1

"One" redirects here. For other uses, see One (disambiguation).

1
Template:Numbers (digits)
Cardinal	1 one
Ordinal	1st first
Numeral system	unary
Factorization	$1$
Divisors	1
Greek numeral	α'
Roman numeral	I
Roman numeral (Unicode)	Ⅰ, ⅰ
Persian	١ - یک
Arabic	١
Ge'ez	፩
Bengali	১
Chinese numeral	一，弌，壹
Korean	일, 하나
Devanāgarī	१
Tamil	௧
Hebrew	א (alef)
Khmer	១
Thai	๑
prefixes	mono- /haplo- (from Greek) uni- (from Latin)
Binary	1
Octal	1
Duodecimal	1
Hexadecimal	1

1 (one, Template:Pron-en, also English pronunciation: /ˈwɒn/) is a number, 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 the 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 empty product, as any number multiplied by one is itself, which produces the same result 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 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. 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 $\mathrm {I}$. 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 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

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

One cannot be used as the base of a positional numeral system; sometimes tallying 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.

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.).

1 is the smallest number of ways n objects can be arranged.

Contrary to popular opinion, 1 is NOT a prime number OR a composite number.

1 is the smallest number that is both triangular and pentagonal.

Also, 1 is the smallest number that is both interesting and boring at the same time

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 singletons (sets with one element);
• in the Von Neumann cardinal assignment of natural numbers, 1 is defined as the set {0}.

In a multiplicative 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, 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.

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 sequence 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 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, 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 (e.g., 4 = 2² = (-1)⁴×1²³×2²).

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. 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 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 or absolute value of a 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 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 and 3/4) in terms of sums 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 that 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, $|x|<1$.

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
$1\times x$	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
$1\div x$	1	0.5	$0.{\overline {3}}$	0.25	0.2	$0.1{\overline {6}}$	$0.{\overline {142857}}$	0.125	$0.{\overline {1}}$	0.1		$0.{\overline {0}}{\overline {9}}$	$0.08{\overline {3}}$	$0.{\overline {076923}}$	$0.0{\overline {714285}}$	$0.0{\overline {6}}$
$x\div 1$	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
$1^{x}\,$	1	1	1	1	1	1	1	1	1	1		1	1	1	1	1	1	1	1	1	1
$x^{1}\,$	1	2	3	4	5	6	7	8	9	10		11	12	13	14	15	16	17	18	19	20

In technology

1 as a resin identification code, used in recycling

• The resin identification code used in recycling to identify polyethylene terephthalate
• Used in binary code

In science

• The atomic number of hydrogen

See also

• −1
• One (word)
• Root of unity