2 (
two) ( ) is a
number,
numeral, and
glyph. It is the
natural number following
1 and preceding
3.
In mathematics
Two has many properties in
mathematics.
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 (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
SmarandacheWellin 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 oddindexed 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 = x^{2} multiplication to exponentiation
 x^{x} = 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}{n1} \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
settheoretical 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
allHarshad number, a
meandric number, a
semimeandric number, and an
open meandric number.
Two is the number of
nQueens
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}^{n1} 2^k = 2^{n}  1
and also
 \sum_{k=a}^{n1} 2^k = 2^n  \sum_{k=0}^{a1} 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}^{d1} (n+k)\quad
\mbox{if}\quad d\ge 2 gives the nth ddimensional 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 = VE+F =
2.
As of 2008, there are only two known
Wieferich primes.
List of basic calculations
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 
Evolution of the glyph
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.
In fonts with
text figures, 2 usually
is of
xheight, for example,
.
In science
Astronomy
In technology
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
"SecondDay Yom Tov" (Yom Tov Sheini
Shebegaliyot) is a rabbinical enactment that mandates a
twoday celebration for each of the oneday 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
Luck
The most common philosophical
dichotomy is
perhaps the one of
good
and
evil, but there are many others. See
dualism for an overview. In
Hegel 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.
In pre1972
Indonesian and
Malay orthography,
2 was
shorthand for the
reduplication that
forms plurals:
orang "person",
orangorang or
orang2 "people".
In North American educational systems, the number 2.00 denotes a
gradepoint
average of "C," which in some colleges and universities is the
minimum required for good academic standing at the
undergraduate level.
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 TransCanada
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 twopoint conversion is a PAT where the
ball crosses the goal line via run or pass. (In sixman 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 to children as releasing solid waste
in the washroom. An example would be: "Do you need to go number 1
or number 2?"
See also
References
 Wells, D. The Penguin Dictionary of Curious and Interesting
Numbers London: Penguin Group. (1987): 41–44
 Bryan Bunch, The Kingdom of Infinite Number. New York:
W. H. Freeman & Company (2000): 31
 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."
 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
 For a typical example, see the University of Oklahoma grading regulations.
External links