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2

−1 0 1 2 3 4 5 6 7 8 9

Ordinal Number 2nd
second
Numeral system binary
Factorisation prime
Gaussian integer factorisation (1 + i)(1 − i)
Divisors 1, 2
Greek numeral β'
Roman numeral II
Roman numeral (Unicode) Ⅱ, ⅱ
Arabic ٢,2
Ge'ez
Bengali
Chinese numeral 二,弍,贰,貳
Devanāgarī
Tamil
Hebrew ב (Bet)
Khmer
Korean
Thai
prefixes di- (from Greek)

duo- bi- (from Latin) twi- (Old English)

Binary 10
Cardinal 2
two
Octal 2
Hexadecimal 2
Duodecimal 2
Place
Two.png

2 (two) (pronounced /ˈtuː/ ( listen)) is a number, numeral, and glyph. It is the natural number following 1 and preceding 3.

Contents

In mathematics

Two has many properties in mathematics.[1] An integer is called even if it is divisible by 2. For integers written in a numeral system based on an even number, such as decimal and hexadecimal, divisibility by 2 is easily tested by merely looking at the last digit. If it is even, then the whole number is even. In particular, when written in the decimal system, all multiples of 2 will end in 0, 2, 4, 6, or 8.

Two is the smallest and the first prime number, and the only even one[2] (for this reason it is sometimes humorously called "the oddest prime"[3]). The next prime is three. Two and three are the only two consecutive prime numbers. 2 is the first Sophie Germain prime, the first factorial prime, the first Lucas prime, and the first Smarandache-Wellin prime. It is an Eisenstein prime with no imaginary part and real part of the form 3n − 1. It is also a Stern prime, a Pell number, and a Markov number, appearing in infinitely many solutions to the Markov Diophantine equation involving odd-indexed Pell numbers.

It is the third Fibonacci number, and the third and fifth Perrin numbers.

Despite being a prime, two is also a highly composite number, because it has more divisors than the number one. The next highly composite number is four.

Vulgar fractions with 2 or 5 in the denominator do not yield infinite decimal expansions, as is the case with most primes, because 2 and 5 are factors of ten, the decimal base.

Two is the base of the simplest numeral system in which natural numbers can be written concisely, being the length of the number a logarithm of the value of the number (whereas in base 1 the length of the number is the value of the number itself); the binary system is used in computers.

For any number x:

x+x = 2·x addition to multiplication
x·x = x2 multiplication to exponentiation
xx = x↑↑2 exponentiation to tetration

Two also has the unique property that 2+2 = 2·2 = 2²=2↑↑2=2↑↑↑2, and so on, no matter how high the operation is.

Two is the only number x such that the sum of the reciprocals of the powers of x equals itself. In symbols

\sum_{k=0}^{\infin}\frac {1}{2^k}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots=2.

This comes from the fact that:

\sum_{k=0}^\infin \frac {1}{n^k}=1+\frac{1}{n-1} \quad\mbox{for all} \quad n\in\mathbb R > 1.

Powers of two are central to the concept of Mersenne primes, and important to computer science. Two is the first Mersenne prime exponent.

Taking the square root of a number is such a common mathematical operation, that the spot on the root sign where the exponent would normally be written for cubic roots and other such roots, is left blank for square roots, as it is considered tacit.

The square root of two was the first known irrational number.

The smallest field has two elements.

In the set-theoretical construction of the natural numbers, 2 is identified with the set \{\{\emptyset\},\emptyset\}. This latter set is important in category theory: it is a subobject classifier in the category of sets.

Two is a primorial, as well as its own factorial. Two often occurs in numerical sequences, such as the Fibonacci number sequence, but not quite as often as one does. Two is also a Motzkin number, a Bell number, an all-Harshad number, a meandric number, a semi-meandric number, and an open meandric number.

Two is the number of n-Queens Problem solutions for n = 4. With one exception, all known solutions to Znám's problem start with 2.

Two also has the unique property such that

\sum_{k=0}^{n-1} 2^k = 2^{n} - 1

and also

\sum_{k=a}^{n-1} 2^k = 2^n - \sum_{k=0}^{a-1} 2^k - 1

for a not equal to zero

Two has a connection to triangular numbers:

\prod_{k=0}^n 2^k= 2^{tri_2(n)}

where tri_d(n)= \frac {1}{d!}\prod_{k=0}^{d-1} (n+k)\quad \mbox{if}\quad d\ge 2 gives the nth d-dimensional simplex number. When d=2,

tri_2(n)=\frac {n^2+n}{2}.

The number of domino tilings of a 2×2 checkerboard is 2.

For any polyhedron homeomorphic to a sphere, the Euler characteristic is

\chi = V-E+F = 2.\

As of 2008, there are only two known Wieferich primes.

List of basic calculations

Multiplication 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 50 100 1000
2 \times x 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 100 200 2000
Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
2 \div x 2 1 0.\overline{6} 0.5 0.4 0.\overline{3} 0.\overline{285714} 0.25 0.\overline{2} 0.2 0.\overline{1}\overline{8} 0.1\overline{6} 0.\overline{1}5384\overline{6} 0.\overline{1}4285\overline{7} 0.1\overline{3}
x \div 2 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5
Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13
2 ^ x\, 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192
x ^ 2\, 1 4 9 16 25 36 49 64 81 100 121 144 169

Evolution of the glyph

Evolution2glyph.png

The glyph we use today in the Western world to represent the number 2 traces its roots back to the Brahmin Indians, who wrote 2 as two horizontal lines (it is still written that way in modern Chinese and Japanese). The Gupta rotated the two lines 45 degrees, making them diagonal, and sometimes also made the top line shorter and made its bottom end curve towards the center of the bottom line. Apparently for speed, the Nagari started making the top line more like a curve and connecting to the bottom line. The Ghubar Arabs made the bottom line completely vertical, and now the glyph 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 glyph.[4]

In fonts with text figures, 2 usually is of x-height, for example, Text figures 256.svg.

In science

Seven-segment 2.svg

Astronomy

In technology

2 as a resin identification code, used in recycling.

In religion

The number 2 is important in Judaism, with one of the earliest reference being that God ordered Noah to put two of every animal in his ark (see Noah's Ark). Later on, the Ten Commandments were given in the form of two tablets (Shnei Luchot HaBrit).

The number also has ceremonial importance, such as the two candles that are traditionally kindled to usher in the Shabbat, recalling the two different ways Shabbat is referred to in the two times the Ten Commandments are recorded in the Torah. These two expressions are known in Hebrew as שמור וזכור ("guard" and "remember"), as in "Guard the Shabbat day to sanctify it" (Deut. 5:12) and "Remember the Shabbat day to sanctify it" (Ex. 20:8) Two challahs (lechem mishnah) are placed on the table for each Shabbat meal and a blessing made over them, to commemorate the double portion of manna which fell in the desert every Friday to cover that day's meals and the Shabbat meals

In Jewish law, the testimony of two witnesses are required to verify and validate events, such as marriage, divorce, and a crime that warrants capital punishment

"Second-Day Yom Tov" (Yom Tov Sheini Shebegaliyot) is a rabbinical enactment that mandates a two-day celebration for each of the one-day Jewish festivals (i.e., the first and seventh day of Passover, the day of Shavuot, the first day of Sukkot, and the day of Shemini Atzeret) outside the land of Israel

Numerological significance

The twos of all four suits in playing cards

The most common philosophical dichotomy is perhaps the one of good and evil, but there are many others. See dualism for an overview. In Hegelian dialectic, the process of synthesis creates two perspectives from one.

Two (二, èr) is a good number in Chinese culture. There is a Chinese saying, "good things come in pairs". It is common to use double symbols in product brandnames, e.g. double happiness, double coin, double elephants etc. Cantonese people like the number two because it sounds the same as the word "easy" (易) in Cantonese. However, it is also used to identify people who act foolish, arrogant and so on.

In Finland, two candles are lit on Independence Day. Putting them on the windowsill invokes the symbolical meaning of division, and thus independence.[citation needed]

In pre-1972 Indonesian and Malay orthography, 2 was shorthand for the reduplication that forms plurals: orang "person", orang-orang or orang2 "people".[citation needed]

In North American educational systems, the number 2.00 denotes a grade-point average of "C," which in some colleges and universities is the minimum required for good academic standing at the undergraduate level.[6]

In Astrology, Taurus is the second sign of the Zodiac.

In other fields

Groups of two:

  • The name of several fictional characters: Number Two.
  • The position of the President of the Mess Committee at the Australian Defence Force Academy, commonly referred to as number 2, which is currently held by OCDT Dale Bogle “flamer”.
  • The designation of the Trans-Canada Highway in most of the province of New Brunswick.
  • U.S. Route 2, two separated highways in the northern tier of the United States, the western segment connecting Everett, Washington to St. Ignace, Michigan and the eastern route connecting Rouses Point, New York to Houlton, Maine.
  • The lowest channel of television in the United States, Canada, and Mexico on which television signals are broadcast.
  • In American football, a two-point conversion is a PAT where the ball crosses the goal line via run or pass. (In six-man football, however, the traditional PAT kick is worth two points, whereas a PAT via pass or run is only one point.)
  • Number 2 is also reference defecation (as opposed to "Number 1" for urination)

See also

References

  1. ^ Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 41–44
  2. ^ Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 31
  3. ^ John Horton Conway & Richard K. Guy, The Book of Numbers. New York: Springer (1996): 25. "Two is celebrated as the only even prime, which in some sense makes it the oddest prime of all."
  4. ^ 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
  5. ^ Xavier Borg, Engineer (2006-02-02). "Magic numbers derived from VPM nuclear model". Blaze Labs Research. http://www.blazelabs.com/magicnumbers.pdf. Retrieved 2007-03-24. 
  6. ^ For a typical example, see the University of Oklahoma grading regulations.

External links


Simple English

2
Cardinal 2
two
Ordinal number 2nd
second
Numeral system binary
Factorization prime
Gaussian integer factorization (1 + i)(1 - i)
Divisors 1, 2
Greek numeral β'
Roman numeral II
Roman numeral (Unicode) Ⅱ, ⅱ
Arabic ٢
Ge'ez
Bengali
Chinese numeral 二,弍,贰,貳
Devanāgarī
Hebrew ב (Bet)
Khmer
Korean
Thai
prefixes di- (from Greek)

duo- bi- (from Latin) twi- (Old English)

Binary 10
Octal 2
Duodecimal 2
Hexadecimal 2

[[File:|right]] 2 (two) is a number, numeral, and glyph. It is the natural number after One and before Three.

In mathematics

Two has many properties in mathematics. For example: 1 + 1 = 2. [1] An integer is called even if it can be divided by 2. For integers written in a numeral system based on an even number, such as decimal and hexadecimal, it can be easily tested if it is a multiple of 2 by merely looking at the one's place digit. If it is even, then the whole number is even. In particular, when written in the decimal system, all multiples of 2 will end in 0, 2, 4, 6, or 8.

Two is the smallest and the first prime number, and the only even one (for this reason it is sometimes humorously called "the oddest prime"). The next prime is three. Two and three are the only two consecutive prime numbers. 2 is the first Sophie Germain prime, the first factorial prime, the first Lucas prime, and the first Smarandache-Wellin prime. It is an Eisenstein prime with no imaginary part and real part of the form 3n - 1. It is also a Stern prime, a Pell number, and a Markov number, appearing in infinitely many solutions to the Markov Diophantine equation involving odd-indexed Pell numbers.

It is the third Fibonacci number, and the third and fifth Perrin numbers.

Even though it is a prime number, two is also a highly composite number, because it has more divisors than the number one. The next highly composite number is four.

Vulgar fractions with 2 or 5 in the denominator do not yield infinite decimal expansions, as is the case with most primes, because 2 and 5 are factors of ten, the decimal base.

Two is the base of the simplest system of numbers in which natural numbers can be written concisely, being the length of the number a logarithm of the value of the number (whereas in base 1 the length of the number is the value of the number itself); the binary system is used in computers.

For any number x:

x+x = 2·x addition to multiplication
x·x = x2 multiplication to exponentiation
xx = x↑↑2 exponentiation to tetration

Two also has the unique property that 2+2 = 2·2 = 2²=2↑↑2=2↑↑↑2, and so on, no matter how high the operation is.

Two is the only number x such that the sum of the reciprocals of the powers of x equals itself. In symbols: \sum_{k=0}^{\infin}\frac {1}{2^k}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots=2.

This comes from the fact that:

\sum_{k=0}^\infin \frac {1}{n^k}=1+\frac{1}{n-1} \quad\mbox{for all} \quad n\in\mathbb R > 1.

Powers of two are central to the concept of Mersenne primes, and important to computer science. Two is the first Mersenne prime exponent.

Taking the square root of a number is such a common mathematical operation, that the spot on the root sign where the exponent would normally be written for cubic roots and other such roots, is left blank for square roots, as it is considered tacit.

The square root of two was the first known irrational number.

The smallest field has two elements.

In the set-theoretical construction of the natural numbers, 2 is identified with the set \{\{\emptyset\},\emptyset\}. This latter set is important in category theory: it is a subobject classifier in the category of sets.

Two is a primorial, as well as its own factorial. Two often occurs in numerical sequences, such as the Fibonacci number sequence, but not quite as often as one does. Two is also a Motzkin number, a Bell number, an all-Harshad number, a meandric number, a semi-meandric number, and an open meandric number.

Two is the number of n-Queens Problem solutions for n = 4. With one exception, all known solutions to Znám's problem start with 2.

Two also has the unique property such that:

\sum_{k=0}^{n-1} 2^k = 2^{n} - 1

and also

\sum_{k=a}^{n-1} 2^k = 2^n - \sum_{k=0}^{a-1} 2^k - 1

for a not equal to zero

Two has a connection to triangular numbers:

\prod_{k=0}^n 2^k= 2^{tri_2(n)}

Where tri_d(n)= \frac {1}{d!}\prod_{k=0}^{d-1} (n+k)\quad \mbox{if}\quad d\ge 2 gives the nth d-dimensional simplex number. When d=2,

tri_2(n)=\frac {n^2+n}{2}

The number of domino tilings of a 2×2 checkerboard is 2.

For any polyhedron homeomorphic to a sphere, the Euler characteristic is \chi = V-E+F = 2.

As of 2008, there are only two known Wieferich primes.

List of basic calculations

Multiplication 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 50 100 1000
2 \times x 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 100 200 2000
Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
2 \div x 2 1 0.\overline{6} 0.5 0.4 0.\overline{3} 0.\overline{2}8571\overline{4} 0.25 0.\overline{2} 0.2 0.\overline{1}\overline{8} 0.1\overline{6} 0.\overline{1}5384\overline{6} 0.\overline{1}4285\overline{7} 0.1\overline{3}
x \div 2 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5
Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13
2 ^ x\, 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192
x ^ 2\, 1 4 9 16 25 36 49 64 81 100 121 144 169

References

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  1. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 41–44
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