General · EdgepediaSeptember 17, 20267 min read

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1 (one), also called unity, is the first positive integer and an odd number. It is the multiplicative identity of the ordinary number systems: multiplying any number by 1 leaves the number unchanged. Although 1 was historically counted as a prime number, modern mathematics places it in a class of its own, neither prime nor composite.¹

Key fact	Detail
Classification	First positive integer; odd; neither prime nor composite in the modern convention¹
Multiplicative identity	The unique element with 1 · x = x · 1 = x in the integers, rationals, reals, and complex numbers²
Foundational role	In the Peano axioms (Dedekind 1888, Peano 1889), the natural numbers are generated from a starting element by repeatedly applying a successor function³
Historical status as prime	Listed as prime by Lebesgue (1859), Joseph Ray (1866), William J. Milne (1892), and D. N. Lehmer's 1914 table of primes to 10,006,721⁴
Pythagorean view	One was not an actual number but the starting point of all numbers⁵
Written form	The European digit 1 is Unicode U+0031 DIGIT ONE, one of the most widely used decimal digits, with separately encoded variants in many scripts⁶
Decimal identity	1 = 0.999…, the repeating decimal¹

Arithmetic properties

As the multiplicative identity, 1 satisfies 1 · x = x · 1 = x for every element x of the integers, the rational, real, and complex numbers, and their extension rings such as the Gaussian integers; the residue class of 1 plays the same role in the quotient rings of the integers.² In abstract algebra the same role is filled by the identity element (also called the unit element) of a group or monoid, an element often denoted 1 precisely because of this arithmetic prototype; the alternative symbol E derives from the German word for unity, Einheit.⁷ In a ring, an element with a multiplicative inverse is called a unit, and the units of an integral domain form a group under multiplication; in the rational numbers this group contains only +1 and −1.⁸

Some consequences of the identity role are visible in elementary notation. Fractions with 1 as numerator are called unit fractions, and the generating function whose coefficients are all 1 is the geometric series 1/(1 − x) = 1 + x + x² + x³ + ⋯.¹ The number 1 also equals the repeating decimal 0.999…, a standard example in discussions of decimal representation.¹

Foundations: the successor construction

Modern foundations treat 1 not as a primitive object but as the second element of a generated sequence. The Peano axioms, proposed by Richard Dedekind in 1888 and Giuseppe Peano in 1889, characterize the natural numbers with five properties: a starting natural number, closure of the naturals under a successor function, the facts that no successor equals the starting element and that two naturals with the same successor are equal, and mathematical induction.³ In a common formulation the constant 0 is a natural number, the successor function S is one-to-one, every nonzero natural is a successor, and any set containing 0 and closed under S contains all naturals; the elements are then written 0, S(0), S(S(0)), and so on, abbreviated 0, 1, 2.⁹ The recursion theorem turns these properties into definitions of addition and multiplication, from which familiar results such as commutativity and the infinitude of the primes follow.³

In the set-theoretic construction, the successor of a set x is the set S(x) = x ∪ {x}, so that 1 = 0 ∪ {0} and each natural number n is the set of all smaller naturals.¹⁰ Dedekind denoted the natural numbers by N in Was sind und was sollen die Zahlen (1888), and Peano adopted the same symbol in his 1889 Arithmetices principia nova methodo exposita.¹¹ Whether the sequence starts at 0 or at 1 remains a convention: some authors use "natural numbers" for the positive integers 1, 2, 3, … and "nonnegative integers" for 0, 1, 2, ….¹²

Was 1 prime?

The prime question has a documented history of reversal. Most textbooks today call 1 neither prime nor composite, but older texts generally considered it prime: Lebesgue stated explicitly that 1 is prime in Exercices d'analyse numérique (1859), it appears as prime in Joseph Ray's Primary Elements of Algebra (1866) and William J. Milne's Standard Arithmetic (1892), and D. N. Lehmer's 1914 list of primes to 10,006,721 includes it.⁴ Goldbach's original 1742 statement of his conjecture, that every natural number greater than 2 is the sum of three primes, relied on counting 1 as prime; the modern formulation says every even number greater than 2 is the sum of two primes.¹³

The modern exclusion is a choice made to protect theorems. If 1 or −1 were prime, an integer such as 6 would have infinitely many factorizations, 6 = 2 · 3, 6 = 1 · 1 · 1 ⋯ 1 · 2 · 3, and so on, destroying the uniqueness asserted by the fundamental theorem of arithmetic; excluding units from the definitions of prime and composite restores a well-defined list of prime factors.¹⁴ The same reasoning applies in polynomial rings, where every nonzero constant is a unit, and in the theory of ideals, where excluding units keeps the study of prime ideals informative.¹⁴

Historical views of unity

The Pythagoreans of ancient Greece did not count 1 among the numbers at all: in their teaching, one was not an actual number but the starting point of all numbers. Numbers were thought of as arrangements of discrete points, and 1, 2, 3, and 4 carried musical and mystical significance, generating the octave, fifth, and fourth through the ratios 2:1, 3:2, and 4:3.⁵ Aristotle treated the one as the first measure of quantity, and medieval schastics debated unity as a "transcendental" property convertible with being, distinct from mathematical unity as the principle of number.¹⁵

The cognitive science of number adds a cultural qualification. The mathematics known today, including concepts as simple as fractions, real numbers, and zero, was unknown through almost all of human history, and some cultures have used counting systems that do not mark exact numerosities at all, at times lacking an unambiguous term for "one"; iconic techniques based on one-to-one correspondence have allowed some groups to solve quantitative problems without symbolic number.¹⁶ The idea of building all numbers by starting with the unit one and repeatedly adding one, formalized in the Peano axioms, is thus one cultural formulation among several rather than a universal human default.¹⁶

The digit 1 and place value

The written symbol 1 descends from the Indian numeral tradition. Historians credit the Indian subcontinent with the first combined use of decimal numerals and place value, with evidence of an early decimal system in the Indus valley by 3000 BC and the system in common use by the 9th century AD; the key contribution was the place-holder zero that lets a single digit 1 mean one, ten, one hundred, or any power of ten by position.¹⁷ The numerals 0 through 9 took forms close to their modern European shapes in the 15th century, when printing drove standardization.¹⁸

In computing, the digit is encoded as U+0031 DIGIT ONE in the Unicode Standard, whose most widely used decimal digits are the European digits U+0030 through U+0039, also known as ASCII, Western, or Latin digits.⁶ Many scripts carry their own digit one, including Arabic-Indic, Devanagari, Bengali, Thai, and fullwidth forms, and mathematics uses separately encoded styled variants such as the mathematical double-struck digit one.⁶ Enclosed variants such as ① CIRCLED DIGIT ONE are compatibility forms decomposable to the digit itself plus an enclosing mark.¹⁹

References

Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Arithmetic and number systems › Integer sequences and partitions › Special and named integers › Named individual integers

Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026

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