1

1 (one, unit, unity) is a

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

Among the earliest known record of a numeral system, is the

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

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]}

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

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${\displaystyle fx=fx}$).

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

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

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

• 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
  alkali metals
  .
• Period 1 of the periodic table consists of just two elements, hydrogen and helium.

In philosophy

In the philosophy of

Wikimedia Commons has media related to:
1 (number) (category)

Wikiquote has quotations related to 1 (number).

• −1
• +1 (disambiguation)
• List of mathematical constants
• One (word)
• Root of unity

References