2

2 (two) is the natural number that follows 1 and precedes 3, and it is the unique even prime number: it is divisible by 2 and has exactly two distinct positive divisors, 1 and itself.^{1 • 2} Every other prime is an odd prime, because any even number greater than 2 is divisible by 1, 2, and itself, giving at least three divisors.^{3 • 4} This double role, as the smallest prime and the base of evenness, makes 2 a dividing line throughout number theory, and its role as the base of the binary system makes it the arithmetic foundation of computing.

Evenness and parity

An even number is an integer of the form $n = 2k$, where $k$ is an integer, so the even numbers run $\ldots, -4, -2, 0, 2, 4, 6, \ldots$. Because even numbers are integrally divisible by two, the congruence $n \equiv 0 \pmod 2$ holds for every even $n$, and an integer is odd exactly when $n \equiv 1 \pmod 2$.^{5 • 10} The oddness or evenness of a number is called its parity, with even numbers assigned parity 0 and odd numbers parity 1.⁵

The number 2 is the modulus that defines this two-way split. Products and sums obey simple parity rules because of it: the product of an even number and an odd number is always even, since $(2k)(2l+1) = 2[k(2l+1)]$ is divisible by 2.⁵ The same structure appears in abstract algebra, where the integers modulo 2 form a two-element ring isomorphic to the parity ring whose elements are "even" and "odd".¹¹

The smallest prime

A prime number has exactly two distinct positive divisors, and 2 meets this condition with the divisors 1 and 2.¹ It is the first prime and the only one that is even; MathWorld notes humorously that 2 is therefore the "oddest" prime of all.³

Its evenness shapes several parts of number theory:

• Prime gaps. The difference between two successive primes $p_{k+1} - p_k$ is always even except at the first prime $p_1 = 2$, so all prime gaps greater than 1 are even.¹²
• Mersenne primes. Numbers of the form $2^{p} - 1$, where $p$ is prime, are the Mersenne numbers, and Euclid's rule for even perfect numbers uses them: an even natural number $N$ is perfect if and only if $N = 2^{k-1}(2^{k} - 1)$ for an integer $k$ such that $2^{k} - 1$ is a Mersenne prime.¹³
• Goldbach's conjecture. The conjecture asserts that every positive even integer $n \ge 4$ can be expressed as the sum of two primes $p + q$, a pair called a Goldbach partition of $2n$; it remains open.¹⁴ Chen's theorem, a proved approximation, states that every sufficiently large even number $N$ can be written as $N = p + m$, where $p$ is a prime and $m$ is either a prime or a semiprime.¹⁵

Another oddity is that 2 stands alone as the unique number $x$ whose sum of reciprocals of all powers equals itself: since $1/2 + 1/4 + 1/8 + \cdots = 1$, the total comes to 2.¹⁶

Base 2 and computing

Because 2 has only two states to track, it serves as the base of the binary number system, in which any number is written using only the characters 0 and 1. Gottfried Wilhelm Leibniz published "Explication de l'Arithmétique Binaire" in the Mémoires of the Paris Academy in 1703, illustrating binary addition, subtraction, multiplication, and division; Knuth's history of positional number systems calls this article the usual reference point for the birth of radix-2 arithmetic.^{6 • 17} Leibniz had developed the ideas earlier, corresponding about the principles of his binary arithmetic from February 1701, and he also connected the system to the ancient Chinese figures of Fuxi.^{6 • 18}

Modern computing is built on this base. A bit, short for binary digit, is a value of 0 or 1, and digital computers encode text, graphics, and other information as strings of bits; the letter "A" is typically represented by the bit string "01000001".^{7 • 19} Powers of 2 organize the units of information: one byte is $2^{3} = 8$ bits, and the International Electrotechnical Commission approved prefixes for binary multiples in December 1998, so that one kibibit is $2^{10} = 1024$ bits and one mebibyte is $2^{20} = 1\,048\,576$ bytes.⁸ In binary floating-point arithmetic, a normalized machine number is written $x = (-1)^{s} \cdot (b_{0}.b_{1}b_{2}\ldots b_{p-1}) \cdot 2^{E}$, with base 2 throughout.²⁰

Foundations: how 2 is defined

In axiomatic treatments the natural numbers are generated from a starting element by a successor operation. In the Peano axioms, the successor of 0 is denoted 1, and the successor of 1 is denoted 2, so that 2 is by definition $S(1)$, the number after 1.⁹ Addition and multiplication are then defined recursively on this basis, with $n + 1 = S(n)$.⁹

Philosophers have also asked what the number 2 is. Bertrand Russell, in An Introduction to Mathematical Philosophy, proposed defining 2 as the class of all couples: "the class of all couples will be the number 2, according to our definition," a definition chosen for its definiteness, since the class of couples is easier to pin down than the number itself. The mathematician E. C. Titchmarsh, discussing this account, noted the difficulties it raises, such as what counts as a member of the class of all couples.²¹

References

1. Even Prime — Definition, Formula & Examples, Mathwords
2. 2 (Number), MetaNumbers
3. Even Prime — from Wolfram MathWorld
4. Prime Number, Encyclopedia.com
5. Even Number — from Wolfram MathWorld
6. Leibniz: Explanation of binary arithmetic (1703), facsimile transcription
7. bit — Glossary, NIST CSRC
8. Definitions of the SI units: The binary prefixes, NIST
9. 2 Construction of Numbers, Introduction to Analysis lecture notes, University of Ljubljana
10. 1.3: Integers modulo n, Mathematics LibreTexts
11. Isomorphism between Ring of Integers Modulo 2 and Parity Ring, ProofWiki
12. Prime Gaps — from Wolfram MathWorld
13. Mathematics of Computation 83 (2014), American Mathematical Society
14. Goldbach Conjecture — from Wolfram MathWorld
15. Chen's Theorem — from Wolfram MathWorld
16. The Power of Two: Two Tips for Mathematicians, AMS e-Mentoring Network
17. Positional Number Systems, The Art of Computer Programming, InformIT
18. Leibniz's invention of binary arithmetic, ILAB catalogue
19. Quantum Computing Explained, NIST
20. DLMF §3.1 Arithmetics and Error Measures, NIST
21. E C Titchmarsh on Counting, MacTutor History of Mathematics

Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Arithmetic and number systems › Number systems › Integers and rational numbers

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

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