# 1 (number)

﻿
1 (number)
1
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
Bengali
Chinese numeral 一，弌，壹
Korean 일, 하나
Devanāgarī
Telugu
Tamil
Hebrew א (alef)
Khmer
Thai
prefixes mono- /haplo- (from Greek)

uni- (from Latin)

Binary 1
Octal 1
Duodecimal 1

1 (one;  /ˈwʌn/ or UK /ˈwɒn/) 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 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.[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 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

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 1x 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.).

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 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)6×123×22).

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 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 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 sports

• Many sports use the number 1 as their standard scoring increment—examples include goals in a large number of sports, runs in baseball and cricket, and points in volleyball. Examples where 1 is a non-standard increment, or used for one of several possible classes of scores, are listed below by sport.
• In Australian rules football, 1 point is awarded to the attacking team for a behind, scored when:
• The ball is kicked by the attacking team and passes between a goal post (taller post) and the nearest behind post (shorter post) on the defensive side of the field without touching the behind post.
• The ball passes between the defending team's goal posts, but either (1) was not kicked by the attacking team or (2) hit a goal post.
• The defending team deliberately forces the ball between any two of its own posts. This particular score is officially called a "rushed behind".
• In baseball scoring, the number 1 is assigned to the pitcher.
• 1 point is awarded for a successful free throw.
• The number 1 is used to designate the point guard position.
• In Gaelic football, hurling and camogie, a "point", with a scoring value of 1, is awarded when the attacking team legally sends the ball over the opponent's crossbar (above the goal).
• In gridiron football codes, one point is awarded under the following circumstances:
• In almost all leagues, for a successful place kick after a touchdown. In American football, the score is formally known as a "try", although the terms "extra point", "conversion", "PAT" (for "point after touchdown"), and "point after" are far more commonly used. In Canadian football, the score is formally and popularly called a "convert". Conversions can also be scored by the now-rare drop kick; in standard American and Canadian football, such a conversion is worth 1 point, while most forms of indoor football, including the Arena Football League, award 2 points for a drop-kicked conversion.
• In six-man football, one point is awarded for a successful conversion from scrimmage after a touchdown. Note that in standard 11-man (American) or 12-man (Canadian) football, place kicks are worth 1 point and conversions from scrimmage worth 2; this is reversed in six-man because the reduced number of players makes kicked conversions much more difficult.
• In Canadian football only, a single or "rouge" is awarded when the ball is legally kicked into the opponent's end zone (except for a successful field goal), and the receiving team does not return, or kick, the ball out of its end zone. (In American football, the same play would result in a touchback and no points.)
• Some forms of indoor football in the U.S. award a "single", similar to the Canadian score, on kickoffs only.
• In rugby league:
• A drop goal is worth 1 point.
• In most competitions (though not the European Super League, which uses static squad numbering), the starting fullback wears jersey number 1.
• In rugby union:
• The starting loosehead prop wears the jersey number 1.
• In the early years of the sport (prior to 1890), conversions, penalties, drop goals, and goals from mark were all worth 1 point. At that time, a try was worth no points, only giving the attacking team the right to attempt a conversion. In 1890–91, tries were rewarded with 1 point, while all other scores were increased in value. After that time, all scores have been worth at least 2 points (the goal from mark was abolished in 1977).
• The jersey number 1 has been retired by several North American sports teams in honor of past playing greats or other key figures (or, in one case, a team's fans):

## Notes

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

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