Perenampuluhan

Sistem angka
Sistem angka Hindu-Arab
Arab barat Arab timur Burma India	Khmer Mongol Thai
Angka Asia timur
Batang pembilang Cina Jepun	Korea Suzhou Vietnam
Angka abjad
Abjad Armenia Āryabhaṭa Cyril	Ge'ez Ibrani Yunani (Ionia)
Sistem lain
Aegean Attic Babylon Brahmi Etruscan Inuit	Maya Mesir Quipu Rom Sumeria Urnfield
Sistem kedudukan dengan asas
Perpuluhan (10)
1, 2, 3, 4, 5, 6, 8, 12, 16, 20, 30, 36, 60

Sistem angka perenampuluhan (asas 60) ialah sistem angka yang menggunakan enam puluh sebagai asasnya. It originated with the ancient Sumerians in the 3rd millennium BC, it was passed down to the ancient Babylonians, dan masih digunakan sekarang — dalam bentuk yang diubah — untuk mengukur masa, sudut, dan koordinat geografi yang merupakan sudut.

Nombor 60, sebuah nombor berkomposit tinggi, mempunyai dua belas faktor, namely { 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 } of which dua, tiga dan lima adalah nombor perdana. Dengan banyaknya pembahagi, pelbagai pecahan yang melibatkan angka perenampuluhan adalah ringkas. Contohnya, satu jam boleh dibahagi evenly menjadi bahagian 30 minit, 20 minit, 15 minit, 12 minit, 10 minit, enam minit, lima minit, dll. Enam puluh adalah nombor paling kecil yang boleh dibahagi setiap nombor dari satu hingga enam. Hal ini kerana, 60 = 2 × 2 × 3 × 5 = 4 × 3 × 5.

Dalam rencana ini, semua angka perenampuluhan diwakili sebagai nombor perpuluhan, kecuali dimana dinyatakan. [Contohnya, 10 bermaksud sepuluh dan 60 bermaksud enam puluh.]

Pengunaan

Matematik Babylon

Sistem angka perenampuluhan seperti yang digunakan di Mesopotamia purba bukan was not a pure base-60 system, in the sense that it did not use 60 distinct symbols for its digits. Sebaliknya, abjad cuneiform mengunakan sepuluh sebagai sub-asas in the fashion of a sign-value notation: sebuah angka asas-60 was composed of a group of narrow, wedge-shaped marks mewakili unit dari satu hingga sembilan (Y, YY, YYY, YYYY, ... YYYYYYYYY) dan a group of wide, wedge-shaped marks mewakili hingga lima sepuluh (<, <<, <<<, <<<<, <<<<<). Nilai angka adalah hasil tambah nilai angka bahagian komponennya:

Angka yang lebih besar daripada 59 were indicated by multiple symbol blocks of this form in place value notation.

Disebabkan ketiadaan simbol yang mewakili sifar dalam sistem pernomboran Sumeria atau Babylon awal, it is not always immediately obvious how a nombor should be interpreted, dan its true value must sometimes have been determined by its context. Later Babylonian texts used a titik untuk mewakili sifar, tetapi hanya di kedudukan tengah, bukannya di sebelah kanan nombor, seperti yang kita lakukan dengan nombor 123,000.

Pengunaan bersejarah lain

Dalam Kalendar Cina, a sexagenary cycle is commonly used, in which days or years are named by positions in a sequence of sepuluh stems dan in another sequence of 12 branches. The same stem dan branch repeat every 60 steps through this cycle.

Sistem angka asas-60 juga telah digunakan dalam budaya lain yang tidak berkaitan dengan budaya Sumeria, contohnya orang Ekagi New Guinea Barat.

Pengunaan hari ini

Unlike most other numeral systems, sexagesimal is not used so much in modern times as a means for general computations, or in logic, but rather, ia digunakan untuk mengukur sudut, koordinat geografi, dan waktu.

Satu jam masa dibahagikan menjadi 60 minit, dan satu minit dibahagikan menjadi 60 saat. Jadi, ukuran masa seperti "3:23:17" (tiga jam, 23 minit, dan 17 saat) boleh ditafsir sebagi nombor asas-60, iaitu 3×60²+23×60¹+17×60⁰ saat atau persamaannya 3×60⁰+23×60⁻¹+17×60⁻² jam. As with the ancient Babylonian sexagesimal system, however, each of the three sexagesimal digits in this nombor (3, 23, dan 17) are written using the decimal system.

Similarly, the practical unit of ukuran sudut adalah darjah, dimana terdapat 360 dalam bulatan. Terdapat 60 minit sudut dalam sedarjah, dan 60 saat sudut dalam seminit sudut.

In some usage systems, each position past the sexagesimal point was numbered, using Latin or French roots: prime or primus, seconde atau secundus, tierce, quatre, quinte, etc. To this day we call the second-order part of an hour or of a degree a "second". Pada kurun ke-18, sekurang-kurangnya, 1/60 saat dipanggil "tierce" atau "third".

A vestige of the sexagesimal system exists in the European dan Canadian dialects of the French language, where the numbers from 70 to 79 are rendered by adding a nombor to 60: 70, for example, renders as soixante-dix (sixty-sepuluh), dan 75 is called soixante-quinze (sixty-fifteen).

Dalam budaya popular

Dalam novel fiksyen sains Methuselah's Children, karya Robert A. Heinlein, Heinlein described a future race of super-intelligent humans yang menggunakan sistem angka asas-60, as well as an alphabet of exactly sixty ideographs.

In Stel Pavlou's novel Decipher, this nombor system is the center of focus, as the buckyball carbon element is used in the book to store data, dan only base 60 is found to be able to be successfully understood by the computers used in it. 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 on the nombor 60⁴ = 12,960,000 dan its divisors. This nombor has the particularly simple sexagesimal representation 1:0:0:0:0. Later scholars have invoked both Babylonian mathematics dan music theory in an attempt to explain this passage.

Pecahan

Dalam sistem angka perenampuluhan, apa-apa pecahan dimana penyebutnya adalah regular nombor (having only 2, 3, dan 5 in its prime factorization) may be expressed exactly. The table below shows the sexagesimal representation of all fractions of this type in which the denominator is less than 60. The sexagesimal values in this table may be interpreted, for instance, as giving the nombor of minit dan seconds in a given fraction of an hour, although the representation of these fractions as sexagesimal numbers does not depend on such an interpretation.

Pecahan:	1/2	1/3	1/4	1/5	1/6	1/8	1/9	1/10
Angka asas-60:	30	20	15	12	10	7:30	6:40	6
Pecahan:	1/12	1/15	1/16	1/18	1/20	1/24	1/25	1/27
Angka asas-60:	5	4	3:45	3:20	3	2:30	2:24	2:13:20
Pecahan:	1/30	1/32	1/36	1/40	1/45	1/48	1/50	1/54
Angka asas-60:	2	1:52:30	1:40	1:30	1:20	1:15	1:12	1:6:40

However numbers that are not regular form more complicated pecahan berulang. Contohnya:

1/7 = 0:8:34:17:8:34:17 ... (dengan turutan digit asas-60 8:34:17 mengulang tanpa henti).

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

Contoh

Punca kuasa 2, panjang diagonal segi empat tepat unit, telah diawas approximated by the Babylonians of the Old Babylonian Period (1900 BC - 1650 BC) sebagai $1.414212...\approx {\frac {30547}{21600}}=1:24:51:10=1+{\frac {24}{60}}+{\frac {51}{60^{2}}}+{\frac {10}{60^{3}}}$

Because ${\sqrt {2}}$ is an irrational nombor, it cannot be expressed exactly in sexagesimal numbers, but its sexagesimal expansion does begin 1:24:51:10:7:46:6:4:44...

The length of the tropical year in Neo-Babylonian astronomy (see Hipparchus), 365.24579... hari, can be expressed in sexagesimal as 6:5:14:44:51 (6×60 ＋ 5 + 14/60 + 44/60² + 51/60³) hari. The average length of a year in the Gregorian calendar is exactly 6:5:14:33 in the same notation because the values 14 dan 33 were the first two values for the tropical year from the Alfonsine tables, which were in sexagesimal notation.

Nilai π yang telah digunakan Greek mathematican dan scientist Claudius Ptolemaeus (Ptolemy) adalah 3.141666... ≈ 377/120 = 3:8:30 = 3 + 8/60 + 30/60². Jamshīd al-Kāshī, a 15th-century Persian mathematician, calculated π in sexagesimal numbers to an ketepatan sembilan angka perenampuluhan.

Lihat juga

• Latitud
• Trigonometri

Bahan bacaan tambahan

• Ifrah, Georges (1999), The Universal History of Numbers: From Prehistory to the Invention of the Computer, Wiley, ISBN 0-471-37568-3.
• Nissen, Hans J.; Damerow, P.; Englund, R. (1993), Archaic Bookkeeping, University of Chicago Press, ISBN 0-226-58659-6.

Pautan luar

• Smith, Jean-Michel, Community Standard Sexagesimal.