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

- zillions more

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; to the ancient Greeks even 2 was sometimes uncertain as to whether it 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

- many many more.

This common occurrence of three is largely due to three being 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

Probably the biggest thing about the number 3 in googology is that many googolisms are 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).

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