Two is most commonly a determiner used with plural countable nouns, as in two daysorI'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
Aninteger 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,
where represents the limit inferior (since there will always exist a larger prime number with a maximum of two divisors).[5] Aside from square numbers and prime powers raised to an even exponent, or integers that are the product of an even number of prime powers with even exponents, an integer will have a that is a multiple of . The two smallest natural numbers have unique properties in this regard: is the only number with a single divisor (itself), where on the other hand, is the only number to have an infinite number of divisors, since dividing zero by any strictly positive or negative integer yields (i.e., aside from division of zero by zero, ).
is the only set of numbers whose distinct divisors (with more than one) are also consecutive integers, when excluding negative integers.[a]
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".[b]
Powers of two are essential in computer science, and important in the constructabilityofregular 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,
Every number ispolygonal by being -gonal (i.e., a natural number), as well as the root of some type of -gonal number. For , being -gonal and -gonal is the same, which make two the only number that is polygonal in only one way.
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.[26]
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 [27] 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.[28] In general, the repetition threshold of an infinite binary-rich word will be [29]
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.[34]
^Since zero has an infinite number of divisors, except for itself, the real line does not divide wholesomely into zero (in parts); as the only integer, zero (itself) is the only divisor that does not strictly map back to itself through division, when the only elementin is zero (only in select cases). Therefore, is seen as in an indeterminate form, since it can behave in various different ways, depending on the context of a function.
^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 (atA003417)."
^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