistic "one, two, many" concept in some primitive languages - in those languages, one and two get their own words, but three and higher are only described with a word for "many". This is even reflected in English with the word "both" - there's no equivalent terms for three and higher, and we instead use "all three", "all four", etc.
This number is so important that in general, terms involving two (and one as well) have special names, such as double, second, square, and half - compare this to quintuple, fifth, fifth power, and fifth, the equivalent terms for five. Common prefixes for two include di- (dioxide, diagonal, dialogue), du- (duet, duplicate), and bi- (bicycle, bilingual, biannual, binary) - those words are just a smattering of the many many words constructed with roots meaning two.
In the real world there are often two types of things, such as good and evil, hot and cold, tall and short, big and small, male and female, and more. That type of division into two gets its own special term, "dichotomy".
Two is used as the base of the binary numeral system, which is very prevalent in computing - it uses only two digits but it works effectively.
For example, a thousand would have to be represented like so in unary (base 1):
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which is very inconvenient.
But binary can represent a thousand as 1111101000, which is a lot more compact. It's also extremely convenient in computing, since binary numbers can be reprsented as "on" for 1's and "off" for zeros. Therefore a thousand can be thought of as "on on on on on off on off off off". Binary remains the framework base of computing to this day, leading to the powers of two being usually associated with computing. Several entries on this list discuss the powers of two, particularly 1024 which discusses the convenience of powers of 2.
Two was also the first number in a list I wrote on paper as a kid (and never finished) called "Very Important Numbers". I remember this list quite well, but I can't seem to find the notebook that had this list. I said that 2 is important because 2+2 = 2*2 = 2^2, and maybe some other stuff. Very Important Numbers can be thought of as an ancestor of this list. Strangely, I didn't put 0 or 1 on that list, or 3 or 4, which are also some very important numbers as I discuss on their respective entries. For more on that list, see my entry for 17.
In general, as a kid I thought of two as a very special number - to this day I can't deny that at all, although I have now found a particular appeal in the number 3 as well, perhaps to a greater extent than 2.
Site note: Two can be called "googoi" using the googo- naming system.
Side note 2: 2