1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of unit length is a line segment of length 1. In conventions of sign where zero is considered neither positive nor negative, 1 is the first and smallest positive integer. It is also sometimes considered the first of the infinite sequence of natural numbers, followed by 2, although by other definitions 1 is the second natural number, following 0.

The fundamental mathematical property of 1 is to be a multiplicative identity, meaning that any number multiplied by 1 equals the same number. Most if not all properties of 1 can be deduced from this. In advanced mathematics, a multiplicative identity is often denoted 1, even if it is not a number. 1 is by convention not considered a prime number; this was not universally accepted until the mid-20th century. Additionally, 1 is the smallest possible difference between two distinct natural numbers.

The unique mathematical properties of the number have led to its unique uses in other fields, ranging from science to sports. It commonly denotes the first, leading, or top thing in a group.

As a word

Etymology

One originates from the Old English word an, derived from the Germanic root *ainaz, from the Proto-Indo-European root *oi-no- (meaning "one, unique").^[1]

Modern usage

Linguistically, one is a cardinal number used for counting and expressing the number of items in a collection of things.^[2] One is commonly used as a determiner for singular countable nouns, as in one day at a time.^[3] One is also a gender-neutral pronoun used to refer to an unspecified person or to people in general as in one should take care of oneself.^[4] Words that derive their meaning from one include alone, which signifies all one in the sense of being by oneself, none meaning not one, once denoting one time, and atone meaning to become at one with the someone. Combining alone with only (implying one-like) leads to lonely, conveying a sense of solitude.^[5] Other common numeral prefixes for the number 1 include uni- (e.g., unicycle, universe, unicorn), sol- (e.g., solo dance), derived from Latin, or mono- (e.g., monorail, monogamy, monopoly) derived from Greek.^[6][7]

Symbols and representation

Among the earliest known record of a numeral system, is the Sumerian decimal-sexagesimal system on clay tablets dating from the first half of the third millennium BCE.^[8] The Archaic Sumerian numerals for 1 and 60 both consisted of horizontal semi-circular symbols.^[9] By c. 2350 BCE, the older Sumerian curviform numerals were replaced with cuneiform symbols, with 1 and 60 both represented by the same symbol . The Sumerian cuneiform system is a direct ancestor to the Eblaite and Assyro-Babylonian Semitic cuneiform decimal systems.^[10] Surviving Babylonian documents date mostly from Old Babylonian (c. 1500 BCE) and the Seleucid (c. 300 BCE) eras.^[8] The Babylonian cuneiform script notation for numbers used the same symbol for 1 and 60 as in the Sumerian system.^[11]

The most commonly used glyph in the modern Western world to represent the number 1 is the Arabic numeral, a vertical line, often with a serif at the top and sometimes a short horizontal line at the bottom. It can be traced back to the Brahmic script of ancient India, as represented by Ashoka as a simple vertical line in his Edicts of Ashoka in c. 250 BCE.^[12] This script's numeral shapes were transmitted to Europe via the Maghreb and Al-Andalus during the Middle Ages, through scholarly works written in Arabic.^{[citation needed]} In some countries, the serif at the top may be extended into a long upstroke as long as the vertical line. This variation can lead to confusion with the glyph used for seven in other countries and so to provide a visual distinction between the two the digit 7 may be written with a horizontal stroke through the vertical line.^{[citation needed]}

In modern typefaces, the shape of the character for the digit 1 is typically typeset as a lining figure with an ascender, such that the digit is the same height and width as a capital letter. However, in typefaces with text figures (also known as Old style numerals or non-lining figures), the glyph usually is of x-height and designed to follow the rhythm of the lowercase, as, for example, in .^[13] In old-style typefaces (e.g., Hoefler Text), the typeface for numeral 1 resembles a small caps version of I, featuring parallel serifs at the top and bottom, while the capital I retains a full-height form. This is a relic from the Roman numerals system where I represents 1.^[14][15] The modern digit '1' did not become widespread until the mid-1950s. As such, many older typewriters do not have dedicated key for the numeral 1 might be absent, requiring the use of the lowercase letter l or uppercase I as substitutes.^[15] The lower case "j" can be considered a swash variant of a lower-case Roman numeral "i", often employed for the final i of a "lower-case" Roman numeral. It is also possible to find historic examples of the use of j or J as a substitute for the Arabic numeral 1.^{[16][17][18][19]}

In mathematics

Mathematically, the number 1 has unique properties and significance. In normal arithmetic (algebra), the number 1 is the first natural number after 0 (zero) and can be used to make up all other integers (e.g., ${\displaystyle 1=1}$; ${\displaystyle 2=1+1}$; ${\displaystyle 3=1+1+1}$ etc.). The product of 0 numbers (the empty product) is 1 and the factorial 0! evaluates to 1, as a special case of the empty product.^[20] Any number ${\displaystyle n}$ multiplied or divided by 1 remains unchanged (${\displaystyle n\times 1=n/1=n}$). This makes it a mathematical unit, and for this reason, 1 is often called unity. Consequently, if ${\displaystyle f(x)}$ is a multiplicative function, then ${\displaystyle f(1)}$ must be equal to 1. This distinctive feature leads to 1 being is its own factorial (${\displaystyle 1!=1}$), its own square (${\displaystyle 1^{2}=1}$) and square root (${\displaystyle {\sqrt {1))=1}$), its own cube (${\displaystyle 1^{3}=1}$) and cube root (${\displaystyle {\sqrt[{3}]{1))=1}$), and so forth. 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). It is the multiplicative identity of the integers, real numbers, and complex numbers. 1 is the only natural number that is neither composite (a number with more than two distinct positive divisors) nor prime (a number with exactly two distinct positive divisors) with respect to division.^[21]

In algebraic structures such as multiplicative groups and monoids the identity element is often denoted 1, but e (from the German Einheit, "unity") is also traditional. However, 1 is especially common for the multiplicative identity of a ring, i.e., when an addition and 0 are also present. Moreover, if a ring has characteristic n not equal to 0, the element represented by 1 has the property that n1 = 1n = 0 (where this 0 denotes the additive identity of the ring). Important examples that involve this concept are finite fields.^{[citation needed]} A matrix of ones or all-ones matrix is defined as a matrix composed entirely of 1s.^[22]

Formalizations of the natural numbers have their own representations of 1. For example, in the original formulation of the Peano axioms, 1 serves as the starting point in the sequence of natural numbers.^[23] Peano later revised his axioms to state 0 as the "first" natural number such that 1 is the successor of 0.^[24] In the Von Neumann cardinal assignment of natural numbers, numbers are defined as the set containing all preceding numbers, with 1 represented as the singleton {0}.^[25] In lambda calculus and computability theory, natural numbers are represented by Church encoding as functions, where the Church numeral for 1 is represented by the function ${\displaystyle f}$ applied to an argument ${\displaystyle x}$ once (1${\displaystyle fx=fx}$).^[26] 1 is both the first and second number in the Fibonacci sequence (0 being the zeroth) and is the first number in many other mathematical sequences. As a pan-polygonal number, 1 is present in every polygonal number sequence as the first figurate number of every kind (e.g., triangular number, pentagonal number, centered hexagonal number).^{[citation needed]}

The simplest way to represent the natural numbers is by the unary numeral system, as used in tallying.^[27] This is often referred to as "base 1", since only one mark – the tally itself – is needed. Unlike base 2 or base 10, this is not a positional notation. Since the base 1 exponential function (1^x) always equals 1, its inverse (i.e., the logarithm base 1) does not exist.^{[citation needed]}

The number 1 can be represented in decimal form by two recurring notations: 1.000..., where the digit 0 repeats infinitely after the decimal point, and 0.999..., which contains an infinite repetition of the digit 9 after the decimal point. The latter arises from the definition of decimal numbers as the limits of their summed components, such that "0.999..." and "1" represent exactly the same number.^[28]

Primality

Although 1 appears to meet the naïve definition of a prime number, being evenly divisible only by 1 and itself (also 1), by convention 1 is neither a prime number nor a composite number. This is because 1 is the only positive integer divisible by exactly one positive integer, whereas prime numbers are divisible by exactly two positive integers and composite numbers by more than two positive integers. As late as the beginnings of the 20th century, some mathematicians considered 1 a prime number.^[29] However, the prevailing and enduring mathematical consensus has been to exclude due to its impact upon the fundamental theorem of arithmetic and other theorems related to prime numbers. For example, the fundamental theorem of arithmetic guarantees unique factorization over the integers only up to units, i.e., 4 = 2² represents a unique factorization. However, if units are included, 4 can also be expressed as (−1)⁶ × 1²³ × 2², among infinitely many similar "factorizations".^[30] Furthermore, Euler's totient function and the sum of divisors function are different for prime numbers than they are for 1.^[31][32]

Other mathematical attributes and uses

In many mathematical and engineering problems, 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. For example, by definition, 1 is the probability of an event that is absolutely or almost certain to occur.^[33] Likewise, vectors are often normalized into unit vectors (i.e., vectors of magnitude one), because these often have more desirable properties. Functions, too, are often normalized by the condition that they have integral one, maximum value one, or square integral one, depending on the application.^[34][35]

In category theory, 1 is the terminal object of a category if there is a unique morphism.^[36] In number theory, 1 is the value of Legendre's constant, which was introduced in 1808 by Adrien-Marie Legendre in expressing the asymptotic behavior of the prime-counting function. The value was originally conjectured by Legendre to be approximately 1.08366, but was proven in 1899 to equal exactly 1 by Charles Jean de la Vallée Poussin.^[37][38]

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.^{[citation needed]}

In numerical data, 1 is the most common leading digit in many sets of data (occurring about 30% of the time), a consequence of Benford's law.^[39]

1 is the only known Tamagawa number for a simply connected algebraic group over a number field.^{[40][failed verification][41][page needed]}

The generating function that has all coefficients equal to 1 is a geometric series, given by ${\displaystyle {\frac {1}{1-x))=1+x+x^{2}+x^{3}+\ldots }$^{[citation needed]}

The zeroth metallic mean is 1, with the golden section equal to the continued fraction [1;1,1,...], and the infinitely nested square root ${\displaystyle \scriptstyle {\sqrt {1+{\sqrt ((\text{ ))1+\cdots {\text{ )))))).}$^{[citation needed]}

The series of unit fractions that most rapidly converge to 1 are the reciprocals of Sylvester's sequence, which generate the infinite Egyptian fraction ${\displaystyle 1={\frac {1}{2))+{\frac {1}{3))+{\frac {1}{7))+{\frac {1}{43))+\cdots }$.^{[citation needed]}

Table of basic calculations

In technology

In digital technology, data is represented by binary code, i.e., a base-2 numeral system with numbers represented by a sequence of 1s and 0s. Digitised data is represented in physical devices, such as computers, as pulses of electricity through switching devices such as transistors or logic gates where "1" represents the value for "on". As such, the numerical value of true is equal to 1 in many programming languages.^[42][43]

In science

• Dimensionless quantities are also known as quantities of dimension one.
• Hydrogen, the first element of the periodic table, has an atomic number of 1.
• Group 1 of the periodic table consists of the alkali metals.
• Period 1 of the periodic table consists of just two elements, hydrogen and helium.

In philosophy

In the philosophy of Plotinus (and that of other neoplatonists), The One is the ultimate reality and source of all existence.^[44] 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]).

The Neopythagorean philosopher Nicomachus of Gerasa affirmed that one is not a number, but the source of number. He also believed the number two is the embodiment of the origin of otherness. His number theory was recovered by Boethius in his Latin translation of Nicomachus's treatise Introduction to Arithmetic.^[45]