1 (number)

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! colspan'2' align'center' style'font: 10em times; background:ccc;' 1
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Cardinal numberCardinal
1 one
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Ordinal numbers (linguistics)Ordinal
1st first
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Numeral system
Unary numeral systemunary
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Factorization
1
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Divisors
1
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Greek numeral
α'
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Roman numeral
I
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Roman numeral (Unicode)
Ⅰ, ⅰ
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Hindu-Arabic numeral systemArabic ١
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Ge'ez alphabetGe'ez ፩
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Bengali languageBengali ১
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Chinese numeral 一，弌，壹
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Korean languageKorean 일, 하나
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Devanāgarī १
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Tamil languageTamil ௧
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Hebrew numeralsHebrew א (Alef)
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Khmer numeralsKhmer ១
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Thai numeralsThai ๑
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numerical prefixprefixes
Wiktionary:mono-mono- /Wiktionary:haplo-haplo- (from Greek languageGreek)
Wiktionary:uni-uni- (from Latin)
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Binary numeral systemBinary
1
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Octal
1
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Duodecimal
1
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Hexadecimal
1
)
, using J as a symbol for 1
"1" ("one") is a number, number namesnumeral, and the name of the glyph representing that number.
It represents a single entity, the unit of counting or measurement. For example, a line segment of 'unit length' is a line segment of length 1.

As a number
One, sometimes referred to as "unity", is the integer before 2 (number)two and after 0 (number)zero. One is the first non-zero number in the natural numbers as well as the first odd number in the natural numbers.

Any number multiplied by one is the number, as one is the Identity elementidentity for multiplication. As a result, one is its own factorial, its own Square (algebra)square, its own Cube (algebra)cube, and so on. One is also the empty product as any number multiplied by one is itself, which is the same as multiplying by no numbers at all.

As a digit

The glyph used today in the Western world to represent the number 1, a vertical line, often with a serif at the top and sometimes a short horizontal line at the bottom, traces its roots back to the Indian subcontinentIndians, who wrote 1 as a horizontal line, much like the Chinese LanguageChinese script “ 一 ”(Yi). The Gupta scriptGupta wrote it as a curved line, and the Nagari sometimes added a small circle on the left (rotated a quarter turn to the right, this 9-look-alike became the present day numeral 1 in the Gujarati languageGujarati and Punjabi languagePunjabi scripts). The Nepali languageNepali also rotated it to the right, but kept the circle small.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): 392, Fig. 24.61 This eventually became the top serif in the modern numeral, but the occasional short horizontal line at the bottom probably originates from similarity with the Roman numeral I. In some European countries (e.g., Germany) the little serif at the top is sometimes extended into a long upstroke, sometimes as long as the vertical line, which can lead to confusion with the glyph for seven in other countries. Where the 1 is written with a long upstroke, the number 7 has a horizontal stroke through the vertical line.

While the shape of the 1 character has an Ascender (typography)ascender in most modern typefaces, in typefaces with text figures the character usually is of x-height, as, for example, in .

Many older typewriters do not have a separate symbol for '1' and use the lowercase 'l' instead. It is possible to find cases when the uppercase 'J' is used, while it may be for decorative purposes.

Mathematics
Mathematically, 1 is
in arithmetic (algebra) and calculus, the natural number that follows 0 (number)0 and precedes 2 (number)2, the multiplicative identity (mathematics)identity of the integers, real numbers and complex numbers;
more generally, in abstract algebra, the multiplicative identity ('unity'), usually of a ring (mathematics)ring.

One cannot be used as the base of a positional numeral system; sometimes tally marktallying is referred to as 'base 1', since only one mark (the tally) is needed, but this is not a positional notation.

The logarithms base 1 are undefined, since the function 1'x' always equals 1 and so has no unique inverse functioninverse.

In the real number system, 1 can be represented in two ways as a recurring decimal: as 1.000... and as 0.999... ('q.v.').

Formalizations of the natural numbers have their own representations of 1:
in the Peano axioms, 1 is the successor of 0;
in Principia Mathematica, 1 is defined as the set of all singleton (mathematics)singletons (sets with one element);
the Von Neumann cardinal assignment of natural numbers, 1 is defined as the Set (mathematics)set (0).

In a multiplicative group (mathematics)group or monoid, the identity element is sometimes denoted '1', but "e" (from the German 'Einheit', unity) is more traditional. However, '1' is especially common for the multiplicative identity of a ring (mathematics)ring, i.e. when an addition and '0' are also present. When such a ring has Characteristic (algebra)characteristic 'n' not equal to 0, the element called 1 has the property that 'n'1 1'n' '0' (where this '0' is the additive identity of the ring). Important examples are general Field (mathematics)fields.

One is the first figurate number of every kind, such as triangular number, pentagonal number and centered hexagonal number to name just a few.

Because of the multiplicative identity, if 'f'('x') is a multiplicative function, then 'f'(1) must equal 1.

It is also the first and second numbers in the Fibonacci numberFibonacci sequence, and is the first number in many mathematical sequences. As a matter of convention, Sloane's early 'Handbook of Integer Sequences' added an initial 1 to any sequence that didn't already have it, and considered these initial 1's in its lexicographic ordering. Sloane's later 'Encyclopedia of Integer Sequences' and its Web counterpart, the 'On-Line Encyclopedia of Integer Sequences', ignore initial ones in their lexicographic ordering of sequences, because such initial ones often correspond to trivial cases.

One is neither a prime number nor a composite number, but a unit (ring theory)unit, like -1 and, in the Gaussian integers, 'imaginary uniti' and '-i'. The fundamental theorem of arithmetic guarantees factorizationunique factorization over the integers only up to units ('e.g.' 4 22 (-1)4×123×22).

The definition of a field (mathematics)field requires that 1 must not be equal to zero0. Thus, there are no fields of characteristic 1. Nevertheless, abstract algebra can consider the field with one element, which is not a singleton though, and is not a set at all.

One is the only positive integer divisible by exactly one positive integer (whereas prime numbers are divisible by exactly two positive integers, composite numbers are divisible by more than two positive integers, and zero is divisible by all positive integers). One was formerly considered prime by some mathematicians, using the definition that a prime is divisible only by one and itself. However, this complicates the fundamental theorem of arithmetic, so modern definitions exclude units. The last professional mathematician to publicly label 1 a prime number was Henri Lebesgue in 1899.

One is one of three possible values of the Möbius function: it takes the value one for square-free integersquare-free integers with an even number of distinct prime factors.

One is the only odd number in the range of Euler's totient function φ('x'), in the cases 'x' 1 and 'x' 2.

One is the only 1-perfect number (see multiply perfect number).

By definition, 1 is the magnitude (mathematics)magnitude or absolute value of a unit vector and a identity matrixunit matrix (more usually called an 'identity matrix'). Note that the term 'unit matrix' is sometimes used to mean something Matrix of onesquite different.

By definition, 1 is the probability of a certain event.

One is the most common leading digit in many sets of data, a consequence of Benford's law.

The ancient Egyptians represented all fractions (with the exception of 2/3 and 3/4) in terms of sum of fractions with numerator 1 and distinct denominators. For example, \frac(2)(5) \frac(1)(3) + \frac(1)(15). Such representations are popularly known as Egyptian Fractions or Unit Fractions.

The Generating Function which has all coefficients 1 is given by

\frac(1)(1-x) 1 + x + x^2 + x^3 + \cdots.

This power series converges and has finite value if, and only if, x .

Table of basic calculations

(class'wikitable' style'text-align: center; background: white'
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!width'105px'Multiplication
!1
!2
!3
!4
!5
!6
!7
!8
!9
!10
!width'5px'
!11

!12
!13
!14
!15
!16
!17
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!width'5px'
!21
!22
!23
!24
!25
!width'5px'
!50
!100
!1000
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1 \times x
"1"
2 (number)2
3 (number)3
4 (number)4
5 (number)5
6 (number)6
7 (number)7
8 (number)8
9 (number)9
10 (number)10
!
11 (number)11
12 (number)12
13 (number)13
14 (number)14
15 (number)15
16 (number)16
17 (number)17
18 (number)18
19 (number)19
20 (number)20
!
21 (number)21
22 (number)22
23 (number)23
24 (number)24
25 (number)25
!
50 (number)50
100 (number)100
1000 (number)1000
)

(class'wikitable' style'text-align: center; background: white'
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!width'105px'Division (mathematics)Division
!1
!2
!3
!4
!5
!6
!7
!8
!9
!10
!width'5px'
!11
!12
!13
!14
!15
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1 \div x
"1"
0.5
0.\overline(3)
0.25
0.2
0.1\overline(6)
0.\overline(142857)
0.125
0.\overline(1)
0.1
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0.\overline(0)\overline(9)
0.08\overline(3)
0.\overline(076923)
0.0\overline(714285)
0.0\overline(6)
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x \div 1
"1"
2
3
4
5
6
7
8
9
10
!
11
12
13
14
15
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(class'wikitable' style'text-align: center; background: white'
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!width'105px'Exponentiation
!1
!2
!3
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!5
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!11
!12
!13
!14
!15
!16
!17
!18
!19
!20
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1 ^ x\,
"1"
"1"
"1"
"1"
"1"
"1"
"1"
"1"
"1"
"1"
!
"1"
"1"
"1"
"1"
"1"
"1"
"1"
"1"
"1"
"1"
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x ^ 1\,
"1"
2
3
4
5
6
7
8
9
10
!
11
12
13
14
15
16
17
18
19
20
)

In technology

The resin identification code used in recycling to identify polyethylene terephthalate.

In science
The atomic number of hydrogen.

See also

−1 (number)−1Notes


en.wikipedia.org