3
Three
3
3
Three is the sum of 1 and 2, defineable with the fundamental successor function as S(2). It's another one of the very few truly small numbers, numbers we can recognize with our primitive number sense alone (the largest is probably around 6). As with two, three of something can be immediately recognized - you don’t even need to count three objects if you see them, as you can look at three apples or other objects and instantly notice their threeness. Three rests in the mind as a clear image and occurs literally everywhere. If you look around, chances are that you will see three of something somewhere.
Three's significance as a number is reflected in that three seems to be the smallest number needed to recognize as a pattern - this goes with the "one, two, many" idea (see the page for 2). For example, if you make a gesture once, it would usually mean doing something; if you make that gesture twice, it would signify doing it twice; but three times is enough to suggest that the action is being done many times.
Properties of the number 3
Three has many numerical properties, once again a lot of them trivial. Here are some examples of the properties of 3 - it is:
the second prime number
the first odd prime number
the only prime that is one less than a square number (22-1)
the first Fermat prime (next is 5)
the first Mersenne prime
the only number that is both a Fermat prime and a Mersenne prime
a Fibonacci number (next is 5)
a rough approximation for both pi and e, two super-important mathematical constants
the only number to be the sum of all previous nonnegative integers
the only prime triangular number
For more on some of these properties, see prime numbers at 19, triangular numbers at 15, Fermat numbers at 257, and Mersenne primes at 127.
Something else notable about three is that it's the smallest number that has always been thought of as a number. It is well-known that nobody in ancient times thought of 0 as a number; the same held to an extent for 1; even some of the ancient Greeks couldn't quite decide whether 2 was a number since it had a beginning and end but no middle.
Also, testing for divisibility by 3 is pretty easy: just add up the number's digits, and repeat the process until you get a number you know is divisible by 3. For more on divisibility tests see the entry for 11.
A division of a set into three groups gets a term called trichotomy, analogous for dichotomy for two - most often, a trichotomy has two sides that are opposites, and one neutral side. A simple example is the trichotomy of real numbers: positive numbers, negative numbers, and zero. A more interesting trichotomy, however, is perfect, abundant, and deficient numbers. An abundant number is a number whose divisors (other than itself) add up to a number larger than the original number; a deficient number is a number whose divisors (other than itself) add up to a number smaller than the original number; and a perfect number is a number whose divisors (other than itself) add up to exactly the original number. Most natural numbers are either abundant or deficient - very few are perfect.
Three in our world
There are many examples of threes in our world, and some of them include:
the three little pigs
three verses in a song
the Three Stooges
the Holy Trinity
Goldilocks and the three bears
the three gods in Hinduism
counting to three (on the count of three, everybody pull)
three legs are needed to stand a table or a tripod
Newton's three laws of motion
three dimensions in our world
three parts of our mind in Freud's model (id, ego, superego)
the hand game rock, paper, scissors; often used among two people to decide which needs to do something neither wants to do
This common occurrence of three is largely because three is the minimum needed to recognize a pattern, as I discussed earlier. But I think three is also chosen for these kinds of things because a group of three can have a feel of symmetry with all three symmetric with each other and not any kind of grouping - two is just too small for that, and four is a little too big and feels like it needs to be grouped.
This goes to show that three is a very special number - that recognition stems from three being the minimum needed to recognize a pattern. There's even a "rule of three" that says that things are inherently more appealing when grouped in threes. Three's ubiquity even is shown directly in mathematics, e.g. three points are needed to determine a plane, or a circle.
Many words in English use the root tri-, a prefix from Greek "tria" and Latin "tres" meaning 3. Trillion, triangle, tricycle, triplet, and trilogy are some words based on three.
Three in googology
If a large number in googology isn't based on tens, it's most often based on threes. This is because 3 is the smallest number that doesn’t create degenerate arrays in Bowers' and Bird's array notation - in googology, three is often the smallest number that doesn't lead to degenerate cases. The root tri- for three also appears in many googolisms' names (like tritri and tridecal).