2
2 is the integer immediately succeeding 1 and preceding 3 in the sequence of positive integers. It is the first prime number and the only even prime, with all other primes being odd. As the smallest even positive integer greater than 0, 2 plays a foundational role in arithmetic, representing duality and serving as the basis for even-odd classification in number theory. In mathematics, 2 is notable for being equal to its own factorial, since $2! = 2 \times [1](/page/1) = 2. It forms the basis of the binary numeral system, a base-2 positional notation using only the digits 0 and 1, which is fundamental to computing and digital electronics. Additionally, 2 determines the operation of squaring, where a quantity raised to the power of 2 yields its square, and it appears in divisibility rules for binary representations, where the number of trailing zeros indicates the highest power of 2 dividing the number. The number 2 also features in various congruences and sequences, such as solutions to $2^n \equiv 3 \pmod{n}, highlighting its unique properties in modular arithmetic. Its status as the sole even prime underscores its exceptional position among primes, influencing theorems like the fundamental theorem of arithmetic.Numeral and Representation
The Arabic Digit 2
The Arabic digit 2 is typically rendered as a curve descending from the top right to the bottom left, with a horizontal base at the bottom extending to the right, and an optional flag at the top.[1] The current shape of the Arabic digit 2 has evolved from ancient Indian numerals used in the Brahmi script around the 3rd century BCE.[1] In handwriting, the stroke order for the digit 2 begins at the top, with a counterclockwise curve descending to the bottom left, followed by a horizontal line from the bottom left to the right, and concluding with an optional short upward flag at the top.[2] Typographic styles vary the appearance of the digit 2 based on font family. In sans-serif fonts like Arial, it appears as a simple, unadorned curve with clean lines and no terminal embellishments.[3] In serif fonts such as Times New Roman, the digit includes small horizontal serifs at the ends of the strokes, including a subtle tick at the top flag and base, providing a more traditional and decorative look.[3] The digit 2 is represented in Unicode as U+0032 (DIGIT TWO), which is a Basic Latin block character with the decimal value 2 and bidirectional class European Number. It is encoded as ASCII value 50 (hex 32), ensuring compatibility across computing standards.[4] Common visual confusions arise with the letter Z, particularly in handwritten or certain sans-serif fonts where the curve and diagonal elements overlap, leading to misidentification in contexts like IDs or medical records.[5]Variations in Writing Systems
In various writing systems beyond the standard Arabic numerals, the number 2 is represented through symbols that reflect cultural and historical conventions for enumeration. The Roman numeral for 2 is II, formed by repeating the symbol I (representing 1) twice according to the additive principle, where the values of the symbols are simply summed.[6] This system remains in use today for numbering clock faces, where Roman numerals provide a traditional aesthetic, and for outlining sections in documents, such as book chapters.[7][8] In Chinese script, the numeral for 2 is 二 (pronounced èr), composed of two parallel horizontal strokes written from top to bottom in a left-to-right sequence, emphasizing balance and simplicity in stroke order.[9] This character appears in both simplified and traditional forms of Chinese writing, but in formal or financial contexts, such as checks or legal documents, it may be replaced by the more complex variant 貳 (traditional) or 贰 (simplified) to prevent forgery through alteration.[10] Other global systems employ distinct representations for 2. In Devanagari script, used for languages like Hindi, the numeral for 2 is २.[11] For Arabic abjad numeration, where letters double as numerical values, 2 is denoted by ب (bāʾ, equivalent to 2 in the system), which takes contextual forms—initial, medial, final, or isolated—depending on its position in cursive Arabic script; additionally, the Eastern Arabic-Indic digit ٢ serves as the positional numeral in modern usage.[12][13] The ancient Mayan system, a vigesimal (base-20) notation, represents 2 with two dots (••), as each dot signifies one unit, while bars represent five; these glyphs were typically arranged vertically or horizontally in codices and monuments.[14] For tactile representation, Braille denotes 2 as ⠃, formed by raised dots in positions 1 (upper left) and 2 (middle left) of the 2x3 cell grid, preceded by the number sign ⠼ when indicating numerals.[15] In electronic digital displays, the seven-segment format renders 2 by illuminating five specific segments: a (top horizontal), b (upper right vertical), g (middle horizontal), e (lower left vertical), and d (bottom horizontal), creating a recognizable angular profile on LED or LCD screens.[16]Mathematics
Fundamental Properties of 2
In mathematics, the number 2 holds a distinctive position as the smallest positive even integer greater than 0.[17] It is also the unique even prime number, defined as a positive integer greater than 1 that has no positive divisors other than 1 and itself, with all other primes being odd.[18] This property stems from the fact that any even integer larger than 2 is divisible by both 1 and 2, rendering it composite.[19] Additionally, 2 serves as the base of the binary numeral system, a positional notation using only the digits 0 and 1 to represent all natural numbers efficiently in computing and digital systems.[20] Regarding divisibility, the positive divisors of 2 are exclusively 1 and 2, confirming its primality.[21] Conversely, 2 divides every even integer, as an even number n can be expressed as n = 2k for some integer k.[22] This divisibility rule underpins the classification of integers into even and odd categories. The parity of 2 reflects its even nature, formalized as $2 \equiv 0 \pmod{2}, meaning 2 leaves a remainder of 0 when divided by itself.[22] In the additive structure of integers, 2 interacts with the additive identity 0 such that $2 + 0 = 2, preserving its value, and it stands as the smallest integer exceeding 1, bridging the gap between unity and the onset of composite numbers.[21] In set theory, 2 denotes the cardinality of a finite set with exactly two distinct elements, such as a pair or doubleton \{a, b\} where a \neq b, representing the "size" of such a set through a bijection to the set \{1, 2\}.[23] This finite cardinality establishes 2 as the foundational measure for pairs in combinatorial contexts.Arithmetic Operations with 2
Addition is one of the fundamental arithmetic operations, where combining the number 2 with another integer yields their sum; for instance, $2 + 1 = 3 and $2 + 2 = 4.[24] This operation underpins the concept of doubling, which represents repeated addition of a number to itself, such that $2 \times n = n + n for any integer n.[25] Subtraction involves finding the difference when removing 2 from a larger number, as in $4 - 2 = 2 or $2 - 1 = 1, but it can also produce negative results when the subtrahend exceeds the minuend, such as $2 - 3 = -1.[26] These examples illustrate how subtraction with 2 extends the number line beyond positive integers.[27] Multiplication by 2 scales a number by combining it with itself, following the commutative property where $2 \times n = n \times 2; a basic example is $2 \times 3 = 6.[28] The multiplication table for 2 up to 10 is as follows:| Multiplicand (n) | Product ($2 \times n) |
|---|---|
| 1 | 2 |
| 2 | 4 |
| 3 | 6 |
| 4 | 8 |
| 5 | 10 |
| 6 | 12 |
| 7 | 14 |
| 8 | 16 |
| 9 | 18 |
| 10 | 20 |