The digit used in the modern Western world to represent the number 2 traces its roots back to the Indic
Brahmic script, where "2" was written as two horizontal lines. The modern
Chinese and
Japanese languages (and Korean
Hanja) still use this method. The
Gupta script rotated the two lines 45 degrees, making them diagonal. The top line was sometimes also shortened and had its bottom end curve towards the center of the bottom line. In the
Nagari script, the top line was written more like a curve connecting to the bottom line. In the Arabic
Ghubar writing, the bottom line was completely vertical, and the digit looked like a dotless closing question mark. Restoring the bottom line to its original horizontal position, but keeping the top line as a curve that connects to the bottom line leads to our modern digit.[1]
Two is most commonly a
determiner used with
plural countable nouns, as in two days or I'll take these two.[2]Two is a
noun when it refers to the number two as in two plus two is four.
Etymology of two
The word two is derived from the
Old English words twā (
feminine), tū (neuter), and twēġen (masculine, which survives today in the form
twain).[3]
The pronunciation /tuː/, like that of who is due to the
labialization of the vowel by the w, which then disappeared before the related sound. The successive stages of pronunciation for the Old English twā would thus be /twɑː/, /twɔː/, /twoː/, /twuː/, and finally /tuː/.[3]
Mathematics
Divisibility rule
An
integer is determined to be even if it is divisible by 2. For integers written in a numeral system based on an even number such as
decimal, divisibility by 2 is easily tested by merely looking at the last digit. If it is even, then the whole number is even. When written in the decimal system, all multiples of 2 will end in
0, 2, 4, 6, or
8.[4]
Characterizations
The number two is the smallest, and only
even,
prime number. As the smallest prime number, two is also the smallest non-zero
pronic number, and the only pronic prime.[5]
Every integer greater than
1 will have at least two distinct factors; by definition, a prime number only has two distinct factors (itself and 1). Therefore, the
number-of-divisors function of positive integers satisfies,
where represents the
limit inferior (since there will always exist a larger prime number with a maximum of two divisors).[6]
A number is perfect if it is equal to its
aliquot sum, or the sum of all of its positive divisors excluding the number itself. This is equivalent to describing a perfect number as having a
sum of divisors equal to .
The
binary system has a
radix of two, and it is the
numeral system with the fewest tokens that allows denoting a natural number substantially more concisely (with tokens) than a direct representation by the corresponding count of a single token (with tokens). This number system is used extensively in
computing.[citation needed]
Two is the first
Mersenne prime exponent, and it is the difference between the first two
Fermat primes (
3 and
5).
Powers of two are essential in
computer science, and important in the
constructability of
regular polygons using basic tools (e.g., through the use of Fermat or
Pierpont primes). is the only number such that the sum of the reciprocals of its natural powers equals itself. In symbols,
The numbers two and
three are the only two prime numbers that are also consecutive
integers. Two is the first prime number that does not have a proper
twin prime with a difference two, while three is the first such prime number to have a twin prime,
five.[9][10] In consequence, three and five encase
four in-between, which is the
square of two, . These are also the two odd prime numbers that lie amongst the only
all-Harshad numbers (
1, 2, 4, and
6)[11] that are also the first four
highly composite numbers,[12] with 2 the only number that is both a prime number and a highly composite number. Furthermore, are the unique pair of
twin primes that yield the second and only
prime quadruplet that is of the form , where is the product of said twin primes.[13]
In the
Thue-Morse sequence, that
successively adjoins the
binaryBoolean complement from onward (in succession), the
critical exponent, or largest number of times an adjoining
subsequence repeats, is , where there exist a vast amount of square
words of the form [30] Furthermore, in , which counts the instances of between consecutive occurrences of in that is instead
square-free, the critical exponent is also , since contains factors of exponents close to due to containing a large factor of squares.[31] In general, the repetition threshold of an
infinite binary-rich word will be [32]
^Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer transl. David Bellos et al. London: The Harvill Press (1998): 393, Fig. 24.62
"{11, 13, 17, 19} is the only prime quadruplet {p, p+2, p+6, p+8} of the form {Q-4, Q-2, Q+2, Q+4} where Q is a product of a pair of twin primes {q, q+2} (for prime q = 3) because numbers Q-2 and Q+4 are for q>3 composites of the form 3*(12*k^2-1) and 3*(12*k^2+1) respectively (k is an integer)."
^Krieger, Dalia (2006). "On critical exponents in fixed points of non-erasing morphisms". In Ibarra, Oscar H.; Dang, Zhe (eds.). Developments in Language Theory: Proceedings 10th International Conference, DLT 2006, Santa Barbara, California, USA, June 26-29, 2006. Lecture Notes in Computer Science. Vol. 4036.
Springer-Verlag. pp. 280–291.
ISBN978-3-540-35428-4.
Zbl1227.68074.
"Only a(1) = 0 prevents this from being a simple continued fraction. The motivation for this alternate representation is that the simple pattern {1, 2*n, 1} (from n=0) may be more mathematically appealing than the pattern in the corresponding simple continued fraction (at
A003417)."