120 (number)

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The 120-cell (or hecatonicosachoron) is a convex regular 4-polytope consisting of 120 dodecahedral cells

120 (one hundred [and] twenty) is the natural number following 119 and preceding 121. 120 was known as "the great hundred", especially prior to the year 1700, from the Teutonic Hundert which equalled 120. The number 100, now known commonly as "one hundred" was then known as "the small hundred".

In mathematics

120 is the factorial of 5, and the sum of a twin prime pair (59 + 61). 120 is the sum of four consecutive prime numbers (23 + 29 + 31 + 37), four consecutive powers of 2 (8+16+32+64), and four consecutive powers of 3 (3 + 9 + 27 + 81). It is highly composite, superabundant, and colossally abundant number, with its 16 divisors being more than any number lower than it has, and it is also the smallest number to have exactly that many divisors. It is also a sparsely totient number. 120 is the smallest number to appear six times in Pascal's triangle. 120 is also the smallest multiple of 6 with no adjacent prime number.

It is the eighth hexagonal number and the fifteenth triangular number, as well as the sum of the first eight triangular numbers, making it also a tetrahedral number. 120 is divisible by the first 5 triangular numbers and the first 4 tetrahedral numbers.

120 is the first multiply perfect number of order three (a 3-perfect or triperfect number). The sum of its factors (including one and itself) sum to 360; exactly three times 120. Note that perfect numbers are order two (2-perfect) by the same definition.

120 is divisible by the number of primes below it, 30 in this case. However there is no integer which has 120 as the sum of its proper divisors, making 120 an untouchable number.

The sum of Euler's totient function φ(x) over the first nineteen integers is 120.

120 figures in Pierre de Fermat's modified Diophantine problem as the largest known integer of the sequence 1, 3, 8, 120. Fermat wanted to find another positive integer that multiplied with any of the other numbers in the sequence yields a number that is one less than a square. Leonhard Euler also searched for this number, but failed to find it, but did find a fractional number that meets the other conditions, 777480 / 28792.

The internal angles of a regular hexagon (one where all sides and all angles are equal) are all 120 degrees.

120 is a Harshad number in base 10.

In science

120 is the atomic number of Unbinilium, an element yet to be discovered.

In history

The existence of a non-decimal base in the earliest traces of the Germanic languages, is attested by the presence of words and glosses meaning that the count is in decimal (cognates to ten-count or tenty-wise), such would be expected if normal counting is not decimal, and unusual if it were. In the Gothic bible, at I Cor 15:6, there is an example of such an marginal, where 'fimf hundram' (five hundred) acquires a marginal to it being 'taihuntewjam' (teentywise). Similar words are known in most other Germanic languages.

Where this counting system is known, it is based on the long hundred of 120 in number, and a long thousand of 1200 in number. The descriptions like 'long' only appear after the small hundred of 100 in number appeared with the Christians. The word for it was twelfty. See, e.g. Zupko,[1]

Gordon's Introduction to Old Norse [2] p 293, gives number names that belong to this system. An expression cognate to 'one hundred and eighty' is translated to 200, and the cognate to 'two hundred' is translated at 240.

Goodare [3] details the use of the long hundred in Scotland in the Middle Ages, giving examples, calculations where the carry implies i C (i.e. one hundred) as 120, etc. That the general population were not alarmed to encounter such numbers suggests common enough use. Goodare gives examples of numbers like vii score, where one avoids the hundred by using extended scores.

Goodare's paper also gives an English form of the roman numbers. This is written as if M and C were units of 1200 and 120 respectively, and a number like v C xlii (for 642). This number appears in a sum where the total implies that the carries from units over C is a column of 120, rather than 10. The year 2013 would be written as i m vi c lxxxxiii.

There is also a paper by W.H. Stevenson,[4] on 'The Long Hundred and its uses in England'. The paper discusses work of Professor Kluge of Jena, which details the long hundred among the Germanic nations, and then proceeds to fill the role of providing the missing information for English. Numbers from 10 to 20 are discussed, particularly the transition from the -lif form to the -teen forms. Likewise, the transition of the decades at 60 are discussed. Sample word use and recorded calculations then follow, the latter make sense where the hundred is reckoned at 120. The final part of the paper is given to discussing Mr Pell's reliance on fairly unbased but scholarly published research, without deep critical thinking, particularly, if the hundred means the decimal hundred, then the english numbers refer to measures 20% larger than the latin ones.

One can avoid using hundreds by using larger intermediate units, such as in the UK, giving weights in stones and pounds, or tons and hundredweight, rather than the long count of pounds as used in the US. Such might well happen when the hundred is in the process of changing its meaning.

The reckoning by long hundreds waned when the Arabic numerals spread throughout Europe during the 14 and later centuries. The Black Death, and the arrival of Hindu-Arabic numerals lead to the replacement of the long hundred with short hundreds, but the tradition died hard.

See also the entry on Hundred at www.sizes.com

In religion

  • The cubits of the height of the Temple building (II Chronicles 3:4)
  • The age at which Moses died (Deut. 34:7).
    • By extension, in Jewish tradition, to wish someone a long life, one says, "until 120."
  • The number of Men of the Great Assembly who canonized the Books of the Tanakh and formulated the Jewish prayers
  • The number of talents of gold Queen Sheba gave to King Solomon in tribute (I Kings 10:10)
  • The number of princes King Darius set over his kingdom (Daniel 6:2)
  • The summed weight in shekels of the gold spoons offered by each tribal prince of Israel (Num. 7:86).
  • In astrology, when two planets in a person's chart are 120 degrees apart from each other, this is called a trine. This is supposed to bring good luck in the person's life.[5]

In sports

  • The height in inches of a regulation basketball hoop{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B=

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In games

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In other fields

120 is also:

See also


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