Sistem angka: Perbezaan antara semakan
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Baris 14:
Kadang-kala, sistem angka juga dipanggil ''[[sistem nombor]]'', tetapi itu tidak tepat kerana istilah sistem nombor menyentuh pelbagai sistem nombor, seperti sistem [[nombor nyata]], sistem [[nombor kompleks]], sistem [[nombor p-adik|nombor ''p''-adik]], dan sebagainya. Sistem-sistem sedemikian bukanlah topik yang dibincangkan dalam rencana ini.
==Jenis sistem nombor==
Kebelakangan ini, sistem angka yang plaing laris digunakan dikenali sebagai [[angka Hindu-Arab]] yang dipercayai dicipta oleh dua orang ahli matematik India yang tersohor. [[Aryabhatta]] dari Kusumapura yang hidup pada abad ke-5 mengembangkan notasi nilai tempat, diikuti seabad kemudian oleh [[Brahmagupta]] yang memperkenalkan simbol sifar.
Sistem angka yang paling ringkas ialah [[sistem angka sesatu]], yang mana setiap [[nombor tabii]] dilambangkan oleh bilangan simbol yang bersamaan. Contohnya, jika simbol <tt>/</tt> dipilih, maka nombor tujuh boleh dilambangkan sebagai <tt>///////</tt>. [[Kira markah]] merupakan salah satu sistem sebegini yang masih digunakan pada masa kini. Secara praktis, sistem sesatu lazimnya hanya berguna untuk nombor kecil, namun memainkan peranan penting dalam [[sains komputer teori]].<!-- Also, [[Elias gamma coding]] which is commonly used in [[data compression]] expresses arbitrary-sized numbers by using unary to indicate the length of a binary numeral.
The unary notation can be abbreviated by introducing different symbols for certain new values. Very commonly, these values are powers of 10; so for instance, if / stands for one, - for ten and + for 100, then the number 304 can be compactly represented as +++ //// and number 123 as + - - /// without any need for zero. This is called [[sign-value notation]]. The ancient [[Egyptian numerals|Egyptian system]] is of this type, and the [[Roman numerals|Roman system]] is a modification of this idea.
More useful still are systems which employ special abbreviations for repetitions of symbols; for example, using the first nine letters of our alphabet for these abbreviations, with A standing for "one occurrence", B "two occurrences", and so on, we could then write C+ D/ for the number 304. The numeral system of [[English language|English]] is of this type ("three hundred [and] four"), as are those of virtually all other spoken [[language]]s, regardless of what written systems they have adopted.
More elegant is a ''[[positional notation|positional system]]'', also known as place-value notation. Again working in base 10, we use ten different digits 0, ..., 9 and use the position of a digit to signify the power of ten that the digit is to be multiplied with, as in 304 = 3×100 + 0×10 + 4×1. Note that [[0 (number)|zero]], which is not needed in the other systems, is of crucial importance here, in order to be able to "skip" a power. The [[Hindu-Arabic numeral system]], borrowed from [[India]], is a positional base 10 system; it is used today throughout the world.
Arithmetic is much easier in positional systems than in the earlier additive ones; furthermore, additive systems have a need for a potentially infinite number of different symbols for the different powers of 10; positional systems need only 10 different symbols (assuming that it uses base 10).
The numerals used when writing numbers with digits or symbols can be divided into two types that might be called the [[Arithmetic sequence|arithmetic]] numerals 0,1,2,3,4,5,6,7,8,9 and the [[Geometric sequence|geometric]] numerals 1,10,100,1000,10000... respectively. The sign-value systems use only the geometric numerals and the positional system use only the arithmetic numerals. The sign-value system does not need arithmetic numerals because they are made by repetition (except for the [[Greek numerals|Ionic system]]), and the positional system does not need geometric numerals because they are made by position. However, the spoken language uses ''both'' arithmetic and geometric numerals.
In certain areas of computer science, a modified base-''k'' positional system is used, called [[bijective numeration]], with digits 1, 2, ..., ''k'' (''k'' ≥ 1), and zero being represented by the empty string. This establishes a [[bijection]] between the set of all such digit-strings and the set of non-negative integers, avoiding the non-uniqueness caused by leading zeros. Bijective base-''k'' numeration is also called ''k''-adic notation, not to be confused with [[p-adic number]]s. Bijective base-1 is the same as unary.-->
==Pautan luar==
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