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Twelve<< 10 11 12 13 14 15 16 17 18 19 >> List of numbers -- Integers 0 10 20 30 40 50 60 70 80 90 >> Cardinal 12 twelve Ordinal 12th twelfth Numeral system duodecimal Factorization Divisors 1, 2, 3, 4, 6, 12 Roman numeral XII Unicode representation of Roman numeral Ⅻ, ⅻ Binary 1100 Octal 14 Duodecimal 10 Hexadecimal C Mathematical properties φ(12) = 4 τ(12) = 6 σ(12) = 28 π(12) = 5 μ(12) = 0 M(12) = -2 12 (twelve) is the natural number following 11 and preceding 13. The word "twelve" is a native English word that presumably arises from the Germanic compound twa-lif "two-leave", meaning that two is left after one takes away the base, ten. This compound meaning may have been transparent to speakers of Old English, but the modern form "twelve" is quite opaque. Only the remaining tw- hints that twelve and two are related. Etymology (Weekley, Skeat) suggests that "twelve" (similar to "eleven") consists of two parts, the first meaning "two" and the second "leftover", so a literal translation would yield "two remaining [after having ten taken]". A group of twelve things is called a Duodecad. The ordinal adjective is duodenary. The number twelve is often used as a sales unit in trade, and is often referred to as a dozen. Twelve dozen are known as a gross. (Note that there are thirteen loaves in a baker's dozen.) As shown below, the number twelve is frequently referenced in the Abrahamic religions and is also central to Western calender and units of time. Twelve is a composite number, the smallest number with exactly six divisors, its proper divisors being 1, 2, 3, 4 and 6. Twelve is also a highly composite number, the next one being 24. The duodecimal system (1210 [twelve] = 1012), which is the use of 12 as a division factor for many ancient and medieval weights and measures, including hours, probably originates from Mesopotamia. As Schoolhouse Rock explains in its song "Little Twelvetoes", if humankind had been born with twelve fingers, they would have counted and multiplied using the duodecimal system. (There is no need to have twelve fingers though, since one will easily arrive at the duodecimal system by simply counting the phalanx bones of fingers with the same hand's thumb, a practice in use with some people until the present day.) Twelve is a superfactorial, being the product of the first three factorials. Twelve being the product of three and four, the first four positive integers show up in the equation 12 = 3 × 4, which can be continued with the equation 56 = 7 × 8. A twelve-sided polygon is a dodecagon. A twelve-sided polyhedron is a dodecahedron. Twelve is a pentagonal number. The densest three-dimensional lattice sphere packing has each sphere touching 12 others, and this is almost certainly true for any arrangement of spheres (the Kepler conjecture). Twelve is also the kissing number in three dimensions. In base thirteen and higher bases (such as hexadecimal), twelve is represented as C. Twelve is superabundant, sparsely totient, a Harshad number and a Pell number. Twelve is the smallest weight for which a cusp form exists. This cusp form is the discriminant Δ(q) whose Fourier coefficients are given by the Ramanujan τ-function and which is (up to a constant multiplier) the 24th power of the Dedekind eta function. This fact is related to a constellation of interesting appearances of the number twelve in mathematics ranging from the value of the Riemann zeta function function at -1 i.e. ζ(-1)=-1/12, the fact that the abelianization of SL(2,Z) has twelve elements, and even the properties of lattice polygons. Check out the following recipes that are tagged "Twelve":
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