0 (số)

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-1 0 1

0 (zero; BrE: /ˈzɪərəʊ/ or AmE: /ˈziːroʊ/) is both a number[1] and the numerical digit used to represent that number in numerals. The number 0 fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems. Names for the number 0 in English include zero, nought or (US) naught ( /ˈnɔːt/), nil, or—in contexts where at least one adjacent digit distinguishes it from the letter "O"—oh or o ( /ˈ/). Informal or slang terms for zero include zilch and zip.[2] Ought and aught ( /ˈɔːt/),[3] as well as cipher,[4] have also been used historically.[5]

Số 0[sửa | sửa mã nguồn]

0số nguyên đứng liền trước số dương 1 và liền sau số -1. Trong hầu hết (không phải tất cả) các hệ thống số, số 0 được xác định trước khái niệm 'số nguyên âm' được chấp nhận.

Số 0 là một số nguyên xác định một số lượng hoặc một lượng hay kích thước có giá trị là rỗng. Nghĩa là nếu số anh em của một người bằng 0 có nghĩa là người đó không có anh em nào, hay nếu vật gì đó có trọng lượng bằng 0 thì nó không có trọng lượng, hoặc là nếu một vật có kích thước bằng 0 thì nó không có kích thước.

Tuy các nhà toán học và phần lớn mọi người đều chấp nhận 0 là một số, nhưng một số người khác có thể cho rằng 0 không phải là một số vì họ cho rằng người ta không thể có 0 thứ gì đó.

Một điều thú vị là số 0 là trục đối xứng của tập hợp số thực

Hầu hết các nhà sử học bỏ năm 0 (năm Công nguyên) ra khỏi lịch Gregory và lịch Julia, nhưng các nhà thiên văn học vẫn giữ nó trong các lịch đó.

Do tập hợp số nguyêntập hợp con của tập hợp số hữu tỷ, số thựcsố phức, số 0 cũng là một số hữu tỷ, thực và phức.

Chữ số 0[sửa | sửa mã nguồn]

Chữ số 0 được dùng để ký hiệu một vị trí trống trong hệ số vị trí-giá trị của chúng ta. Chẳng hạn, trong số 2106, chữ số 0 được dùng với mục đích để hai chữ số 2 và 1 nằm đúng vị trí. Rõ ràng, số 216 có giá trị hoàn toàn khác. Trong các hệ thống số cổ, chẳng hạn hệ thống số Babilon và hệ thống số Maya, một ký hiệu khác hoặc một chỗ trống được dùng với vai trò của chữ số 0.

Một số đặc tính của số 0[sửa | sửa mã nguồn]

0 is the integer immediately preceding 1. Zero is an even number,[6] because it is divisible by 2 with no remainder. 0 is neither positive nor negative. By most definitions[7] 0 is a natural number, and then the only natural number not to be positive. Zero is a number which quantifies a count or an amount of null size. In most cultures, 0 was identified before the idea of negative things, or quantities less than zero, was accepted. The value, or number, zero is not the same as the digit zero, used in numeral systems using positional notation. Successive positions of digits have higher weights, so inside a numeral the digit zero is used to skip a position and give appropriate weights to the preceding and following digits. A zero digit is not always necessary in a positional number system, for example, in the number 02. In some instances, a leading zero may be used to distinguish a number. === Elementary algebra === The number 0 is the smallest non-negative integer. The natural number following 0 is 1 and no natural number precedes 0. The number 0 may or may not be considered a natural number, but it is a whole number and hence a rational number and a real number (as well as an algebraic number and a complex number). The number 0 is neither positive nor negative and is usually displayed as the central number in a number line. It is neither a prime number nor a composite number. It cannot be prime because it has an infinite number of factors and cannot be composite because it cannot be expressed by multiplying prime numbers (0 must always be one of the factors).[8] Zero is, however, even. The following are some basic (elementary) rules for dealing with the number 0. These rules apply for any real or complex number x, unless otherwise stated.
    • Addition: x + 0 = 0 + x = x. That is, 0 is an identity element (or neutral element) with respect to addition.
    • Subtraction: x − 0 = x and 0 − x = −x.
    • Multiplication: x · 0 = 0 · x = 0.
    • Division: 0x = 0, for nonzero x. But x0 is undefined, because 0 has no multiplicative inverse (no real number multiplied by 0 produces 1), a consequence of the previous rule.
    • Exponentiation: x0 = x/x = 1, except that the case x = 0 may be left undefined in some contexts. For all positive real x, 0x = 0. The expression 00, which may be obtained in an attempt to determine the limit of an expression of the form f(x)g(x) as a result of applying the lim operator independently to both operands of the fraction, is a so-called "indeterminate form". That does not simply mean that the limit sought is necessarily undefined; rather, it means that the limit of f(x)g(x), if it exists, must be found by another method, such as l'Hôpital's rule. The sum of 0 numbers is 0, and the product of 0 numbers is 1. The factorial 0! evaluates to 1. === Other branches of mathematics ===
    • In set theory, 0 is the cardinality of the empty set: if one does not have any apples, then one has 0 apples. In fact, in certain axiomatic developments of mathematics from set theory, 0 is defined to be the empty set. When this is done, the empty set is the Von Neumann cardinal assignment for a set with no elements, which is the empty set. The cardinality function, applied to the empty set, returns the empty set as a value, thereby assigning it 0 elements.
    • Also in set theory, 0 is the lowest ordinal number, corresponding to the empty set viewed as a well-ordered set.
    • In propositional logic, 0 may be used to denote the truth value false.
    • In abstract algebra, 0 is commonly used to denote a zero element, which is a neutral element for addition (if defined on the structure under consideration) and an absorbing element for multiplication (if defined).
    • In lattice theory, 0 may denote the bottom element of a bounded lattice.
    • In category theory, 0 is sometimes used to denote an initial object of a category.
    • In recursion theory, 0 can be used to denote the Turing degree of the partial computable functions. === Related mathematical terms ===
    • A zero of a function f is a point x in the domain of the function such that f(x) = 0. When there are finitely many zeros these are called the roots of the function. This is related to zeros of a holomorphic function.
    • The zero function (or zero map) on a domain D is the constant function with 0 as its only possible output value, i.e., the function f defined by f(x) = 0 for all x in D. The zero function is the only function that is both even and odd. A particular zero function is a zero morphism in category theory; e.g., a zero map is the identity in the additive group of functions. The determinant on non-invertible square matrices is a zero map.
    • Several branches of mathematics have zero elements, which generalise either the property 0 + x = x, or the property 0 × x = 0, or both.

Lịch sử của số 0[sửa | sửa mã nguồn]

 [sửa | sửa mã nguồn]

Ancient Near East[sửa | sửa mã nguồn]



heart with trachea

beautiful, pleasant, good


Ancient Egyptian numerals were base 10. They used hieroglyphs for the digits and were not positional. By 1740 BC, the Egyptians had a symbol for zero in accounting texts. The symbol nfr, meaning beautiful, was also used to indicate the base level in drawings of tombs and pyramids and distances were measured relative to the base line as being above or below this line.[9] By the middle of the 2nd millennium BC, the Babylonian mathematics had a sophisticated sexagesimal positional numeral system. The lack of a positional value (or zero) was indicated by a space between sexagesimal numerals. By 300 BC, a punctuation symbol (two slanted wedges) was co-opted as a placeholder in the same Babylonian system. In a tablet unearthed at Kish (dating from about 700 BC), the scribe Bêl-bân-aplu wrote his zeros with three hooks, rather than two slanted wedges.[10]

The Babylonian placeholder was not a true zero because it was not used alone. Nor was it used at the end of a number. Thus numbers like 2 and 120 (2×60), 3 and 180 (3×60), 4 and 240 (4×60), looked the same because the larger numbers lacked a final sexagesimal placeholder. Only context could differentiate them.

Classical antiquity[sửa | sửa mã nguồn]

The ancient Greeks did not have a name for zero and did not use a placeholder.[11] They seemed unsure about the status of zero as a number. They asked themselves, "How can nothing be something?", leading to philosophical and, by the medieval period, religious arguments about the nature and existence of zero and the vacuum. The paradoxes of Zeno of Elea depend in large part on the uncertain interpretation of zero.

Example of the early Greek symbol for zero (lower right corner) from a 2nd-century papyrus

By 130 AD, Ptolemy, influenced by Hipparchus and the Babylonians, was using a symbol for zero (a small circle with a long overbar) in his work on mathematical astronomy called the Syntaxis Mathematica, also known as the Almagest. The way in which it is used can be seen in his table of chords in that book. Ptolemy's zero was used within a sexagesimal numeral system otherwise using alphabetic Greek numerals. Because it was used alone, not just as a placeholder, this Hellenistic zero was perhaps the first documented use of a number zero in the Old World.[12] However, the positions were usually limited to the fractional part of a number (called minutes, seconds, thirds, fourths, etc.)—they were not used for the integral part of a number. In later Byzantine manuscripts of Ptolemy's Almagest, the Hellenistic zero had morphed into the Greek letter omicron (otherwise meaning 70). Another zero was used in tables alongside Roman numerals by 525 (first known use by Dionysius Exiguus), but as a word, nulla meaning "nothing", not as a symbol.[13] When division produced zero as a remainder, nihil, also meaning "nothing", was used. These medieval zeros were used by all future medieval calculators of Easter. The initial "N" was used as a zero symbol in a table of Roman numerals by Bede or his colleagues around 725.

India and Southeast Asia[sửa | sửa mã nguồn]

The concept of zero as a digit in the decimal place value notation was developed in India, presumably as early as during the Gupta period (c. 5th century), with the oldest unambiguous evidence dating to the 7th century.[14] The Indian scholar Pingala (c. 200 BC) used binary numbers in the form of short and long syllables (the latter equal in length to two short syllables), a notation similar to Morse code.[15] Pingala used the Sanskrit word śūnya explicitly to refer to zero.[16]

The earliest text to use a decimal place-value system, including a zero, the Lokavibhāga, a Jain text surviving in a medieval Sanskrit translation of the Prakrit original, which is internally dated to AD 458 (Saka era 380). In this text, śūnya ("void, empty") is also used to refer to zero.[17] The origin of the modern decimal-based place value notation can be traced to the Aryabhatiya (c. 500), which states sthānāt sthānaṁ daśaguṇaṁ syāt "from place to place each is ten times the preceding."[18][18][19][20]

The rules governing the use of zero appeared for the first time in the Brahmasputha Siddhanta (7th century). This work considers not only zero, but negative numbers, and the algebraic rules for the elementary operations of arithmetic with such numbers. In some instances, his rules differ from the modern standard, specifically the definition of the value of zero divided by zero as zero.[21]

Epigraphy[sửa | sửa mã nguồn]

The number 605 in Khmer numerals, from the Sambor inscription (Saka era 605 corresponds to AD 683). The earliest known material use of zero as a decimal figure.

There are numerous copper plate inscriptions, with the same small o in them, some of them possibly dated to the 6th century, but their date or authenticity may be open to doubt.[10] A stone tablet found in the ruins of a temple near Sambor on the Mekong, Kratié Province, Cambodia, includes the inscription of "605" in Khmer numerals (a set of numeral glyphs of the Hindu numerals family). The number is the year of the inscription in the Saka era, corresponding to a date of AD 683.[22]

The first known use of special glyphs for the decimal digits that includes the indubitable appearance of a symbol for the digit zero, a small circle, appears on a stone inscription found at the Chaturbhuja Temple at Gwalior in India, dated 876.[23][24]

China[sửa | sửa mã nguồn]

This is a depiction of zero expressed in Chinese counting rods, based on the example provided by A History of Mathematics. An empty space is used to represent zero.[25]

The Sunzi Suanjing, of unknown date but estimated to be dated from the 1st to 5th centuries, and Japanese records dated from the eighteenth century, describe how counting rods were used for calculations. According to A History of Mathematics, the rods "gave the decimal representation of a number, with an empty space denoting zero."[25] The counting rod system is considered a positional notation system.[26] Zero was not treated as a number at that time, but as a "vacant position", unlike the Indian mathematicians who developed the numerical zero.[27] Ch'in Chiu-shao's 1247 Mathematical Treatise in Nine Sections is the oldest surviving Chinese mathematical text using a round symbol for zero.[28] Chinese authors had been familiar with the idea of negative numbers by the Han Dynasty (2nd century AD), as seen in the The Nine Chapters on the Mathematical Art,[29] much earlier than the fifteenth century when they became well established in Europe.[28]

Middle Ages[sửa | sửa mã nguồn]

Transmission to Islamic culture[sửa | sửa mã nguồn]

The Arabic-language inheritance of science was largely Greek,[30] followed by Hindu influences.[31] In 773, at Al-Mansur's behest, translations were made of many ancient treatises including Greek, Latin, Indian, and others.

In AD 813, astronomical tables were prepared by a Persian mathematician, Muḥammad ibn Mūsā al-Khwārizmī, using Hindu numerals;[31] and about 825, he published a book synthesizing Greek and Hindu knowledge and also contained his own contribution to mathematics including an explanation of the use of zero.[32] This book was later translated into Latin in the 12th century under the title Algoritmi de numero Indorum. This title means "al-Khwarizmi on the Numerals of the Indians". The word "Algoritmi" was the translator's Latinization of Al-Khwarizmi's name, and the word "Algorithm" or "Algorism" started meaning any arithmetic based on decimals.[31] Muhammad ibn Ahmad al-Khwarizmi, in 976, stated that if no number appears in the place of tens in a calculation, a little circle should be used "to keep the rows". This circle was called ṣifr.[33]

Transmission to Europe[sửa | sửa mã nguồn]

The Hindu–Arabic numeral system (base 10) reached Europe in the 11th century, via the Iberian Peninsula through Spanish Muslims, the Moors, together with knowledge of astronomy and instruments like the astrolabe, first imported by Gerbert of Aurillac. For this reason, the numerals came to be known in Europe as "Arabic numerals". The Italian mathematician Fibonacci or Leonardo of Pisa was instrumental in bringing the system into European mathematics in 1202, stating:

After my father's appointment by his homeland as state official in the customs house of Bugia for the Pisan merchants who thronged to it, he took charge; and in view of its future usefulness and convenience, had me in my boyhood come to him and there wanted me to devote myself to and be instructed in the study of calculation for some days. There, following my introduction, as a consequence of marvelous instruction in the art, to the nine digits of the Hindus, the knowledge of the art very much appealed to me before all others, and for it I realized that all its aspects were studied in Egypt, Syria, Greece, Sicily, and Provence, with their varying methods; and at these places thereafter, while on business. I pursued my study in depth and learned the give-and-take of disputation. But all this even, and the algorism, as well as the art of Pythagoras, I considered as almost a mistake in respect to the method of the Hindus (Modus Indorum). Therefore, embracing more stringently that method of the Hindus, and taking stricter pains in its study, while adding certain things from my own understanding and inserting also certain things from the niceties of Euclid's geometric art. I have striven to compose this book in its entirety as understandably as I could, dividing it into fifteen chapters. Almost everything which I have introduced I have displayed with exact proof, in order that those further seeking this knowledge, with its pre-eminent method, might be instructed, and further, in order that the Latin people might not be discovered to be without it, as they have been up to now. If I have perchance omitted anything more or less proper or necessary, I beg indulgence, since there is no one who is blameless and utterly provident in all things. The nine Indian figures are: 9 8 7 6 5 4 3 2 1. With these nine figures, and with the sign 0 ... any number may be written.

Here Leonardo of Pisa uses the phrase "sign 0", indicating it is like a sign to do operations like addition or multiplication. From the 13th century, manuals on calculation (adding, multiplying, extracting roots, etc.) became common in Europe where they were called algorismus after the Persian mathematician al-Khwārizmī. The most popular was written by Johannes de Sacrobosco, about 1235 and was one of the earliest scientific books to be printed in 1488. Until the late 15th century, Hindu–Arabic numerals seem to have predominated among mathematicians, while merchants preferred to use the Roman numerals. In the 16th century, they became commonly used in Europe.

Pre-Columbian Americas[sửa | sửa mã nguồn]

The back of Olmec stela C from Tres Zapotes, the second oldest Long Count date discovered. The numerals translate to September, 32 BC (Julian). The glyphs surrounding the date are thought to be one of the few surviving examples of Epi-Olmec script.

The Mesoamerican Long Count calendar developed in south-central Mexico and Central America required the use of zero as a place-holder within its vigesimal (base-20) positional numeral system. Many different glyphs, including this partial quatrefoilMAYA-g-num-0-inc-v1.svg—were used as a zero symbol for these Long Count dates, the earliest of which (on Stela 2 at Chiapa de Corzo, Chiapas) has a date of 36 BC.[34] Since the eight earliest Long Count dates appear outside the Maya homeland,[35] it is generally believed that the use of zero in the Americas predated the Maya and was possibly the invention of the Olmecs.[36] Many of the earliest Long Count dates were found within the Olmec heartland, although the Olmec civilization ended by the 4th century BC, several centuries before the earliest known Long Count dates.

Although zero became an integral part of Maya numerals, with a different, empty tortoise-like "shell shape" used for many depictions of the "zero" numeral, it is assumed to have not influenced Old World numeral systems. Quipu, a knotted cord device, used in the Inca Empire and its predecessor societies in the Andean region to record accounting and other digital data, is encoded in a base ten positional system. Zero is represented by the absence of a knot in the appropriate position.

Kinh Dịch[sửa | sửa mã nguồn]

Không "0" chỉ đến trạng thái hỗn nguyên của vũ trụ, là trạng thái mọi vật chất đều ở nguyên dạng sơ khai. Không là chỉ đến trạng thái âm, cũng là chỉ đến người phụ nữ sơ khai...

Tham khảo[sửa | sửa mã nguồn]

  1. ^ Matson, John (21 tháng 8 năm 2009). “The Origin of Zero”. Scientific American. Springer Nature. Truy cập ngày 24 tháng 4 năm 2016. 
  2. ^ Soanes, Catherine; Waite, Maurice; Hawker, Sara biên tập (2001). The Oxford Dictionary, Thesaurus and Wordpower Guide (Hardback) (ấn bản 2). New York: Oxford University Press. ISBN 978-0-19-860373-3. 
  3. ^ "aught, Also ought" in Webster's Collegiate Dictionary (1927), Third Edition, Springfield, MA: G. & C. Merriam.
  4. ^ "cipher", in Webster's Collegiate Dictionary (1927), Third Edition, Springfield, MA: G. & C. Merriam.
  5. ^ aught at etymonline.com
  6. ^ Lemma B.2.2, The integer 0 is even and is not odd, in Penner, Robert C. (1999). Discrete Mathematics: Proof Techniques and Mathematical Structures. World Scientific. tr. 34. ISBN 981-02-4088-0. 
  7. ^ Bunt, Lucas Nicolaas Hendrik; Jones, Phillip S.; Bedient, Jack D. (1976). The historical roots of elementary mathematics. Courier Dover Publications. tr. 254–255. ISBN 0-486-13968-9. , Extract of pages 254–255
  8. ^ Reid, Constance (1992). From zero to infinity: what makes numbers interesting (ấn bản 4). Mathematical Association of America. tr. 23. ISBN 978-0-88385-505-8. 
  9. ^ Joseph, George Gheverghese (2011). The Crest of the Peacock: Non-European Roots of Mathematics (Third Edition). Princeton. tr. 86. ISBN 978-0-691-13526-7. 
  10. ^ a ă Kaplan, Robert. (2000). The Nothing That Is: A Natural History of Zero. Oxford: Oxford University Press.
  11. ^ Wallin, Nils-Bertil (19 tháng 11 năm 2002). “The History of Zero”. YaleGlobal online. The Whitney and Betty Macmillan Center for International and Area Studies at Yale. Truy cập ngày 1 tháng 9 năm 2016. 
  12. ^ O'Connor, John J.; Robertson, Edmund F., “A history of Zero”, Dữ liệu Lịch sử Toán học MacTutor, Đại học St. Andrews 
  13. ^ “Zero and Fractions”. Know the Romans. Truy cập ngày 21 tháng 9 năm 2016. 
  14. ^ Bourbaki, Nicolas Elements of the History of Mathematics (1998), p. 46. Britannica Concise Encyclopedia (2007), entry "Algebra"[cần giải thích]
  15. ^ “Math for Poets and Drummers” (pdf). people.sju.edu. 
  16. ^ Kim Plofker (2009), Mathematics in India, Princeton University Press, ISBN 978-0691120676, page 54–56. Quote – "In the Chandah-sutra of Pingala, dating perhaps the third or second century BC, [...] Pingala's use of a zero symbol [śūnya] as a marker seems to be the first known explicit reference to zero." Kim Plofker (2009), Mathematics in India, Princeton University Press, ISBN 978-0691120676, 55–56. "In the Chandah-sutra of Pingala, dating perhaps the third or second century BC, there are five questions concerning the possible meters for any value “n”. [...] The answer is (2)7 = 128, as expected, but instead of seven doublings, the process (explained by the sutra) required only three doublings and two squarings – a handy time saver where “n” is large. Pingala’s use of a zero symbol as a marker seems to be the first known explicit reference to zero.
  17. ^ Ifrah, Georges (2000), p. 416.
  18. ^ a ă Aryabhatiya of Aryabhata, translated by Walter Eugene Clark.
  19. ^ O'Connor, Robertson, J.J., E. F. “Aryabhata the Elder”. School of Mathematics and Statistics University of St Andrews, Scotland. Truy cập ngày 26 tháng 5 năm 2013. 
  20. ^ William L. Hosch biên tập (15 tháng 8 năm 2010). The Britannica Guide to Numbers and Measurement (Math Explained). books.google.com.my (The Rosen Publishing Group). tr. 97–98. ISBN 9781615301089. 
  21. ^ Algebra with Arithmetic of Brahmagupta and Bhaskara, translated to English by Henry Thomas Colebrooke (1817) London
  22. ^ Cœdès, Georges, "A propos de l'origine des chiffres arabes," Bulletin of the School of Oriental Studies, University of London, Vol. 6, No. 2, 1931, pp. 323–328. Diller, Anthony, "New Zeros and Old Khmer," The Mon-Khmer Studies Journal, Vol. 25, 1996, pp. 125–132.
  23. ^ Casselman, Bill. “All for Nought”. ams.org. University of British Columbia), American Mathematical Society. 
  24. ^ Ifrah, Georges (2000), p. 400.
  25. ^ a ă Hodgkin, Luke (2 tháng 6 năm 2005). A History of Mathematics : From Mesopotamia to Modernity: From Mesopotamia to Modernity. Oxford University Press. tr. 85. ISBN 978-0-19-152383-0. 
  26. ^ Crossley, Lun. 1999, p.12 "the ancient Chinese system is a place notation system"
  27. ^ Kang-Shen Shen; John N. Crossley; Anthony W. C. Lun; Hui Liu (1999). The Nine Chapters on the Mathematical Art: Companion and Commentary. Oxford University Press. tr. 35. ISBN 978-0-19-853936-0. zero was regarded as a number in India... whereas the Chinese employed a vacant position 
  28. ^ a ă “Mathematics in the Near and Far East” (pdf). grmath4.phpnet.us. tr. 262. 
  29. ^ Struik, Dirk J. (1987). A Concise History of Mathematics. New York: Dover Publications. pp. 32–33. "In these matrices we find negative numbers, which appear here for the first time in history."
  30. ^ Pannekoek, A. (1961). A History of Astronomy. George Allen & Unwin. tr. 165. 
  31. ^ a ă â Will Durant (1950), The Story of Civilization, Volume 4, The Age of Faith: Constantine to Dante – A.D. 325–1300, Simon & Schuster, ISBN 978-0965000758, p. 241, Quote = "The Arabic inheritance of science was overwhelmingly Greek, but Hindu influences ranked next. In 773, at Mansur's behest, translations were made of the Siddhantas – Indian astronomical treatises dating as far back as 425 BC; these versions may have the vehicle through which the "Arabic" numerals and the zero were brought from India into Islam. In 813, al-Khwarizmi used the Hindu numerals in his astronomical tables."
  32. ^ Brezina, Corona (2006). Al-Khwarizmi: The Inventor Of Algebra. The Rosen Publishing Group. ISBN 978-1-4042-0513-0. 
  33. ^ Will Durant (1950), The Story of Civilization, Volume 4, The Age of Faith, Simon & Schuster, ISBN 978-0965000758, p. 241, Quote = "In 976, Muhammad ibn Ahmad, in his Keys of the Sciences, remarked that if, in a calculation, no number appears in the place of tens, a little circle should be used "to keep the rows". This circle the Mosloems called ṣifr, "empty" whence our cipher."
  34. ^ No long count date actually using the number 0 has been found before the 3rd century AD, but since the long count system would make no sense without some placeholder, and since Mesoamerican glyphs do not typically leave empty spaces, these earlier dates are taken as indirect evidence that the concept of 0 already existed at the time.
  35. ^ Diehl, p. 186
  36. ^ Mortaigne, Véronique (28 tháng 11 năm 2014). “The golden age of Mayan civilisation – exhibition review”. The Guardian. Bản gốc lưu trữ ngày 28 tháng 11 năm 2014. Truy cập ngày 10 tháng 10 năm 2015. 

Liên kết ngoài[sửa | sửa mã nguồn]

Chú giải 1:  Ngày nay Omicron viết là O (Mai Thanh Hiền-Nguyễn Du)