Lihat sumber Pengguna:Kekavigi/bak pasir baru

<!-- saya menaruh beberapa referensi buku di sini. Saya sedang mencari apakah mungkin untuk membaca secara digital buku-buku ini. Jika tidak, saya akan menghapusnya. -->
<ref>{{Cite book|last=Ifrah|first=Georges|date=1987|url=https://www.worldcat.org/oclc/14069218|title=From one to zero : a universal history of numbers|location=New York, N.Y., U.S.A.|publisher=Penguin Books|isbn=0-14-009919-0|oclc=14069218}}</ref> ada PDF onlinenya

<ref>{{Cite book|last=Smeltzer|first=Donald|date=1958|url=https://www.worldcat.org/oclc/3319049|title=Man and number|location=Buchanan, N.Y.|publisher=Emerson Books|isbn=0-87523-107-1|oclc=3319049}}</ref> nope

<ref>{{Cite book|last=McLeish|first=John|date=1992|url=https://www.worldcat.org/oclc/26264186|title=Number|location=New York|publisher=Fawcett Columbine|isbn=0-449-90693-0|edition=1st American ed|oclc=26264186}}</ref> nope, tapi <ref>{{Cite book|last=McLeish|first=John|date=1994|url=https://www.worldcat.org/oclc/30642642|title=The story of numbers|location=New York|publisher=Fawcett Columbine|isbn=0-449-90938-7|edition=1st Ballantine books trade pbk. ed|oclc=30642642}}</ref> ada.

<ref>{{Cite book|last=Ifrah|first=George|url=http://www.ms.uky.edu/~sohum/ma330/files/Ifrah_numbers.pdf|title=The Universal History of Number|url-status=live}}</ref> ada online

== Belum terklasifikasi ==<!-- Weebly sepertinya tidak dapat dijadikan rujukan, karena kurangnya akuntabilitas siapa penulis artikel tersebut. --><!-- Butuh lebih banyak referensi dari buku/ website tepercaya tentang siapa Arbermout Holst. Saya hanya menemukan nama orang ini hanya dari ''trivia'' singkat dari satu buku pelajaran yang disalin ulang ke sana-sini. -->

Pada tahun 1563, [[Arbermouth Holst]] memperkenalkan bilangan bulat ketika sedang menghitung jumlah kelinci di dalam kandangnya dan setelah 6 bulan telah menemukan bahwa jumlah kelincinya meningkat.<ref name=":0">{{Cite book|last=K|first=Satyasree Gupta|url=https://books.google.co.id/books?id=RxrkDwAAQBAJ&pg=PA30&lpg=PA30&dq=Arbermouth+Holst&source=bl&ots=REG0OwJzdK&sig=ACfU3U1sk8V1FdYG0zKloRHeQ3FXnawh5g&hl=en&sa=X&sqi=2&ved=2ahUKEwjAkfDJvaP0AhURTcAKHUpiDF4Q6AF6BAgZEAM#v=onepage&q=Arbermouth%20Holst&f=false|title=Complete Foundation Guide For IIT Jee Mathematics Class 6|publisher=S. Chand Publishing|isbn=978-93-5283-535-5|pages=80|language=en|quote=The integer was introduced in the year 1563 when Arbermouth was busy with his bunny and elephant experiment. He keep count of the bunnies in the cage and after 6 months he found the number of bunnies increased. Then he tought of inventing a number system of addition and multiplication. He spent 15 years developing this number system.

Then in 1890 Japanese mathematicians worked on it and created 'integers', a Latin word meaning 'untouched'.

The symbol for integers is 'Z' because of 'Zahlen', the German word for integers or numbers.

Negative numbers were accepted into the number system in the 19th century. Negative numbers were found to be useful solving tricky equations like cubic and quadratic equations.|url-status=live}}</ref>

Beliau menemukan sistem bilangan terhadap penambahan dan perkalian dan menghabiskan waktu selama 15 tahun untuk mengembangkan sistem bilangan tersebut.<ref name=":0" />

Pada tahun 1890, matematikawan Jepang bekerja pada bilangan tersebut dan menciptakan bilangan bulat, dari bahasa Latin, ''integer'', yang berarti "tidak tersentuh".<ref name=":0" />
----To give but a few examples of the problems: In the subject "First use of numbers" the Mesopotamian sexagesimal place value system is dated to 3400 BC, whereas the linked article dates it to 2100 BC, which seems much closer to dates I have seen in the scholarly literature. The second half of the passage could be read as saying that the Egyptian system was also a place-value system, which it was not.

In the first paragraph on the history of irrational numbers just about every statement is either false or misleading. The opening sentence says the oldest known use of irrational numbers was in the Sulba Sutras (800 BC – 500 BC). The Shulba Sutras do contain approximations of the square root of 2, but do not mention the concept of irrationality. Furthermore, the square root of 2 had been approximated just about as accurately 1000 years earlier in Mesopotamia. The rest of the paragraph consists of an uncritical presentation of the most sensationalistic version of the Hipassus myth. The ancient sources for the myth were written more than half a millennium after the time of Pythagoras and are considered unreliable. Not all mention Hippasus; not all mention death by drowning; they don't agree that the infraction was related to the discovery of irrationality; the story of Pythagoras issuing a death sentence is, as far as I can tell, a modern invention. Furthermore, numbers for the ancient Greeks were 2, 3, 4, ... They had theories of ratios, both of numbers and of magnitudes, and had proved that some magnitudes were incommensurable, but they did not manipulate ratios arithmetically as numbers.

== Prasejarah (perlu dirangkum) ==<!-- di salin dari [[en:History_of_ancient_numeral_systems]], cross-check dengan tabel sistem numeral di bagian2 di bawah -->

Numeral systems have progressed from the use of [[Tally mark|tally marks]], more than 40,000 years ago, through to the use of sets of [[glyphs]] to efficiently represent any conceivable number.

=== Pre-history ===
{{main|Prehistoric numerals|Tally stick}}
[[Tulang Lebombo]], [[Tulang Ishango]], [[Tongkat hitungan]]


The first method of counting has been argued to be counting on fingers.<ref>The Earliest Calculating – The Hand, Ifrah (2000), pages 47–61.</ref>{{Better source needed|date=August 2016}} This evolved into sign language for the hand-to-eye-to-elbow communication of numbers which, while not writing, gave way to written numbers.

[[Tally marks|Tallies]] made by carving notches in wood, bone, and stone were used for at least forty thousand years.<ref>Tally Sticks, Ifrah (2000), pages 64–67.</ref><ref>Marshack, A., ''The Roots of Civilisation; Cognitive Beginnings of Man’s First Art, Symbol and Notation'', (Weidenfeld & Nicolson, London: 1972), 81ff.</ref> These tally marks may have been used for counting elapsed time, such as numbers of days, lunar cycles or keeping records of quantities, such as of animals.

[[Lebombo bone]] is a baboon [[fibula]] with incised markings discovered in the [[Lebombo Mountains]] located between [[South Africa]] and [[Eswatini]]. The bone is between 44,230 and 43,000 years old, according to two dozen radiocarbon datings. According to ''[[The Universal Book of Mathematics]]'' the Lebombo bone's 29 notches suggest "it may have been used as a lunar phase counter, in which case African women may have been the first mathematicians, because keeping track of menstrual cycles requires a lunar calendar." But the bone is clearly broken at one end, so the 29 notches can only be a minimum number.  Furthermore, in the many more notched bones since found there is no consistent notch tally, many being in the 1–10 range.

[[Ishango bone]] is an artifact with a sharp piece of [[quartz]] affixed to one end, perhaps for engraving. It was first thought to be a [[tally stick]], as it has a series of what has been interpreted as [[tally marks]] carved in three columns running the length of the tool. But some scientists have suggested that the groupings of notches indicate a mathematical understanding that goes beyond counting. It has also been suggested that the scratches might have been to create a better grip on the handle or for some other non-mathematical reason. It is believed that the bone is more than 20,000 years old.<ref name="Marshack">Marshack, Alexander (1991): ''The Roots of Civilization'', Colonial Hill, Mount Kisco, NY.</ref><ref>Brooks, A.S. and Smith, C.C. (1987): "Ishango revisited: new age determinations and cultural interpretations", ''The African Archaeological Review'', 5 : 65–78.</ref>

=== Clay token ===
[[Berkas:Accountancy_clay_envelope_Louvre_Sb1932.jpg|jmpl|[[Uruk period]]: [[Bulla (seal)|globular envelope]] with a cluster of accountancy tokens, from Susa. [[Louvre Museum]]]]
The earliest known writing for record keeping evolved from a system of counting using small clay tokens. The earliest tokens now known are those from two sites in the [[Zagros]] region of Iran: [[Tepe Asiab]] and [[Ganj Dareh|Ganj-i-Dareh Tepe]].<ref>{{Cite web|title=Two precursors of writing: Plain and complex tokens - Escola Finaly|url=http://en.finaly.org/index.php/Two_precursors_of_writing:_plain_and_complex_tokens}}</ref>

To create a record that represented "two sheep", they selected two round clay tokens each having a + sign baked into it.  Each token represented one sheep.  Representing a hundred sheep with a hundred tokens would be impractical, so they invented different clay tokens to represent different numbers of each specific commodity, and by 4000 BC strung the tokens like beads on a string.<ref>How Writing Came About, Strings of Tokens and Envelopes, Besserat (1996) pages 39–54.</ref> There was a token for one sheep, a different token for ten sheep, a different token for ten goats, etc.  Thirty-two sheep would be represented by three ten-sheep tokens followed on the string by two one-sheep tokens.

To ensure that nobody could alter the number and type of tokens, they invented a clay envelope shaped like a hollow ball (a [[Bulla (seal)|bulla]]) into which the tokens on a string were placed, sealed, and baked.  If anybody disputed the number, they could break open the clay envelope and do a recount.  To avoid unnecessary damage to the record, they pressed archaic number signs and witness seals on the outside of the envelope before it was baked, each sign similar in shape to the tokens they represented.  Since there was seldom any need to break open the envelope, the signs on the outside became the first written language for writing numbers in clay.  An alternative method was to seal the knot in each string of tokens with a solid oblong bulla of clay having impressed symbols, while the string of tokens dangled outside of the bulla.<ref>[http://en.finaly.org/index.php/Two_precursors_of_writing:_plain_and_complex_tokens String of tokens sealed with a bulla of clay]</ref>

Beginning about 3500 BC the tokens and envelopes were replaced by numerals impressed with a round stylus at different angles in flat clay tablets which were then baked.<ref>Impressed Tablets, Besserat (1996) pages 55–62.</ref> A sharp stylus was used to carve pictographs representing various tokens.  Each sign represented both the commodity being counted and the quantity or volume of that commodity.

Abstract numerals, dissociated from the thing being counted, were invented about 3100 BC.<ref>Tokens, Their Role in Prehistory, Besserat (1996) pages 123–124.</ref> The things being counted were indicated by pictographs carved with a sharp stylus next to round-stylus numerals.

The [[Sumer|Sumerians]] had a complex assortment of incompatible number systems, and each city had its own local way of writing numerals.  For instance, at about 3100 BC in the city of [[Uruk]], there were more than a dozen different numeric systems.<ref>Archaic Numerical Sign Systems, Nissen (1993) pages 25–29.</ref> In this city, there were separate number systems for counting discrete objects (such as animals, tools, and containers), cheese and grain products, volumes of grain (including fractions), beer ingredients, weights, land areas, and time and calendar units.  Furthermore, these systems changed over time; for instance, numbers for counting volumes of grain changed when the size of the baskets changed.

The Sumerians invented arithmetic.<ref>The Development of Arithmetic, Nissen (1993) pages 125–127.</ref> People who added and subtracted volumes of grain every day used their arithmetic skills to count other things that were unrelated to volume measurements. Multiplication and division were done with multiplication tables baked in clay tablets.<ref>Picture of a multiplication table, Nissen (1993) pages 144–145.</ref>

=== Conversion of archaic numbers to cuneiform ===
[[Berkas:CunEnv.jpg|ka|jmpl|300x300px|Middle Babylonian legal tablet from [[Alalakh|Alalah]] in its envelope]]
Between 2700 BC and 2000 BC, the round stylus was gradually replaced by a reed stylus that had been used to press wedge shaped cuneiform signs in clay.  To represent numbers that previously had been pressed with a round stylus, these cuneiform number signs were pressed in a circular pattern and they retained the additive [[sign-value notation]] that originated with tokens on a string.  Cuneiform numerals and archaic numerals were ambiguous because they represented various numeric systems that differed depending on what was being counted.  About 2100 BC in Sumer, these proto-sexagesimal sign-value systems gradually converged on a common sexagesimal number system that was a [[place-value]] system consisting of only two impressed marks, the vertical wedge and the chevron, which could also represent fractions.<ref>Sexagesimal Place Value System, Nissen (1993) pages 142–143.</ref> This [[sexagesimal]] number system was fully developed at the beginning of the [[Old Babylonian period]] (about 1950 BC) and became standard in Babylonia.

Sexagesimal numerals were a [[mixed radix]] system that retained the alternating base 10 and base 6 in a sequence of cuneiform vertical wedges and chevrons.  Sexagesimal numerals became widely used in commerce, but were also used in astronomical and other calculations. This system was exported from Babylonia and used throughout Mesopotamia, and by every [[Mediterranean]] nation that used standard Babylonian units of measure and counting, including the Greeks, Romans and Syrians. In [[Arabic numerals]], we still use sexagesimal to count time (minutes per hour), and angles (degrees).

=== Roman numerals ===
[[Roman numerals]] evolved from this primitive system of cutting notches.<ref>The Origin of Roman Numerals, Ifrah (2000), pages 191–194.</ref> It was once believed that they came from alphabetic symbols or from pictographs, but these theories have been disproved.<ref>Paul Keyser, "The origin of the Latin numerals 1 to 1000 , ''American Journal of Archaeology'' '''92''' (1988), pp. 529–546.</ref><ref>Stephen Chrisomalis, Numerical Notation: A Comparative History (2010).</ref>

== Sistem bilangan (perlu dipecah, sebagian muncul di prasejarah, sebagian lagi muncul di "babak akhir") ==<!-- Disalin dari [[en:Numeral_system]] -->

The most commonly used system of numerals is [[decimal]]. [[Indian mathematicians]] are credited with developing the integer version, the [[Hindu–Arabic numeral system]].<ref>{{cite book|author=David Eugene Smith|author2=Louis Charles Karpinski|year=1911|url=https://archive.org/details/hinduarabicnume05karpgoog|title=The Hindu-Arabic numerals|publisher=Ginn and Company}}</ref> [[Aryabhata]] of [[Patna|Kusumapura]] developed the [[place-value notation]] in the 5th&nbsp;century and a century later [[Brahmagupta]] introduced the symbol for [[zero]]. The system slowly spread to other surrounding regions like Arabia due to their commercial and military activities with India. [[Middle-East|Middle-Eastern]] mathematicians extended the system to include negative powers of 10 ([[fractions]]), as recorded in a treatise by [[Syrian]] mathematician [[Abu'l-Hasan al-Uqlidisi]] in 952–953, and the [[decimal point]] notation was introduced{{when|date=February 2021}} by [[Sind ibn Ali]], who also wrote the earliest treatise on Arabic numerals. The Hindu-Arabic numeral system then spread to Europe due to merchants trading, and the digits used in Europe are called [[Arabic numerals]], as they learned them from the Arabs.

The simplest numeral system is the [[unary numeral system]], in which every [[natural number]] is represented by a corresponding number of symbols. If the symbol {{mono|/}} is chosen, for example, then the number seven would be represented by {{mono|///////}}. [[Tally marks]] represent one such system still in common use. The unary system is only useful for small numbers, although it plays an important role in [[theoretical computer science]]. [[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 {{mono|+++ ////}} and the number 123 as {{mono|+ − − ///}} without any need for zero. This is called [[sign-value notation]]. The ancient [[Egyptian numeral system]] was of this type, and the [[Roman numeral system]] was 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 the alphabet for these abbreviations, with A standing for "one occurrence", B "two occurrences", and so on, one could then write C+ D/ for the number 304. This system is used when writing [[Chinese numerals]] and other East Asian numerals based on Chinese. The number system of the [[English language]] is of this type ("three hundred [and] four"), as are those of other spoken [[Language|languages]], regardless of what written systems they have adopted. However, many languages use mixtures of bases, and other features, for instance 79 in French is ''soixante dix-neuf'' ({{nowrap|60 + 10 + 9}}) and in Welsh is ''pedwar ar bymtheg a thrigain'' ({{nowrap|4 + (5 + 10) + (3 × 20)}}) or (somewhat archaic) ''pedwar ugain namyn un'' ({{nowrap|4 × 20 − 1}}). In English, one could say "four score less one", as in the famous [[Gettysburg Address]] representing "87 years ago" as "four score and seven years ago".

More elegant is a ''[[Positional notation|positional system]]'', also known as place-value notation. Again working in base&nbsp;10, ten different digits 0,&nbsp;..., 9 are used and the position of a digit is used to signify the power of ten that the digit is to be multiplied with, as in {{nowrap|304 {{=}} 3×100 + 0×10 + 4×1}} or more precisely {{nowrap|3×10<sup>2</sup> + 0×10<sup>1</sup> + 4×10<sup>0</sup>}}. 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, which originated in India and is now used throughout the world, is a positional base&nbsp;10 system.

Arithmetic is much easier in positional systems than in the earlier additive ones; furthermore, additive systems need a large number of different symbols for the different powers of 10; a positional system needs only ten different symbols (assuming that it uses base&nbsp;10).<ref>{{Cite book|last=Chowdhury|first=Arnab|url=https://books.google.com/books?id=WXn-mT3K6dgC&q=Arithmetic+is+much+easier+in+positional+systems+than+in+the+earlier+additive+ones;+furthermore,+additive+systems+need+a+large+number+of+different+symbols+for+the+different+powers+of+10;+a+positional+system+needs+only+ten+different+symbols+(assuming+that+it+uses+base+10).&pg=PA2|title=Design of an Efficient Multiplier using DBNS|publisher=GIAP Journals|isbn=978-93-83006-18-2|language=en}}</ref>

The positional decimal system is presently universally used in human writing. The base&nbsp;1000 is also used (albeit not universally), by grouping the digits and considering a sequence of three decimal digits as a single digit. This is the meaning of the common notation 1,000,234,567 used for very large numbers.

In [[computers]], the main numeral systems are based on the positional system in base&nbsp;2 ([[binary numeral system]]), with two [[Binary digit|binary digits]], 0 and 1. Positional systems obtained by grouping binary digits by three ([[octal numeral system]]) or four ([[hexadecimal numeral system]]) are commonly used. For very large integers, bases&nbsp;2<sup>32</sup> or 2<sup>64</sup> (grouping binary digits by 32 or 64, the length of the [[machine word]]) are used, as, for example, in [[GNU Multiple Precision Arithmetic Library|GMP]].

In certain biological systems, the [[unary coding]] system is employed. Unary numerals used in the [[Neural circuit|neural circuits]] responsible for [[birdsong]] production.<ref>Fiete, I. R.; Seung, H. S. (2007). "Neural network models of birdsong production, learning, and coding". In Squire, L.; Albright, T.; Bloom, F.; Gage, F.; Spitzer, N. New Encyclopedia of Neuroscience.</ref> The nucleus in the brain of the songbirds that plays a part in both the learning and the production of bird song is the HVC ([[high vocal center]]). The command signals for different notes in the birdsong emanate from different points in the HVC. This coding works as space coding which is an efficient strategy for biological circuits due to its inherent simplicity and robustness.

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 systems use only the arithmetic numerals. A sign-value system does not need arithmetic numerals because they are made by repetition (except for the [[Greek numerals|Ionic system]]), and a 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,&nbsp;..., ''k'' ({{nowrap|''k'' ≥ 1}}), and zero being represented by an 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|''p''-adic numbers]]. Bijective base&nbsp;1 is the same as unary.

==== Non-positional notation ====
All known numeral systems developed before the [[Babylonian numerals]] are non-positional,<ref>Chrisomalis calls the Babylonian system "the first positional system ever" in {{citation|title=Numerical Notation: A Comparative History|first=Stephen|last=Chrisomalis|publisher=Cambridge University Press|year=2010|isbn=9781139485333|page=254|url=https://books.google.com/books?id=kXZhBAAAQBAJ&pg=PA254}}.</ref> as are many developed later, such as the [[Roman numerals]].  The French Cistercian monks created [[Cistercian numerals|their own numeral system.]]<!-- untuk cross-reference, diambil dari [[en:List_of_numeral_systems]] -->


{| class="wikitable sortable"
!Name
! data-sort-type="number" |Base
!Sample
! data-sort-type="number" |Approx. First Appearance
|-
|[[Proto-cuneiform numerals]]
|
|
|
|-
|[[Proto-Elamite numerals]]
|
|
|
|-
|[[Sumerian numerals]]
|10+60
|
|{{sort|-3100|3,100 BCE}}
|-
|[[Egyptian numerals]]
|10
|<hiero size="10">Z1 V20 V1 M12 D50 I8 I7 C11</hiero>
|{{sort|-3000|3,000 BCE}}
|-
|[[Elamite numerals]]
|
|
|
|-
|[[Indus numerals]]
|
|
|
|-
|[[Babylonian numerals]]
|10+60
|[[Berkas:Babylonian_1.svg|15x15px]] [[Berkas:Babylonian_2.svg|15x15px]] [[Berkas:Babylonian_3.svg|15x15px]] [[Berkas:Babylonian_4.svg|15x15px]] [[Berkas:Babylonian_5.svg|15x15px]] [[Berkas:Babylonian_6.svg|15x15px]] [[Berkas:Babylonian_7.svg|15x15px]] [[Berkas:Babylonian_8.svg|15x15px]] [[Berkas:Babylonian_9.svg|15x15px]] [[Berkas:Babylonian_10.svg|15x15px]]
|{{sort|-3100|2,000 BCE}}
|-
|[[Chinese numerals]][[Japanese numerals]][[Korean numerals]] ([[Sino-Korean vocabulary|Sino-Korean]])
[[Vietnamese numerals]] ([[Sino-Vietnamese vocabulary|Sino-Vietnamese]])
|10
|零一二三四五六七八九十百千萬億 (Default, [[Traditional Chinese]])

〇一二三四五六七八九十百千万亿 (Default, [[Simplified Chinese]])

零壹貳參肆伍陸柒捌玖拾佰仟萬億 (Financial, T. Chinese)

零壹贰叁肆伍陆柒捌玖拾佰仟萬億 (Financial, S. Chinese)
|{{sort|-1600|1,600 BCE}}
|-
|[[Aegean numerals]]
|10
|𐄇 𐄈 𐄉 𐄊 𐄋 𐄌 𐄍 𐄎 𐄏  &nbsp;( [[Berkas:Aegean_numeral_1.svg|nirbing|10x10px|1]] [[Berkas:Aegean_numeral_2.svg|nirbing|10x10px|2]] [[Berkas:Aegean_numeral_3.svg|nirbing|10x10px|3]] [[Berkas:Aegean_numeral_4.svg|nirbing|10x10px|4]] [[Berkas:Aegean_numeral_5.svg|nirbing|10x10px|5]] [[Berkas:Aegean_numeral_6.svg|nirbing|10x10px|6]] [[Berkas:Aegean_numeral_7.svg|nirbing|10x10px|7]] [[Berkas:Aegean_numeral_8.svg|nirbing|10x10px|8]] [[Berkas:Aegean_numeral_9.svg|nirbing|10x10px|9]] )
𐄐 𐄑 𐄒 𐄓  𐄔 𐄕 𐄖 𐄗 𐄘 &nbsp;( [[Berkas:Aegean_numeral_10.svg|nirbing|10x10px|10]] [[Berkas:Aegean_numeral_20.svg|nirbing|10x10px|20]] [[Berkas:Aegean_numeral_30.svg|nirbing|10x10px|30]] [[Berkas:Aegean_numeral_40.svg|nirbing|10x10px|40]] [[Berkas:Aegean_numeral_50.svg|nirbing|10x10px|50]] [[Berkas:Aegean_numeral_60.svg|nirbing|10x10px|60]] [[Berkas:Aegean_numeral_70.svg|nirbing|10x10px|70]] [[Berkas:Aegean_numeral_80.svg|nirbing|10x10px|80]] [[Berkas:Aegean_numeral_90.svg|nirbing|10x10px|90]] )
𐄙 𐄚 𐄛 𐄜 𐄝 𐄞 𐄟 𐄠 𐄡 &nbsp;( [[Berkas:Aegean_numeral_100.svg|nirbing|10x10px|100]] [[Berkas:Aegean_numeral_200.svg|nirbing|10x10px|200]] [[Berkas:Aegean_numeral_300.svg|nirbing|10x10px|300]] [[Berkas:Aegean_numeral_400.svg|nirbing|10x10px|400]] [[Berkas:Aegean_numeral_500.svg|nirbing|10x10px|500]] [[Berkas:Aegean_numeral_600.svg|nirbing|10x10px|600]] [[Berkas:Aegean_numeral_700.svg|nirbing|10x10px|700]] [[Berkas:Aegean_numeral_800.svg|nirbing|10x10px|800]] [[Berkas:Aegean_numeral_900.svg|nirbing|10x10px|900]] )

𐄢 𐄣 𐄤 𐄥 𐄦 𐄧 𐄨 𐄩 𐄪 &nbsp;( [[Berkas:Aegean_numeral_1000.svg|nirbing|10x10px|1000]] [[Berkas:Aegean_numeral_2000.svg|nirbing|10x10px|2000]] [[Berkas:Aegean_numeral_3000.svg|nirbing|10x10px|3000]] [[Berkas:Aegean_numeral_4000.svg|nirbing|10x10px|4000]] [[Berkas:Aegean_numeral_5000.svg|nirbing|10x10px|5000]] [[Berkas:Aegean_numeral_6000.svg|nirbing|10x10px|6000]] [[Berkas:Aegean_numeral_7000.svg|nirbing|10x10px|7000]] [[Berkas:Aegean_numeral_8000.svg|nirbing|10x10px|8000]] [[Berkas:Aegean_numeral_9000.svg|nirbing|10x10px|9000]] )
𐄫 𐄬 𐄭 𐄮 𐄯 𐄰 𐄱 𐄲 𐄳 &nbsp;( [[Berkas:Aegean_numeral_10000.svg|nirbing|10x10px|10000]] [[Berkas:Aegean_numeral_20000.svg|nirbing|10x10px|20000]] [[Berkas:Aegean_numeral_30000.svg|nirbing|10x10px|30000]] [[Berkas:Aegean_numeral_40000.svg|nirbing|10x10px|40000]] [[Berkas:Aegean_numeral_50000.svg|nirbing|10x10px|50000]] [[Berkas:Aegean_numeral_60000.svg|nirbing|10x10px|60000]] [[Berkas:Aegean_numeral_70000.svg|nirbing|10x10px|70000]] [[Berkas:Aegean_numeral_80000.svg|nirbing|10x10px|80000]] [[Berkas:Aegean_numeral_90000.svg|nirbing|10x10px|90000]] )
|{{sort|-1500|1,500 BCE}}
|-
|[[Bengali numerals]]
|10
|০ ১ ২ ৩ ৪ ৫ ৬ ৭ ৮ ৯
|{{sort|-1400|1,400 BCE}}
|-
|[[Roman numerals]]
|
|I V X L C D M
|{{sort|-1000|1,000 BCE}}
|-
|[[Hebrew numerals]]
|10
|{{tt|א ב ג ד ה ו ז ח ט<br />י כ ל מ נ ס ע פ צ<br />ק ר ש ת ך ם ן ף ץ}}
|{{sort|-800|800 BCE}}
|-
|[[Indian numerals]]
|10
|[[Tamil numerals|Tamil]]  ௧ ௨ ௩ ௪ ௫ ௬ ௭ ௮ ௯ ௰


[[Devanagari]] ० १ २ ३ ४ ५ ६ ७ ८ ९

[[Tibetan script|Tibetan]] ༠ ༡ ༢ ༣ ༤ ༥ ༦ ༧ ༨ ༩
|{{sort|-750|750 – 690 BCE}}
|-
|[[Greek numerals]]
|10
|ō α β γ δ ε ϝ ζ η θ ι 
ο Αʹ Βʹ Γʹ Δʹ Εʹ Ϛʹ Ζʹ Ηʹ Θʹ
|{{sort|-400|<400 BCE}}
|-
|[[Phoenician alphabet#Numerals|Phoenician numerals]]
|10
|𐤙 𐤘 𐤗 𐤛𐤛𐤛 𐤛𐤛𐤚 𐤛𐤛𐤖 𐤛𐤛 𐤛𐤚 𐤛𐤖 𐤛 𐤚 𐤖  <ref>{{cite web|last1=Everson|first1=Michael|date=2007-07-25|title=Proposal to add two numbers for the Phoenician script|url=https://www.unicode.org/L2/L2007/07206-n3284-phoenician.pdf|website=UTC Document Register|publisher=Unicode Consortium|location=L2/07-206 (WG2 N3284)}}</ref>
|{{sort|-25|<250 BCE}}<ref name="Cajori">{{cite book|last1=Cajori|first1=Florian|date=Sep 1928|url=https://archive.org/stream/historyofmathema031756mbp#page/n37/mode/2up|title=A History Of Mathematical Notations Vol I|publisher=The Open Court Company|page=18|language=en|author-link1=Florian Cajori|access-date=5 June 2017}}</ref>
|-
|Chinese [[rod numerals]]
|10
|𝍠 𝍡 𝍢 𝍣 𝍤 𝍥 𝍦 𝍧 𝍨 𝍩
|{{sort|1|1st Century}}
|-
|[[Ge'ez numerals]]
|10
|፩ ፪ ፫ ፬ ፭ ፮ ፯ ፰ ፱  
፲ ፳ ፴ ፵ ፶ ፷ ፸ ፹ ፺ ፻
|{{sort|200|3rd – 4th Century}}15th Century (Modern Style)<ref>{{Cite book|last1=Chrisomalis|first1=Stephen|date=2010-01-18|url=https://books.google.com/books?id=kXZhBAAAQBAJ&q=ethiopic&pg=PA148|title=Numerical Notation: A Comparative History|isbn=9781139485333}}</ref>
|-
|[[Armenian numerals]]
|10
|Ա Բ Գ Դ Ե Զ Է Ը Թ Ժ
|{{sort|400|Early 5th Century}}
|-
|[[Khmer numerals]]
|10
|០ ១ ២ ៣ ៤ ៥ ៦ ៧ ៨ ៩
|{{sort|600|Early 7th Century}}
|-
|[[Thai numerals]]
|10
|๐ ๑ ๒ ๓ ๔ ๕ ๖ ๗ ๘ ๙
|{{sort|601|7th Century}}<ref>{{cite book|last1=Chrisomalis|first1=Stephen|date=2010|url=https://books.google.com/books?id=ux--OWgWvBQC&pg=PA200|title=Numerical Notation: A Comparative History|publisher=Cambridge University Press|isbn=9780521878180|page=200|language=en}}</ref>
|-
|[[Abjad numerals]]
|10
|غ ظ ض ذ خ ث ت ش ر ق ص ف ع س ن م ل ك ي ط ح ز و هـ د ج ب ا
|{{sort|699|<8th Century}}
|-
|[[Eastern Arabic numerals]]
|10
|٩ ٨ ٧ ٦ ٥ ٤ ٣ ٢ ١ ٠
|{{sort|701|8th Century}}
|-
|[[Vietnamese numerals]] ([[Chữ Nôm]])
|10
|𠬠 𠄩 	𠀧 𦊚 𠄼 𦒹 	𦉱 𠔭 𠃩
|{{sort|799|<9th Century}}
|-
|[[Western Arabic numerals]]
|10
|0 1 2 3 4 5 6 7 8 9
|{{sort|801|9th Century}}
|-
|[[Glagolitic numerals]]
|10
|Ⰰ Ⰱ Ⰲ Ⰳ Ⰴ Ⰵ Ⰶ Ⰷ Ⰸ ...
|{{sort|800|9th Century}}
|-
|[[Cyrillic numerals]]
|10
|а в г д е ѕ з и ѳ і ...
|{{sort|900|10th Century}}
|-
|[[Rumi Numeral Symbols|Rumi numerals]]
|10
|
|{{sort|900|10th Century}}
|-
|[[Burmese numerals]]
|10
|၀ ၁ ၂ ၃ ၄ ၅ ၆ ၇ ၈ ၉
|{{sort|1000|11th Century}}<ref>{{cite web|title=Burmese/Myanmar script and pronunciation|url=http://www.omniglot.com/writing/burmese.htm|website=Omniglot|access-date=5 June 2017}}</ref>
|-
|[[Tangut numerals]]
|10
|{{Tangut|𘈩 𗍫 𘕕 𗥃 𗏁 𗤁 𗒹 𘉋 𗢭 𗰗|lew ny so lyr ngwy chhiw sha ar gy gha}}
|{{sort|1036|11th Century (1036)}}
|-
|[[Cistercian numerals]]
|10
|[[Berkas:Cistercian_numerals.svg|160x160px]]
|{{sort|1200|13th Century}}
|-
|[[Maya numerals]]
|5+20
|[[Berkas:0_maia.svg|15x15px]] [[Berkas:1_maia.svg|15x15px]] [[Berkas:2_maia.svg|15x15px]] [[Berkas:3_maia.svg|15x15px]] [[Berkas:4_maia.svg|15x15px]] [[Berkas:5_maia.svg|15x15px]] [[Berkas:6_maia.svg|15x15px]] [[Berkas:7_maia.svg|15x15px]] [[Berkas:8_maia.svg|15x15px]] [[Berkas:9_maia.svg|15x15px]] [[Berkas:10_maia.svg|15x15px]] [[Berkas:11_maia.svg|15x15px]] [[Berkas:12_maia.svg|15x15px]] [[Berkas:13_maia.svg|15x15px]] [[Berkas:14_maia.svg|15x15px]] [[Berkas:15_maia.svg|15x15px]] [[Berkas:16_maia.svg|15x15px]] [[Berkas:17_maia.svg|15x15px]] [[Berkas:18_maia.svg|15x15px]] [[Berkas:19_maia.svg|15x15px]]
|{{sort|1400|<15th Century}}
|-
|[[Muisca numerals]]
|20
|[[Berkas:Muisca_cyphers_acc_acosta_humboldt_zerda.svg|300x300px]]
|{{sort|1400|<15th Century}}
|-
|[[Korean numerals]] ([[Hangul]])
|10
|하나  둘  셋  넷  다섯  여섯  일곱  여덟  아홉  열
|{{sort|1443|15th Century (1443)}}
|-
|[[Aztec writing#Numerals|Aztec numerals]]
|20
|
|{{sort|1500|16th Century}}
|-
|[[Sinhala numerals]]
|10
|෦ ෧ ෨ ෩ ෪ ෫ ෬ ෭ ෮ ෯ 𑇡 𑇢 𑇣
𑇤 𑇥 𑇦 𑇧 𑇨 𑇩 𑇪 𑇫 𑇬 𑇭 𑇮 𑇯  𑇰 𑇱 𑇲 𑇳 𑇴
|{{sort|1699|<18th Century}}
|-
|[[Pentimal system|Pentimal runes]]
|10
|[[Berkas:Pentimal_Runes_1_through_10.svg|150x150px]]
|{{sort|1800|19th Century}}
|-
|[[Cherokee syllabary#Numerals|Cherokee numerals]]
|10
|
|{{sort|1820|19th Century (1820s)}}
|-
|[[Kaktovik Inupiaq numerals]]
|5+20
|[[Berkas:Kaktovik_Inupiaq_Numerals.svg|394x394px]]
|{{sort|1900|20th Century (1994)}}
|}

=== Nol ===
----Angka nol ({{Lang-la|zephirum}}, yang berarti [[kosong]] atau hampa<ref>{{Cite book|last=Wahyudin|first=Sudrajat|date=2003|url=https://www.google.co.id/books/edition/Ensiklopedi_sains_dan_kehidupan/76oANQAACAAJ?hl=id|title=Ensiklopedia Matematika dan Peradaban Manusia|publisher=Tarity Samudra Berlian|isbn=979-8855-06-X|pages=40|url-status=live}}</ref>) muncul di Mesopotamia sekitar 3 SM.

The first recorded zero appeared in Mesopotamia around 3 B.C.
----[[Peradaban Maya]] menemukannya secara independen sekitar 4 SM dan dirancang di [[India]] pada pertengahan abad ke-5 M.

The Mayans invented it independently circa 4 A.D. It was later devised in India in the mid-fifth century, spread to Cambodia near the end of the seventh century, and into China and the Islamic countries at the end of the eighth. Zero reached western Europe in the 12th century.<ref>{{Cite web|title=What is the origin of zero? How did we indicate nothingness before zero?|url=https://www.scientificamerican.com/article/what-is-the-origin-of-zer/|website=Scientific American|language=en|access-date=2021-11-14}}</ref>
----Angka nol tersebut amiug sekitar akhir abad ke-7 M. ??
----There was an Indian Hindu mathematician who is said to have first endorsed the number “zero”. Prior to its “official” creation, no other countries used the number until it reached Indian subcontinent.
----<!-- disalin dari bagian "Nol" @ [[en:Number]] -->

The first known documented use of [[zero]] dates to AD 628, and appeared in the ''[[Brāhmasphuṭasiddhānta]]'', the main work of the [[Indian mathematician]] [[Brahmagupta]]. He treated&nbsp;0 as a number and discussed operations involving it, including [[Division by zero|division]]. By this time (the 7th&nbsp;century) the concept had clearly reached Cambodia as [[Khmer numerals]], and documentation shows the idea later spreading to China and the [[Islamic world]].
[[Berkas:Khmer_Numerals_-_605_from_the_Sambor_inscriptions.jpg|jmpl|The number 605 in [[Khmer numerals]], from an inscription from 683 AD. Early use of zero as a decimal figure.]]
Brahmagupta's ''Brāhmasphuṭasiddhānta'' is the first book that mentions zero as a number, hence Brahmagupta is usually considered the first to formulate the concept of zero. He gave rules of using zero with negative and positive numbers, such as "zero plus a positive number is a positive number, and a negative number plus zero is the negative number." The ''Brāhmasphuṭasiddhānta'' is the earliest known text to treat zero as a number in its own right, rather than as simply a placeholder digit in representing another number as was done by the Babylonians or as a symbol for a lack of quantity as was done by Ptolemy and the Romans.

The use of 0 as a number should be distinguished from its use as a placeholder numeral in [[Place-value system|place-value systems]]. Many ancient texts used&nbsp;0. Babylonian and Egyptian texts used it. Egyptians used the word ''nfr'' to denote zero&nbsp;balance in [[Double-entry bookkeeping system|double entry accounting]]. Indian texts used a [[Sanskrit]] word {{lang|sa-Latn|Shunye}} or {{lang|sa|shunya}} to refer to the concept of ''void''. In mathematics texts this word often refers to the number zero.<ref>{{cite web|date=1999-04-26|title=Historia Matematica Mailing List Archive: Re: [HM&#93; The Zero Story: a question|url=http://sunsite.utk.edu/math_archives/.http/hypermail/historia/apr99/0197.html|publisher=Sunsite.utk.edu|archive-url=https://web.archive.org/web/20120112073735/http://sunsite.utk.edu/math_archives/.http/hypermail/historia/apr99/0197.html|archive-date=2012-01-12|access-date=2012-01-30|url-status=dead}}</ref> In a similar vein, [[Pāṇini]] (5th century BC) used the null (zero) operator in the ''[[Ashtadhyayi]]'', an early example of an [[Formal grammar|algebraic grammar]] for the Sanskrit language (also see [[Pingala]]).

There are other uses of zero before Brahmagupta, though the documentation is not as complete as it is in the ''Brāhmasphuṭasiddhānta''.

Records show that the [[Ancient Greece|Ancient Greeks]] seemed unsure about the status of&nbsp;0 as a number: they asked themselves "how can 'nothing' be something?" leading to interesting [[philosophical]] and, by the Medieval period, religious arguments about the nature and existence of&nbsp;0 and the [[vacuum]]. The [[Zeno's paradoxes|paradoxes]] of [[Zeno of Elea]] depend in part on the uncertain interpretation of&nbsp;0. (The ancient Greeks even questioned whether&nbsp;{{num|1}} was a number.)

The late [[Olmec]] people of south-central Mexico began to use a symbol for zero, a shell [[glyph]], in the New World, possibly by the {{nowrap|4th century BC}} but certainly by 40&nbsp;BC, which became an integral part of [[Maya numerals]] and the [[Maya calendar]]. Maya arithmetic used base&nbsp;4 and base&nbsp;5 written as base&nbsp;20. [[George I. Sánchez]] in 1961 reported a base&nbsp;4, base&nbsp;5 "finger" abacus.<ref>{{Cite book|last=Sánchez|first=George I.|year=1961|title=Arithmetic in Maya|place=Austin, Texas|publisher=self published|author-link=George I. Sánchez}}</ref>{{Better source needed|reason=The only source is a self-published book, albeit one by a respected educator. According to the (favorable) review by David H. Kelley in 'American Anthropologist', Sánchez was neither a Mayanist nor a mathematician. The review does not mention the abacus.|date=September 2020}}

By 130 AD, [[Ptolemy]], influenced by [[Hipparchus]] and the Babylonians, was using a symbol for&nbsp;0 (a small circle with a long overbar) within a [[sexagesimal]] numeral system otherwise using alphabetic [[Greek numerals]]. Because it was used alone, not as just a placeholder, this [[Greek numerals#Hellenistic zero|Hellenistic zero]] was the first ''documented'' use of a true zero in the Old World. In later [[Byzantine Empire|Byzantine]] manuscripts of his ''Syntaxis Mathematica'' (''Almagest''), the Hellenistic zero had morphed into the [[Greek alphabet|Greek letter]] [[Omicron]] (otherwise meaning&nbsp;70).

Another true zero was used in tables alongside [[Roman numerals#Zero|Roman numerals]] by 525 (first known use by [[Dionysius Exiguus]]), but as a word, {{lang|la|nulla}} meaning ''nothing'', not as a symbol. When division produced&nbsp;0 as a remainder, {{lang|la|nihil}}, also meaning ''nothing'', was used. These medieval zeros were used by all future medieval [[Computus|computists]] (calculators of [[Easter]]). An isolated use of their initial, N, was used in a table of Roman numerals by [[Bede]] or a colleague about 725, a true zero symbol.

=== Bilangan negatif ===
Bilangan negatif, pada pertama kali diciptakan oleh Tiongkok dan menggunakan batang merah untuk mewakilinya. Sekitar 630 SM, matematikawan India Brahmagupta menggunakan bilangan positif dan bilangan negatif, masing-masing merepresentasikan aset dan utang.<ref>{{Cite web|date=2020-01-31|title=7: The Integers|url=https://math.libretexts.org/Courses/Barton_Community_College/Book%3A_Technical_Mathematics_(Turner)/07%3A_The_Integers|website=Mathematics LibreTexts|language=en|access-date=2021-11-14}}</ref> Pada abad keempat, Diophantus menemukan bilangan negatif yang mengacu pada notasi persamaan <math>4x + 20 = -4</math> sebagai penyelesaian yang tidak masuk akal, bila solusinya adalah <math>x = -4</math>.<ref>{{Cite web|title=History of Negative Numbers|url=https://www.basic-mathematics.com/history-of-negative-numbers.html|website=Basic-mathematics.com|access-date=2021-11-14}}</ref>

Before any number system was created people used rocks, sticks or counted on their fingers, matematikawan Italia bernama Girolamo Cardano menjelaskan bilangan negatif hanyalah "fiktif" and accepted the possibility of them being useful.

Chinese:

The Chinese are credited with being the first known culture to recognize and use negative numbers. They used red rods to represent negative numbers.

Europe started using negative numbers in 1545.

Before any number system was created people used rocks, sticks or counted on their fingers. There was an Italian mathematician by the name of Girolamo Cardano, who described negative numbers as ‘fictitious’ and accepted the possibility of them being useful.

Negative numbers were finally accepted into the number system in the 19th century. Negative numbers were found to be useful solving tricky equations like cubic and quartic equations.

Although the first set of rules for dealing with negative numbers was stated in the 7th century by the Indian mathematician Brahmagupta, it is surprising that in 1758 the British mathematician Francis Maseres was claiming that negative numbers

"... darken the very whole doctrines of the equations and make dark of the things which are in their nature excessively obvious and simple" .<ref>{{Cite web|title=The History of Negative Numbers|url=https://nrich.maths.org/5961|website=nrich.maths.org|access-date=2021-11-14}}</ref>
----<!-- This section is linked from [[History of negative numbers]] (R to section) -->

{{anchor|First_usage_of_negative_numbers}}<!-- linked from a 2007 article on the Web --> For a long time, negative solutions to problems were considered "false". In [[Hellenistic Egypt]], the [[Greek mathematics|Greek]] mathematician [[Diophantus]] in the 3rd century AD referred to an equation that was equivalent to 4''x'' + 20 = 4 (which has a negative solution) in ''[[Arithmetica]]'', saying that the equation was absurd.<ref name="Needham volume 3 p90">{{cite book|author1=Needham, Joseph|author2=Wang, Ling|year=1995|url=https://books.google.com/books?id=jfQ9E0u4pLAC|title=Science and Civilisation in China: Volume 3; Mathematics and the Sciences of the Heavens and the Earth|location=Cambridge|publisher=Cambridge University Press|isbn=0-521-05801-5|edition=reprint|page=90|orig-year=1959}}</ref> For this reason Greek geometers were able to solve geometrically all forms of the quadratic equation which give positive roots; while they could take no account of others.<ref>{{cite book|last1=Heath|first1=Thomas L.|date=1897|url=https://archive.org/details/worksofarchimede029517mbp/page/n73/mode/2up|title=The works of Archimedes|publisher=Cambridge University Press|pages=cxxiii}}</ref>

Negative numbers appear for the first time in history in the ''[[Nine Chapters on the Mathematical Art]]'' (''Jiu zhang suan-shu''), which in its present form dates from the period of the [[Han Dynasty]] (202 BC – AD 220), but may well contain much older material.<ref name="struik33" /> The mathematician [[Liu Hui]] (c. 3rd century) established rules for the addition and subtraction of negative numbers. The historian Jean-Claude Martzloff theorized that the importance of duality in Chinese natural philosophy made it easier for the Chinese to accept the idea of negative numbers.<ref name="Hodgkin">{{cite book|author=Luke Hodgkin|year=2005|url=https://archive.org/details/historyofmathema0000hodg|title=A History of Mathematics: From Mesopotamia to Modernity|publisher=Oxford University Press|isbn=978-0-19-152383-0|page=[https://archive.org/details/historyofmathema0000hodg/page/88 88]|quote=Liu is explicit on this; at the point where the ''Nine Chapters'' give a detailed and helpful 'Sign Rule'|url-access=registration}}</ref> The Chinese were able to solve simultaneous equations involving negative numbers. The ''Nine Chapters'' used red [[counting rods]] to denote positive [[Coefficient|coefficients]] and black rods for negative.<ref name="Hodgkin" /><ref name="Needham volume 3 pp90-91">{{cite book|author1=Needham, Joseph|author2=Wang, Ling|year=1995|url=https://books.google.com/books?id=jfQ9E0u4pLAC|title=Science and Civilisation in China: Volume 3; Mathematics and the Sciences of the Heavens and the Earth|location=Cambridge|publisher=Cambridge University Press|isbn=0-521-05801-5|edition=reprint|pages=90–91|orig-year=1959}}</ref> This system is the exact opposite of contemporary printing of positive and negative numbers in the fields of banking, accounting, and commerce, wherein red numbers denote negative values and black numbers signify positive values. Liu Hui writes:
{{quote|Now there are two opposite kinds of counting rods for gains and losses, let them be called positive and negative. Red counting rods are positive, black counting rods are negative.<ref name="Hodgkin"/>}}
The ancient Indian ''[[Bakhshali Manuscript]]'' carried out calculations with negative numbers, using "+" as a negative sign.<ref>Teresi, Dick. (2002). ''Lost Discoveries: The Ancient Roots of Modern Science–from the Babylonians to the Mayas''. New York: Simon & Schuster. {{isbn|0-684-83718-8}}. Page 65.</ref> The date of the manuscript is uncertain. LV Gurjar dates it no later than the 4th century,<ref>{{cite web|author=Pearce, Ian|date=May 2002|title=The Bakhshali manuscript|url=http://www-history.mcs.st-andrews.ac.uk/HistTopics/Bakhshali_manuscript.html|publisher=The MacTutor History of Mathematics archive|access-date=2007-07-24}}</ref> Hoernle dates it between the third and fourth centuries, Ayyangar and Pingree dates it to the 8th or 9th centuries,<ref name="HayashiEncy">{{citation|author=Takao Hayashi|title=Bakhshālī Manuscript|encyclopedia=Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures|volume=1|year=2008|publisher=Springer|isbn=9781402045592|editor=Helaine Selin|editor-link=Helaine Selin|page=B2|url=https://books.google.com/books?id=kt9DIY1g9HYC&pg=RA1-PA1}}</ref> and George Gheverghese Joseph dates it to about AD 400 and no later than the early 7th century,<ref>Teresi, Dick. (2002). ''Lost Discoveries: The Ancient Roots of Modern Science–from the Babylonians to the Mayas''. New York: Simon & Schuster. {{isbn|0-684-83718-8}}. Page 65–66.</ref>

During the 7th century AD, negative numbers were used in India to represent debts. The [[Indian mathematics|Indian mathematician]] [[Brahmagupta]], in ''[[Brahmasphutasiddhanta|Brahma-Sphuta-Siddhanta]]'' (written c. AD 630), discussed the use of negative numbers to produce the general form [[quadratic formula]] that remains in use today.<ref name="Needham volume 3 p90" /> He also found negative solutions of [[Quadratic equation|quadratic equations]] and gave rules regarding operations involving negative numbers and [[0 (number)|zero]], such as "A debt cut off from nothingness becomes a credit; a credit cut off from nothingness becomes a debt." He called positive numbers "fortunes", zero "a cipher", and negative numbers "debts".<ref>[[Colva Roney-Dougal|Colva M. Roney-Dougal]], Lecturer in Pure Mathematics at the University of St Andrews, stated this on the BBC Radio 4 programme "In Our Time," on 9 March 2006.</ref><ref>''Knowledge Transfer and Perceptions of the Passage of Time'', ICEE-2002 Keynote Address by Colin Adamson-Macedo. "Referring again to Brahmagupta's great work, all the necessary rules for algebra, including the 'rule of signs', were stipulated, but in a form which used the language and imagery of commerce and the market place. Thus 'dhana' (= fortunes) is used to represent positive numbers, whereas 'rina' (= debts) were negative".</ref>

In the 9th century, [[Islamic mathematicians]] were familiar with negative numbers from the works of Indian mathematicians, but the recognition and use of negative numbers during this period remained timid.<ref name="Rashed">{{Cite book|last=Rashed|first=R.|date=1994-06-30|title=The Development of Arabic Mathematics: Between Arithmetic and Algebra|publisher=Springer|isbn=9780792325659|pages=36–37}}</ref> [[Al-Khwarizmi]] in his ''Al-jabr wa'l-muqabala'' (from which we get the word "algebra") did not use negative numbers or negative coefficients.<ref name="Rashed" /> But within fifty years, [[Abu Kamil]] illustrated the rules of signs for expanding the multiplication <math>(a \pm b)(c \pm d)</math>,<ref name="Ismail">{{citation|author=Mat Rofa Bin Ismail|title=Algebra in Islamic Mathematics|encyclopedia=Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures|volume=1|year=2008|publisher=Springer|isbn=9781402045592|editor=Helaine Selin|editor-link=Helaine Selin|page=115|edition=2nd}}</ref> and [[al-Karaji]] wrote in his ''al-Fakhrī'' that "negative quantities must be counted as terms".<ref name="Rashed" /> In the 10th century, [[Abū al-Wafā' al-Būzjānī]] considered debts as negative numbers in ''A Book on What Is Necessary from the Science of Arithmetic for Scribes and Businessmen''.<ref name="Ismail" />

By the 12th century, al-Karaji's successors were to state the general rules of signs and use them to solve [[Polynomial division|polynomial divisions]].<ref name="Rashed" /> As [[al-Samaw'al]] writes:<blockquote>the product of a negative number—''al-nāqiṣ''—by a positive number—''al-zāʾid''—is negative, and by a negative number is positive. If we subtract a negative number from a higher negative number, the remainder is their negative difference. The difference remains positive if we subtract a negative number from a lower negative number. If we subtract a negative number from a positive number, the remainder is their positive sum. If we subtract a positive number from an empty power (''martaba khāliyya''), the remainder is the same negative, and if we subtract a negative number from an empty power, the remainder is the same positive number.<ref name="Rashed" /></blockquote>In the 12th century in India, [[Bhāskara II]] gave negative roots for quadratic equations but rejected them because they were inappropriate in the context of the problem. He stated that a negative value is "in this case not to be taken, for it is inadequate; people do not approve of negative roots."

European mathematicians, for the most part, resisted the concept of negative numbers until the middle of the 19th century (!)<ref>{{cite book|last=Martinez|first=Alberto|date=2014|title=Negative Math|publisher=Princeton University Press|pages=80-109}}</ref> In the 18th century it was common practice to ignore any negative results derived from equations, on the assumption that they were meaningless.<ref>{{cite book|last=Martinez|first=Alberto A.|year=2006|title=Negative Math: How Mathematical Rules Can Be Positively Bent|publisher=[[Princeton University Press]]}} a history of controversies on negative numbers, mainly from the 1600s until the early 1900s.</ref> In A.D. 1759, [[Francis Maseres]], an English mathematician, wrote that negative numbers "darken the very whole doctrines of the equations and make dark of the things which are in their nature excessively obvious and simple". He came to the conclusion that negative numbers were nonsensical.<ref>{{cite book|last=Maseres|first=Francis|year=1758|title=A dissertation on the use of the negative sign in algebra: containing a demonstration of the rules usually given concerning it; and shewing how quadratic and cubic equations may be explained, without the consideration of negative roots. To which is added, as an appendix, Mr. Machin's Quadrature of the Circle|work=Quoting from Maseres' work: If any single quantity is marked either with the sign + or the sign − without affecting some other quantity, the mark will have no meaning or significance, thus if it be said that the square of −5, or the product of −5 into −5, is equal to +25, such an assertion must either signify no more than 5 times 5 is equal to 25 without any regard for the signs, or it must be mere nonsense or unintelligible jargon|author-link=Francis Maseres}}</ref>

[[Leonardo of Pisa#Important publications|Fibonacci]] allowed negative solutions in financial problems where they could be interpreted as debits (chapter 13 of ''[[Liber Abaci]]'', AD 1202) and later as losses (in ''[[Leonardo of Pisa|Flos]]''). In the 15th century, [[Nicolas Chuquet]], a Frenchman, used negative numbers as [[Exponentiation|exponents]]<ref>{{citation|title=Nicolas Chuquet, Renaissance Mathematician: a study with extensive translations of Chuquet's mathematical manuscript completed in 1484|first1=Graham|last1=Flegg|first2=C.|last2=Hay|first3=B.|last3=Moss|publisher=D. Reidel Publishing Co.|year=1985|isbn=9789027718723|page=354|url=https://books.google.com/books?id=_rO6lVwdbjcC&pg=PA354}}.</ref> but referred to them as "absurd numbers".<ref>{{citation|title=Famous Problems and Their Mathematicians|publisher=Greenwood Publishing Group|year=1999|isbn=9781563084461|page=56|url=https://books.google.com/books?id=STKX4qadFTkC&pg=PA56}}.</ref> In his 1544 ''Arithmetica Integra'' [[Michael Stifel]] also dealt with negative numbers, also calling them ''numeri absurdi''. In 1545, [[Gerolamo Cardano]], in his [[Ars Magna (Gerolamo Cardano)|''Ars Magna'']], provided the first satisfactory treatment of negative numbers in Europe.<ref name="Needham volume 3 p90" /> He did not allow negative numbers in his consideration of [[Cubic equation|cubic equations]], so he had to treat, for example, ''x''<sup>3</sup>&nbsp;+&nbsp;''ax''&nbsp;=&nbsp;''b'' separately from ''x''<sup>3</sup>&nbsp;=&nbsp;''ax''&nbsp;+&nbsp;''b'' (with ''a'',''b''&nbsp;&#x3E;&nbsp;0 in both cases). In all, Cardano was driven to the study of thirteen different types of cubic equations, each expressed purely in terms of positive numbers. (Cardano also dealt with [[complex numbers]], but understandably liked them even less.)

== Bilangan rasional, irasional, transenden dan real, ketakhinggaan, kompleks, p-adic (dan sistem bilangan non-standar yang lain) ==
[[:en:Number#History]]

[[en:n-standard_positional_numeral_systems

<nowiki>[[en:List_of_numeral_systems#By_type_of_notation]]</nowiki>

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== Referensi ==