# Dictionary Definition

sexagesimal adj : of or relating to or reckoning
in sixtieths; "the sexagesimal divisions of hours and
degrees"

# User Contributed Dictionary

## English

### Adjective

sexagesimal- of, pertaining by, proceeding by, or based on the number sixty

### Noun

- a sexagesimal fraction

# Extensive Definition

Sexagesimal (base-sixty) is a numeral
system with sixty as the
base. It originated with
the ancient Sumerians in the
2000s
BC, was transmitted to the Babylonians, and
is still used in modified form nowadays for measuring time, angles,
and geographic coordinates. Base-60 number systems have also been
used in some other cultures, for instance the Ekagi of Western
New Guinea.

The number 60 has twelve factors, 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60, of which 2, 3, and 5
are prime. With so many
factors, many simple fractions of sexagesimal numbers are simple.
For example, an hour can be divided evenly into segments of any of
twelve lengths: 60 minutes, 30 minutes, 20 minutes, etc. Note that
60 is the smallest number divisible by every number from 1 to
6.

In this article places are represented in modern
decimal, except where otherwise noted (for example, "10" means
ten
and "60" means sixty).

## Sexagesimal in Babylonia

Sexagesimal as used in ancient Mesopotamia was not a pure base 60 system, in the sense that they didn't use 60 distinct symbols for their digits. Instead, their cuneiform digits used ten as a sub-base in the fashion of a sign-value notation: a digit was composed of a number of narrow wedge-shaped marks representing units up to nine (Y, YY, YYY, YYYY, ... YYYYYYYYY) and a number of wide wedge-shaped marks representing tens up to five (<, <<, <<<, <<<<, <<<<<); the value of the digit was the sum of the values of its component parts:Numbers larger than sixty were indicated by
multiple symbol blocks of this form in place
value notation.

Because there was no symbol for zero with
either the Sumerians or the early Babylonians, it is not always
immediately obvious how a number should be interpreted, and the
true value must sometimes be determined by the context; later
Babylonian texts used a dot to represent zero.

## Usage

Unlike most other numeral systems, sexagesimal is
not used so much nowadays as a means of general computation or
logic, but is used in measuring angles, geographic coordinates,
and time.

One hour
of time is divided into 60 minutes, and one minute is
divided into 60 seconds. Thus, a measurement of time such as
"3:23:17" (three hours, 23 minutes, and 17 seconds) can be
interpreted as a sexagesimal number, meaning
3×602+23×601+17×600 seconds or
equivalently
3×600+23×60−1+17×60−2
hours. As with the ancient Babylonian sexagesimal system, however,
each of the three sexagesimal digits in this number (3, 23, and 17)
are written using the decimal system.

Similarly, the fundamental unit of angular
measure is the degree, of
which there are 360 in a
circle. There are 60 minutes of
arc in a degree, and 60 seconds of arc in a minute.

In the Chinese
calendar, a sexagenary
cycle is commonly used, in which days or years are named by
positions in a sequence of ten stems and in another sequence of 12
branches; the same stem and branch repeat every 60 steps through
this cycle.

### Popular culture

In Stel Pavlou's novel Decipher, this number system is the center of focus, as the bucky ball carbon element is used in the book to store data, and only base 60 proved able to be successfully understood by the computer used. At least one popular book uses the spelling "sexigesimal" instead of "sexagesimal," with the latter being the more common spelling of the word.Book VIII of Plato's Republic
involves an allegory of marriage centered around the number 604 =
12,960,000 and its divisors. This number has the particularly
simple sexagesimal representation 1:0:0:0. Later scholars have
invoked both Babylonian mathematics and music theory in an attempt
to explain this passage.

## Fractions

In the sexagesimal system, any fraction in which the denominator is a regular number (having only 2, 3, and 5 in its prime factorization) may be expressed exactly. Here, for instance, the values may be interpreted as the number of minutes and seconds in a given fraction of an hour, although the representation of these fractions as sexagesimal numbers does not depend on such an interpretation.However numbers that are not regular form more
complicated repeating
fractions. For example:

- 1/7 = 0:8:34:17:8:34:17 ... (with the sequence of sexagesimal digits 17:8:34 repeating infinitely often).

The fact that the adjacent numbers to 60, 59 and
61, are both prime implies that simple repeating fractions that
repeat with a period of one or two sexagesimal digits can only have
59 or 61 as denominators, and that other non-regular primes have
fractions that repeat with a longer period.

In some usage systems, each position past the
sexagesimal point was numbered, using Latin or French roots: prime
or primus, seconde or secundus, tierce, quatre, quinte, etc. To
this day we call the second-order part of an hour or of a
degree a "second". In the 1700s, at least, 1/60 of a second was
called a "tierce" or "third".
http://books.google.com/books?id=aLiyFCm5ylMC&pg=PA193&lpg=PA193&ots=5tYt8d3YPn&sig=KZ8o0lE4rwqwdIFoAulNJPYpU8o#PPA193,M1,
http://books.google.com/books?id=UKXz8y-gSZ8C&pg=PA231&lpg=PA231&ots=S9Cmx_4epc&sig=JKmgLeUBPw_WS3cvvb0zDi255PE

## Examples

The square
root of 2, the length of the diagonal of a unit square,
was approximated by the Babylonians of the Old Babylonian Period
(1900 BC
- 1650
BC) as

- 1.414212... ≈ 30547/21600 = 1:24:51:10 = 1 + 24/60 + 51/602 + 10/603

The length of the tropical
year in Neo-Babylonian
astronomy (see Hipparchus),
365.24579... days, can be expressed in sexagesimal as 6:5:14:44:51
(6×60 ＋ 5 + 14/60 + 44/602 + 51/603) days. The average length of a
year in the Gregorian
calendar is exactly 6:5:14:33 in the same notation.

## See also

## References

- .

- .

## External links

sexagesimal in Catalan: Sistema
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sexagesimal in German: Sexagesimalsystem

sexagesimal in Spanish: Sistema
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sexagesimal in Esperanto: Sesdekuma
sistemo

sexagesimal in French: Système sexagésimal

sexagesimal in Korean: 육십진법

sexagesimal in Croatian: Seksagezimalni brojevni
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sexagesimal in Indonesian: Seksagesimal

sexagesimal in Icelandic: Sextugakerfi

sexagesimal in Hebrew: בסיס סקסגסימלי

sexagesimal in Haitian: Sistèm
seksajezimal

sexagesimal in Dutch: Sexagesimaal

sexagesimal in Japanese: 六十進法

sexagesimal in Polish: Sześćdziesiątkowy system
liczbowy

sexagesimal in Portuguese: Sistema
sexagesimal

sexagesimal in Russian: Шестидесятеричная
система счисления

sexagesimal in Finnish:
Seksagesimaalijärjestelmä

sexagesimal in Swedish: Sexagesimala
talsystemet