Two is most commonly a
determiner used with
plural countable nouns, as in two days or I'll take these two.[1]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).[2]
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/.[2]
Characterizations of the number
Parity
An
integer is determined to be
even if it is
divisible by two. For integers written in a numeral system based on an even number such as
decimal, divisibility by two 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
multiple of 2 will end in
0, 2, 4, 6, or
8.[3]
1 is neither prime nor
composite yet
odd.
0, which is an
origin to the
integers in the
real line, especially when considered alongside
negative integers, is neither prime nor composite, however it is distinctively even (as a multiple of two) since if it were to be odd, then for some integer there would be that yields a of , which is a contradiction (however, for a
function, the zero function is the only function to both be even and odd).
Primality
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.[4]
The divisor function
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,
This means that is the only
set of numbers whose distinct divisors are also consecutive integers (when excluding
negative integers).
Twin primes
Meanwhile, the numbers two and
three are the only two prime numbers that are consecutive
integers, where the number two is also adjacent to the
unit. 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.[6][7] 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)[8] that are also the first four
highly composite numbers,[9] with the only number that is both a prime number and a "highly composite number".[a]
In the smallest
Cunningham chains of nearly doubled primes (of the first and second kind) two is the first member, as part of the sets and .
The first fifteen prime numbers between and are also consecutive primes that are part of
Bhargava’s seventeen-integer quadratic matrix representative of all prime numbers (only two other numbers are part of this set of prime integers, namely the nineteenth and twenty-first prime numbers
67 and
73).[12] The seventh
square number, , is in equivalence with the sum of the first and fifteenth primes, where also .
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,
In
decimal representation, after the first two, three, four and five
digits in the
approximation of pi () the number is the only digit greater than zero not yet represented (overall, up to the largest appearing digit).[c]
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]
Thue-Morse sequence
In the
Thue-Morse sequence, that
successively adjoins the binary
Boolean 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 [24] 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.[25] In general, the repetition threshold of an
infinite binary-rich word will be [26]
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.[29]
^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.[10]
^Where is strictly the first prime number, and the only even prime number, the sum between the second prime number
3 and the second
composite number6 (that is twice 3, or thrice 2) is the first
oddcomposite number, . At nine, the ratio of composite numbers to prime numbers is one-to-one, a proportion that is only repeated again at
11 and
13.
"{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)."
"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)."
^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.
^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