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The quantity twelve (12) is sometimes known as a dozen.It is in turn one twelfth of a gross.Base-12 is known as duodecimal.The Schoolhouse Rock segment "Little Twelvetoes" discusses the usefulness of multiplying by 12: "Well, with twelve digits, I mean fingers, He probably would've invented two more digits When he invented his number system. Then, if he'd saved the zero for the end, He could count and multiply by 12's, Just as easily as you and I do by 10's. Now, if man Had been born with six fingers on each hand, He's probably count: 1, 2, 3, 4, 5, 6, 7, 8, 9, dek, el, do. Dek and el being two entirely new signs meaning 10 and 11 - single digits. And his 12, do, would've been written: one - zero. Get it? That'd be swell, to multiply by 12."

The base 8 notational system for representing real numbers. The digits used are 0, 1, 2, 3, 4, 5, 6, and 7, so that (8 in base 10) is represented as () in base 8. The following table gives the octal equivalents of the first few decimal numbers.11111321252212142226331315232744141624305515172531661620263277172127338101822283491119232935101220243036The song "New Math" by Tom Lehrer (That Was the Year That Was, 1965) explains how to compute in octal. (The answer is .)

In simple algebra, multiplication is the process of calculating the result when a number is taken times. The result of a multiplication is called the product of and , and each of the numbers and is called a factor of the product . Multiplication is denoted , , , or simply . The symbol is known as the multiplication sign. Normal multiplication is associative, commutative, and distributive.More generally, multiplication can also be defined for other mathematical objects such as groups, matrices, sets, and tensors.Karatsuba and Ofman (1962) discovered that multiplication of two digit numbers can be done with a bit complexity of less than using an algorithm now known as Karatsuba multiplication.Eddy Grant's pop song "Electric Avenue" (Electric Avenue, 2001) includes the commentary: "Who is to blame in one country; Never can get to the one; Dealin' in multiplication; And they still can't feed everyone, oh no."..

Long division is an algorithm for dividing two numbers, obtaining the quotient one digit at a time. The example above shows how the division of 123456/17 is performed to obtain the result 7262.11....The term "long division" is also used to refer to the method of dividing one polynomial by another, as illustrated above. This example illustrates the resultThe symbol separating the dividend from the divisor seems to have no established name, so can be simply referred to as the long division symbol (or sometimes the division bracket).The chorus of the song "Singular Girl" by Rhett Miller (The Believer, 2006) contains the slightly cryptic line "Talking to you girl is like long division, yeah." Coincidentally, Long Division (1995) is also the name of the second album by the band Low...

The number two (2) is the second positive integer and the first prime number. It is even, and is the only even prime (the primes other than 2 are called the odd primes). The number 2 is also equal to its factorial since . A quantity taken to the power 2 is said to be squared. The number of times a given binary number is divisible by 2 is given by the position of the first , counting from the right. For example, is divisible by 2 twice, and is divisible by 2 zero times.The only known solutions to the congruenceare summarized in the following table (OEIS A050259). M. Alekseyev explored all solutions below on Jan. 27 2007, finding no other solutions in this range.reference4700063497Guy (1994)3468371109448915M. Alekseyev (pers. comm., Nov. 13, 2006)8365386194032363Crump (pers. comm., 2000)10991007971508067Crump (2007)63130707451134435989380140059866138830623361447484274774099906755Montgomery (1999)In general,..

The hypotenuse of a right triangle is the triangle's longest side, i.e., the side opposite the right angle. The word derives from the Greek hypo- ("under") and teinein ("to stretch").The length of the hypotenuse of a right trianglecan be found using the Pythagorean theorem.Among his many other talents, Major General Stanley in Gilbert and Sullivan's operetta The Pirates of Penzance impresses the pirates with his knowledge of the hypotenuse in "The Major General's Song" as follows: "I am the very model of a modern Major-General, I've information vegetable, animal, and mineral, I know the kings of England, and I quote the fights historical, From Marathon to Waterloo, in order categorical; I'm very well acquainted too with matters mathematical, I understand equations, both the simple and quadratical, About binomial theorem I'm teeming with a lot o' news-- With many cheerful facts about the square of..

A die (plural "dice") is a solid with markings on each of its faces. The faces are usually all the same shape, making Platonic solids and Archimedean duals the obvious choices. The die can be "rolled" by throwing it in the air and allowing it to come to rest on one of its faces. Dice are used in many games of chance as a way of picking random numbers on which to bet, and are used in board or role-playing games to determine the number of spaces to move, results of a conflict, etc. A coin can be viewed as a degenerate 2-sided case of a die.In 1787, Mozart wrote the measures and instructions for a musical composition dice game. The idea is to cut and paste pre-written measures of music together to create a Minuet (Chuang).The most common type of die is a six-sided cube with the numbers 1-6 placed on the faces. The value of the roll is indicated by the number of "spots" showing on the top. For the six-sided die, opposite faces are arranged..

Jenny's constant is the name given (Munroe 2012) to the positive real constant defined by(1)(2)(OEIS A182369), the first few digits of which are 867-5309, corresponding to the fictitious phone number in the song "867-5309/Jenny" performed by Tommy Tutone in 1982.Other "simple" expressions that might vie for that moniker include(3)(4)(5)(6)(7)(8)(9)(10)where is the hard hexagon entropy constant, the first three of which are "better" than the canonical Jenny expression (E. Weisstein, Jul. 12, 2013).

A simple group is a group whose only normal subgroups are the trivial subgroup of order one and the improper subgroup consisting of the entire original group. Simple groups include the infinite families of alternating groups of degree , cyclic groups of prime order, Lie-type groups, and the 26 sporadic groups.Since all subgroups of an Abelian group are normal and all cyclic groups are Abelian, the only simple cyclic groups are those which have no subgroups other than the trivial subgroup and the improper subgroup consisting of the entire original group. And since cyclic groups of composite order can be written as a group direct product of factor groups, this means that only prime cyclic groups lack nontrivial subgroups. Therefore, the only simple cyclic groups are the prime cyclic groups. Furthermore, these are the only Abelian simple groups.In fact, the classification theorem of finite groups states that such groups can be classified completely..

The group is the unique group of group order 2. is both Abelian and cyclic. Examples include the point groups , , and , the integers modulo 2 under addition (), and the modulo multiplication groups , , and (which are the only modulo multiplication groups isomorphic to ).The group is also trivially simple, and forms the subject for the humorous a capella song "Finite Simple Group (of Order 2)" by the Northwestern University mathematics department a capella group "The Klein Four."The cycle graph is shown above, and the cycleindex isThe elements satisfy , where 1 is the identity element.Its multiplication table is illustrated aboveand enumerated below. 1111The conjugacy classes are and . The only subgroups of are the trivial group and entire group , both of which are trivially normal.The irreducible representation for the group is ...

A finite group is a group having finite group order. Examples of finite groups are the modulo multiplication groups, point groups, cyclic groups, dihedral groups, symmetric groups, alternating groups, and so on.Properties of finite groups are implemented in the Wolfram Language as FiniteGroupData[group, prop].The classification theorem of finite groups states that the finite simple groups can be classified completely into one of five types.A convenient way to visualize groups is using so-called cycle graphs, which show the cycle structure of a given abstract group. For example, cycle graphs of the 5 nonisomorphic groups of order 8 are illustrated above (Shanks 1993, p. 85).Frucht's theorem states that every finite group is the graph automorphism group of a finite undirected graph.The finite (cyclic) group forms the subject for the humorous a capella song "Finite Simple Group (of Order 2)" by the Northwestern University..

There are several closely related results that are variously known as the binomial theorem depending on the source. Even more confusingly a number of these (and other) related results are variously known as the binomial formula, binomial expansion, and binomial identity, and the identity itself is sometimes simply called the "binomial series" rather than "binomial theorem."The most general case of the binomial theorem is the binomialseries identity(1)where is a binomial coefficient and is a real number. This series converges for an integer, or . This general form is what Graham et al. (1994, p. 162). Arfken (1985, p. 307) calls the special case of this formula with the binomial theorem.When is a positive integer , the series terminates at and can be written in the form(2)This form of the identity is called the binomial theorem by Abramowitz and Stegun (1972, p. 10).The differing terminologies are..

A strange loop is a phenomenon in which, whenever movement is made upwards or downwards through the levels of some hierarchical system, the system unexpectedly arrives back where it started. Hofstadter (1989) uses the strange loop as a paradigm in which to interpret paradoxes in logic (such as Grelling's paradox, the liar's paradox, and Russell's antinomy) and calls a system in which a strange loop appears a tangled hierarchy.Canon 5 from Bach's Musical Offering (sometimes known as Bach's endlessly rising canon) is a musical piece that continues to rise in key, modulating through the entire chromatic scale until it ends in the same key in which it began. This is the first example cited by Hofstadter (1989) as a strange loop.Other examples include the endlessly rising stairs in M. C. Escher 1960 lithograph Ascending and Descending, the endlessly falling waterfall in his 1961 lithograph Waterfall, and the pair of hands drawing each..

Zero is the integer denoted 0 that, when used as a counting number, means that no objects are present. It is the only integer (and, in fact, the only real number) that is neither negative nor positive. A number which is not zero is said to be nonzero. A root of a function is also sometimes known as "a zero of ."The Schoolhouse Rock segment "My Hero, Zero" extols the virtues of zero with such praises as, "My hero, zero Such a funny little hero But till you came along We counted on our fingers and toes Now you're here to stay And nobody really knows How wonderful you are Why we could never reach a star Without you, zero, my hero How wonderful you are."Zero is commonly taken to have the factorization (e.g., in the Wolfram Language's FactorInteger[n] command). On the other hand, the divisors and divisor function are generally taken to be undefined, since by convention, (i.e., divides 0) for every except zero.Because the number of..

The constant pi, denoted , is a real number defined as the ratio of a circle's circumference to its diameter ,(1)(2) has decimal expansion given by(3)(OEIS A000796). Pi's digits have many interesting properties, although not very much is known about their analytic properties. However, spigot (Rabinowitz and Wagon 1995; Arndt and Haenel 2001; Borwein and Bailey 2003, pp. 140-141) and digit-extraction algorithms (the BBP formula) are known for .A brief history of notation for pi is given by Castellanos (1988ab). is sometimes known as Archimedes' constant or Ludolph's constant after Ludolph van Ceulen (1539-1610), a Dutch calculator. The symbol was first used by Welsh mathematician William Jones in 1706, and subsequently adopted by Euler. In Measurement of a Circle, Archimedes (ca. 225 BC) obtained the first rigorous approximation by inscribing and circumscribing -gons on a circle using the Archimedes algorithm. Using (a 96-gon),..

A quadratic equation is a second-order polynomial equation in a single variable (1)with . Because it is a second-order polynomial equation, the fundamental theorem of algebra guarantees that it has two solutions. These solutions may be both real, or both complex.Among his many other talents, Major General Stanley in Gilbert and Sullivan's operetta the Pirates of Penzance impresses the pirates with his knowledge of quadratic equations in "The Major General's Song" as follows: "I am the very model of a modern Major-General, I've information vegetable, animal, and mineral, I know the kings of England, and I quote the fights historical, From Marathon to Waterloo, in order categorical; I'm very well acquainted too with matters mathematical, I understand equations, both the simple and quadratical, About binomial theorem I'm teeming with a lot o' news-- With many cheerful facts about the square of the hypotenuse."The..

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