10 (ten) is the even natural number following 9 and preceding 11. Ten is the base of the decimalnumeral system, the most common system of denoting numbers in both spoken and written language.
Anthropology
Usage and terms
A collection of ten items (most often ten years) is called a decade.
The ordinal adjective is decimal; the distributive adjective is denary.
Increasing a quantity by one order of magnitude is most widely understood to mean multiplying the quantity by ten.
To reduce something by one tenth is to decimate. (In ancient Rome, the killing of one in ten soldiers in a cohort was the punishment for cowardice or mutiny; or, one-tenth of the able-bodied men in a village as a form of retribution, thus causing a labor shortage and threat of starvation in agrarian societies.)
Mathematics
Ten is the fifth composite number, and the smallest noncototient, which is a number that cannot be expressed as the difference between any integer and the total number of coprimes below it.[1] Ten is the eighth Perrin number, preceded by 5, 5, and 7.[2]
, the smallest number that can be written as the sum of two prime numbers in two different ways[5][6]
, the sum of the first three prime numbers, and the smallest semiprime that is the sum of all the distinct prime numbers from its lower factor through its higher factor[7]
The factorial of ten is equal to the product of the factorials of the first four odd numbers as well: ,[8] and 10 is the only number whose sum and difference of its prime divisors yield prime numbers and .
10 is also the first number whose fourth power (10,000) can be written as a sum of two squares in two different ways, and
According to conjecture, ten is the average sum of the proper divisors of the natural numbers if the size of the numbers approaches infinity,[11] and it is the smallest number whose status as a possible friendly number is unknown.[12]
The smallest integer with exactly ten divisors is 48, while the least integer with exactly eleven divisors is 1024, which sets a new record.[13][a]
The metric system is based on the number 10, so converting units is done by adding or removing zeros (e.g. 1 centimetre = 10 millimetres, 1 decimetre = 10 centimetres, 1 meter = 100 centimetres, 1 dekametre = 10 meters, 1 kilometre = 1,000 meters).
Music
The interval of a major tenth is an octave plus a major third.
The interval of a minor tenth is an octave plus a minor third.
^The initial largest span of numbers for a new maximum record of divisors to appear lies between numbers with 1 and 5 divisors, respectively. This is also the next greatest such span, set by the numbers with 7 and 11 divisors, and followed by numbers with 13 and 17 divisors; these are maximal records set by successive prime counts. Powers of 10 contain divisors, where is the number of digits: 10 has 22 = 4 divisors, 102 has 32 = 9 divisors, 103 has 42 = 16 divisors, and so forth.
"... reading the segment (1, 10) together with the line from 10, in the direction 10, 34, ..., in the square spiral whose vertices are the generalized hexagonal numbers (A000217)."[17]
Aside from the zeroth term, this sequence matches the sums of squares of consecutive odd numbers.[3]
^55 is also the fourth doubly triangular number.[20] In the sequence of triangular numbers, indexed powers of 10 in this sequence generate the following sequence of triangular numbers, in decimal representation: 55 (10th), 5,050 (100th), 500,500 (1,000th), ...[21] 19 is another number that is the first member of a sequence displaying a similar uniform property, where the 19th triangular number is 190, the 199th triangular number is 19900, etc.[22]
^Where 55 is the sum of the first four terms in Sylvester's sequence (2, 3, 7, and 43), the product of these is 1806, whose sum with the fifth term 1807 yields the 505th indexed prime number and 42nd centered square number, 3613.[24][25] Unit fractions from terms in this sequence form an infinite series that converges to 1, where successive terms from Sylvester's sequence will always multiply to one less the value of the following term (i.e., 42 and 43 for the first three and fourth terms).
^Sloane, N. J. A. (ed.). "Sequence A055233 (Composite numbers equal to the sum of the primes from their smallest prime factor to their largest prime factor.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-08.
For n = 1; a(1) = 1 + 2 + ...... + 10 = 11 * 5 = 55
For n = 2; a(2) = 1 + 2 + .... + 100 = 101 * 50 = 5050
For n = 3; a(3) = 1 + 2 + .. + 1000 = 1001 * 500 = 500500
...
^Sloane, N. J. A. (ed.). "Sequence A186076 (Numbers m such that m equal to Sum_{i equal to x..y} i being (10^k)*y + x, where 0 is less than or equal to x less than y, 0 less than or equal to x less than 10^k for some positive integers k.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-07.
^Andrews, W.S. (1917). Magic Squares and Cubes (2nd ed.). Open Court Publishing. p. 30.
^Sloane, N. J. A. (ed.). "Sequence A002322 (Reduced totient function psi(n): least k such that x^k is congruent 1 (mod n) for all x prime to n; also known as the Carmichael lambda function (exponent of unit group mod n); also called the universal exponent of n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-07.
^Sloane, N. J. A. (ed.). "Sequence A000682 (Semi-meanders: number of ways a semi-infinite directed curve can cross a straight line n times.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-08.
^Sloane, N. J. A. (ed.). "Sequence A003601 (Numbers j such that the average of the divisors of j is an integer: sigma_0(j) divides sigma_1(j). Alternatively, numbers j such that tau(j) (A000005(j)) divides sigma(j) (A000203(j)).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-13.
^Sloane, N. J. A. (ed.). "Sequence A102187 (Arithmetic means of divisors of arithmetic numbers (arithmetic numbers, A003601, are those for which the average of the divisors is an integer).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-13.
^Sloane, N. J. A. (ed.). "Sequence A000085 (Number of self-inverse permutations on n letters, also known as involutions; number of standard Young tableaux with four cells;)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-02-17.
^M.J. Bertin; A. Decomps-Guilloux; M. Grandet-Hugot; M. Pathiaux-Delefosse; J.P. Schreiber (1992). Pisot and Salem Numbers. Birkhäuser. ISBN 3-7643-2648-4.
^Grünbaum, Branko; Shephard, G. C. (1987). "Section 2.1: Regular and uniform tilings". Tilings and Patterns. New York: W. H. Freeman and Company. p. 64. doi:10.2307/2323457. ISBN 0-7167-1193-1. JSTOR 2323457. OCLC 13092426. S2CID 119730123.
^Gummelt, Petra (1996). "Penrose tilings as coverings of congruent decagons". Geometriae Dedicata. 62 (1). Berlin: Springer: 1–17. doi:10.1007/BF00239998. MR 1400977. S2CID 120127686. Zbl 0893.52011.
^Coxeter, H. S. M (1948). "Chapter 14: Star-polytopes". Regular Polytopes. London: Methuen & Co. LTD. p. 263.
External links
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