1 (number)

"One" and "No. 1" redirect here. For other uses, see One (disambiguation).
← 0 1 2 →
−1 0 1 2 3 4 5 6 7 8 9 →
Cardinal one
Ordinal 1st
(first)
Numeral system unary
Factorization 1
Divisors 1
Roman numeral I
Roman numeral (unicode) Ⅰ, ⅰ
Greek prefix mono- /haplo-
Latin prefix uni-
Binary 12
Ternary 13
Quaternary 14
Quinary 15
Senary 16
Octal 18
Duodecimal 112
Hexadecimal 116
Vigesimal 120
Base 36 136
Greek numeral α'
Persian ١ - یک
Arabic ١
Urdu ۱
Sindhi ١
Ge'ez ፩
Bengali & Assamese ১
Georgian Ⴁ/ⴁ/ბ(Bani)
Chinese numeral 一，弌，壹
Korean 일, 하나
Devanāgarī १ (ek)
Telugu ೧
Tamil ௧
Kannada ೧
Hebrew א (alef)
Khmer ១

1 (one; /ˈwʌn/ or UK /ˈwɒn/, also called unit, unity, and (multiplicative) identity), is a number, a numeral, 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 two and after 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 that number, as one is the identity for multiplication. As a result, one is its own factorial, its own square, its own cube, and so on. One is also the result of the empty product, as any number multiplied by one is itself. It is also the only natural number that is neither composite nor prime with respect to division, but instead considered a unit.

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 Indians, who wrote 1 as a horizontal line, much like the Chinese character 一. The Gupta 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 and Punjabi scripts). The Nepali also rotated it to the right but kept the circle small.[1] 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 countries, 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 in most modern typefaces, in typefaces with text figures, the character usually is of x-height, as, for example, in .

The 24-hour tower clock in Venice, using J as a symbol for 1.

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:

One cannot be used as the base of a positional numeral system, as the only digit that would be permitted in such a system would be 0. (Sometimes tallying is referred to as "base 1", since only one mark — the tally itself — is needed, but this is not a positional notation.)

Since the base 1 exponential function (1x) always equals 1, its inverse does not exist (which would be called the logarithm base 1 if it did exist).

There are two ways to write the real number 1 as a recurring decimal: as 1.000..., and as 0.999.... There is only one way to represent the real number 1 as a Dedekind cut:

(\left\{x|x<1\right\},\left\{x|x\geq 1\right\}).

Formalizations of the natural numbers have their own representations of 1:

In a multiplicative group or monoid, the identity element is sometimes denoted 1, especially in abelian groups, but e (from the German Einheit, "unity") is more traditional. However, 1 is especially common for the multiplicative identity of a ring, i.e., when an addition and 0 are also present. When such a ring has characteristic n not equal to 0, the element called 1 has the property that n1 = 1n = 0 (where this 0 is the additive identity of the ring). Important examples are general 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.

In many mathematical and engineering equations, numeric values are typically normalized to fall within the unit interval from 0 to 1, where 1 usually represents the maximum possible value in the range of parameters.

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 number in the Fibonacci sequence (0 is the zeroth) and is the first number in many other mathematical sequences. As a matter of convention, Sloane's early Handbook of Integer Sequences added an initial 1 to any sequence that did not already have it and considered these initial 1s 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, like −1 and, in the Gaussian integers, i and −i. The fundamental theorem of arithmetic guarantees unique factorization over the integers only up to units. (For example, 4 = 22, but if units are included, is also equal to, say, (−1)6 × 123 × 22, among infinitely many similar "factorizations".)

The definition of a field requires that 1 must not be equal to 0. Thus, there are no fields of characteristic 1. Nevertheless, abstract algebra can consider the field with one element, which is not a singleton 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.

One is one of three possible values of the Möbius function: it takes the value one for square-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, absolute value, or norm of a unit complex number, unit vector, and a unit matrix (more usually called an identity matrix). Note that the term unit matrix is sometimes used to mean something quite different.

By definition, 1 is the probability of an event that is almost certain to occur.

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) in terms of sums of fractions with numerator 1 and distinct denominators. For example, 2/5 = 1/3 + 1/15. Such representations are popularly known as Egyptian Fractions or Unit Fractions.

The generating function that has all coefficients 1 is given by

1/1 − x = 1 + x + x2 + x3 + …

This power series converges and has finite value if and only if, |x| < 1.

Table 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
1 × x 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
Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1 ÷ x 1 0.5 0.3 0.25 0.2 0.16 0.142857 0.125 0.1 0.1 0.09 0.083 0.076923 0.0714285 0.06
x ÷ 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1x 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

In technology

In science

In astronomy

In philosophy

In the philosophy of Plotinus and a number of other neoplatonists, The One is the ultimate reality and source of all existence. Philo of Alexandria (20 BC – AD 50) regarded the number one as God's number, and the basis for all numbers ("De Allegoriis Legum," ii.12 [i.66]).

In literature

In comics

In sports

Etymology

The word one can be used as a noun, an adjective and a pronoun.[3]

It comes from the Old English word an,[3] which comes from the Proto-Germanic root *ainaz.[3] The Proto-Germanic root *ainaz comes from the Proto-Indo-European root *oi-no-.[3]

Compare the Proto-Germanic root *ainaz to Old Frisian an, Gothic ains, Danish een, Dutch een, German eins and Old Norse einn.

Compare the Proto-Indo-European root *oi-no- (which means one, single[3]) to Greek oinos (which means "ace" on dice[3]), Latin unus (one[3]), Old Persian aivam, Old Church Slavonic -inu and ino-, Lithuanian vienas, Old Irish oin and Breton un (one[3]).

See also

Notes

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  1. ↑ 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
  2. ↑ "Plastic Packaging Resins" (PDF). American Chemistry Council.
  3. 1 2 3 4 5 6 7 8 "Online Etymology Dictionary". Douglas Harper.

External links

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