10
10 (ten) is the even natural number following 9 and preceding 11, and the smallest two-digit positive integer.[1] As a composite number, 10 factors into the product of the first two prime numbers, 2 and 5, making it the fundamental unit in the decimal place-value system where each position denotes a power of 10.[1] Powers of 10 span immense scales in nature and science, from ≈10^{-43} s (Planck time)[2] to ≈10^{80} m³ (observable universe volume), and this factorization underlies their structure, such as 10n = 2n × 5n, which underpins scientific notation (a × 10^b where 1 ≤ a < 10) for compactly expressing vast or tiny quantities.[1] The decimal numeral system, based on 10, originated from ancient counting practices likely influenced by humans having ten fingers, and it remains the standard for arithmetic, commerce, and science worldwide due to its positional notation using digits 0 through 9.[3] In geometry, 10 defines the decagon, a ten-sided polygon that is constructible with straightedge and compass and appears in star polygons like the decagram.[4] Beyond pure mathematics, 10's role extends to measurement prefixes in the metric system (e.g., deci- for one-tenth) and modular arithmetic modulo 10, which is used in various computational contexts.[1]Linguistics
Etymology
The English word "ten" originates from Old English tīen, which evolved from Proto-Germanic *tehun, ultimately deriving from the Proto-Indo-European (PIE) root *déḱm̥(t).[5][6] The development of base-10 numeral systems across many cultures is often linked to human finger-counting, though this is separate from the etymology of the word itself. In non-Indo-European language families, equivalents for "ten" also reflect numeral origins. For instance, in Semitic languages, the term stems from Proto-Semitic *ʕaśr-, appearing as Hebrew ʿeśer.[7] Similarly, in Sino-Tibetan languages, the Chinese character shí (十) depicts a cross, possibly symbolizing the completion of a decimal count.[8] Historical linguistic shifts further illustrate this pattern within Indo-European branches. The Latin decem, also from PIE *déḱm̥, influenced Romance languages, yielding forms like French dix and Spanish diez, preserving the ancient root while adapting phonetically over centuries. These evolutions underscore the word's deep ties to the decimal system's development.Representation in languages
In alphabetic writing systems, the number 10 is represented by words derived from various linguistic roots. For example, in English, it is "ten," reflecting a base-10 counting tradition shared with many Indo-European languages. In Greek, the term is "δέκα" (deka), used in modern and classical contexts to denote the quantity. In Arabic, it appears as "عشر" (ʿashr), the root form underlying numerals like ʿashara for the cardinal ten.[9] Logographic systems employ distinct characters to symbolize 10 without phonetic components. In Chinese, the character 十 (shí) visually represents a cross, signifying ten as a foundational numeral in the Sino-Tibetan script.[10] Similarly, in Japanese kanji, which borrows from Chinese, 十 (jū) denotes ten and is used in compounds like jūni for twelve.[11] Roman numerals use X to represent 10, a symbol originating from Etruscan influences and possibly stylized as intersecting strokes for simplicity in ancient inscriptions.[12] In sign languages, representations of 10 vary by cultural adaptation, often linking to manual counting methods. In American Sign Language (ASL), the sign for ten is made by holding the thumb upright and shaking it from side to side, though numeral "10" may be signed as the combination of one and zero handshapes.[13][14] Unique cultural notations highlight 10's role in non-decimal systems. In the Mayan vigesimal (base-20) numeral system, 10 functions as a subunit, depicted as two horizontal bars stacked vertically, facilitating calculations in a positional framework where it halves the base.[15] These representations across languages often relate to historical counting practices.Mathematics
Numeral properties
In mathematics, 10 is classified as a composite number because it has more than two distinct positive divisors: 1, 2, 5, and 10.[16] It is also an even integer, as it is divisible by 2, making it the product of 2 and the integer 5.[17] Furthermore, 10 serves as the smallest positive two-digit integer in the base-10 numeral system, where numbers with two digits range from 10 to 99 according to the positional inequality $10^{1} \leq n < 10^{2}.[18] The prime factorization of 10 is $10 = 2 \times 5, with 2 and 5 as its distinct prime factors.[19] A key property related to this is the divisibility rule for 10 in base 10: a number is divisible by 10 if and only if its last digit is 0, reflecting the structure of the decimal system where 10 acts as the radix or base. This base-10 system employs place values based on powers of 10, such as $10^1 = 10 for the tens place and $10^2 = 100 for the hundreds place, facilitating the representation of larger integers efficiently.[20] Additionally, 10 qualifies as a Harshad number (also known as a Niven number) in base 10, since it is divisible by the sum of its digits: $1 + 0 = 1, and $10 \div 1 = 10, an integer.[21]Geometric applications
In geometry, the number 10 frequently appears in the study of polygons and polyhedra, particularly through the decagon, a ten-sided polygon. A regular decagon features ten equal sides and ten equal interior angles, each measuring 144°.<grok:render type="render_inline_citation">Arithmetic operations
Multiplication by 10 in the decimal system follows a simple pattern where the product is obtained by appending a zero to the multiplicand, facilitating quick mental arithmetic. The multiplication table for 10 up to 12 is as follows:| Multiplier | Product |
|---|---|
| 10 × 1 | 10 |
| 10 × 2 | 20 |
| 10 × 3 | 30 |
| 10 × 4 | 40 |
| 10 × 5 | 50 |
| 10 × 6 | 60 |
| 10 × 7 | 70 |
| 10 × 8 | 80 |
| 10 × 9 | 90 |
| 10 × 10 | 100 |
| 10 × 11 | 110 |
| 10 × 12 | 120 |