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0 | |
---|---|

Cardinal | 0, zero, "oh", nought, naught, nil. |

Ordinal | 0th, zeroth, noughth |

Factorization | 0 |

Divisors | all numbers |

Arabic | ٠,0 |

Bengali | ০ |

Devanāgarī | ० |

Chinese | 零, 〇 |

Japanese | 零, 〇 |

Khmer | ០ |

Thai | ๐ |

Binary | 0 |

Octal | 0 |

Duodecimal | 0 |

Hexadecimal | 0 |

The word "**zero**" came via French *zéro* from Venetian *zero*, which (together with *cypher*) came via Italian *zefiro* from Arabic صفر, *ṣafira* = "it was empty", *ṣifr* = "zero", "nothing". This was a translation of the Sanskrit word *shoonya* (śūnya), meaning "empty".^{[4]} ^{[5]} ^{[6]} ^{[7]}

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.^{[8]}

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.

The concept of zero as a number and not merely a symbol for separation is attributed to India, where, by the 9th century AD, practical calculations were carried out using zero, which was treated like any other number, even in case of division.^{[9]} ^{[10]} The Indian scholar Pingala (circa 5th-2nd century BC) used binary numbers in the form of short and long syllables (the latter equal in length to two short syllables), making it similar to Morse code.^{[11]} ^{[12]} He and his contemporary Indian scholars used the Sanskrit word *śūnya* to refer to zero or *void*.The use of a blank on a counting board to represent 0 dated back in India to 4th century BC.^{[13]} In 498 AD, Indian mathematician and astronomer Aryabhata stated that "Sthanam sthanam dasa gunam" or place to place in ten times in value, which is the origin of the modern decimal-based place value notation.^{[14]}

The oldest known text to use a decimal place-value system, including a zero, is the Jain text from India entitled the *Lokavibhâga*, dated 458 AD, where *shunya* ("void" or "empty") was employed for this purpose.^{[15]} 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 AD.^{[16]} ^{[17]} There are many documents on copper plates, with the same small *o* in them, dated back as far as the sixth century AD, but their authenticity may be doubted.^{[8]}

Since the 4th century BC, counting rods were used in China for decimal calculation s including the use of blank spaces. Chinese mathematicians understood negative numbers and zero, some mathematicians indicated the for the latter with *wúrù* (無入 "no entry"), *kōng* (空 "empty") and the frame-like symbol 口/囗, until Gautama Siddha introduced the symbol 0 in the 8th century.^{[18]} ^{[19]}

Prior to that, *The Nine Chapters on the Mathematical Art*, composed in the 1st century AD, had already explicitly stated "[when subtracting] subtract same signed numbers, add differently signed numbers, subtract a positive number from zero to make a negative number, and subtract a negative number from zero to make a positive number."^{[20]}

The Hindu-Arabic numerals and the positional number system were introduced around 500 AD, and in 825 AD, it was introduced by a Persian scientist, al-Khwārizmī, in his book on arithmetic. This book synthesized Greek and Hindu knowledge and also contained his own fundamental contribution to mathematics and science including an explanation of the use of zero. It was only centuries later, in the 12th century, that the Arabic numeral system was introduced to the Western world through Latin translations of his treatise *Arithmetic*.

Records show that the ancient Greeks 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.

By 130 AD, Ptolemy, influenced by Hipparchus and the Babylonians, was using a symbol for zero (a small circle with a long overbar) 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. 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 *Syntaxis Mathematica* (also known as the *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. When division produced zero as a remainder, *nihil*, also meaning "nothing", was used. These medieval zeros were used by all future medieval computists (calculators of Easter). The initial "N" was used as a zero symbol in a table of Roman numerals by Bede or his colleague around 725.

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 quatrefoil——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.^{[21]} Since the eight earliest Long Count dates appear outside the Maya homeland,^{[22]} it is assumed that the use of zero in the Americas predated the Maya and was possibly the invention of the Olmecs. 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.s lacked a final sexagesimal placeholder. Only context could differentiate them.

Although zero became an integral part of Maya numerals, it did not influence 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.

**0** is the integer immediately preceding 1. In most cultures, 0 was identified before the idea of negative things (quantities) that go lower than zero was accepted. Zero is an even number,^{[23]} because it is divisible by 2. 0 is neither positive nor negative. By most definitions^{[24]} 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.

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.

See main article: 0 (year). In the BC calendar era, the year 1 BC is the first year before AD 1; no room is reserved for a year zero. By contrast, in astronomical year numbering, the year 1 BC is numbered 0, the year 2 BC is numbered −1, and so on.

See main article: Names for the number 0 and Symbols for zero.

In 976 AD the Persian encyclopedist Muhammad ibn Ahmad al-Khwarizmi, in his "Keys of the Sciences", remarked that if, in a calculation, no number appears in the place of tens, then a little circle should be used "to keep the rows". This circle the Arabs called صفر *ṣifr*, "empty". That was the earliest mention of the name *ṣifr* that eventually became *zero*.^{[25]}

Italian *zefiro* already meant "west wind" from Latin and Greek *zephyrus*; this may have influenced the spelling when transcribing Arabic *ṣifr*.^{[26]} The Italian mathematician Fibonacci (c.1170–1250), who grew up in North Africa and is credited with introducing the decimal system to Europe, used the term *zephyrum*. This became *zefiro* in Italian, which was contracted to *zero* in Venetian.

As the decimal zero and its new mathematics spread from the Arab world to Europe in the Middle Ages, words derived from *ṣifr* and *zephyrus* came to refer to calculation, as well as to privileged knowledge and secret codes. According to Ifrah, "in thirteenth-century Paris, a 'worthless fellow' was called a '... cifre en algorisme', i.e., an 'arithmetical nothing'."^{[26]} From *ṣifr* also came French *chiffre* = "digit", "figure", "number", *chiffrer* = "to calculate or compute", *chiffré* = "encrypted". Today, the word in Arabic is still *ṣifr*, and cognates of *ṣifr* are common in the languages of Europe and southwest Asia.

The modern numerical digit 0 is usually written as a circle or ellipse. Traditionally, many print typefaces made the capital letter O more rounded than the narrower, elliptical digit 0. Typewriters originally made no distinction in shape between O and 0; some models did not even have a separate key for the digit 0. The distinction came into prominence on modern character displays.^{[27]}

A slashed zero can be used to distinguish the number from the letter. The digit 0 with a dot in the center seems to have originated as an option on IBM 3270 displays and has continued with the some modern computer typefaces such as Andalé Mono. One variation uses a short vertical bar instead of the dot. Some fonts designed for use with computers made one of the capital-O–digit-0 pair more rounded and the other more angular (closer to a rectangle). A further distinction is made in falsification-hindering typeface as used on German car number plates by slitting open the digit 0 on the upper right side. Sometimes the digit 0 is used either exclusively, or not at all, to avoid confusion altogether.

The rules governing the use of zero appeared for the first time in Brahmagupta's book *Brahmasputha Siddhanta (The Opening of the Universe)*,^{[28]} written in 628 AD. Here Brahmagupta 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. Here are the rules of Brahmagupta:^{[28]}

- The sum of zero and a negative number is negative.
- The sum of zero and a positive number is positive.
- The sum of zero and zero is zero.
- The sum of a positive and a negative is their difference; or, if their absolute values are equal, zero.
- A positive or negative number when divided by zero is a fraction with the zero as denominator.
- Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator.
- Zero divided by zero is zero.

In saying zero divided by zero is zero, Brahmagupta differs from the modern position. Mathematicians normally do not assign a value to this, whereas computers and calculators sometimes assign NaN, which means "not a number." Moreover, non-zero positive or negative numbers when divided by zero are either assigned no value, or a value of unsigned infinity, positive infinity, or negative infinity. Once again, these assignments are not numbers, and are associated more with computer science than pure mathematics, where in most contexts no assignment is done.

See also: History of the Hindu-Arabic numeral system.

Positional notation without the use of zero (using an empty space in tabular arrangements, or the word *kha* "emptiness") is known to have been in use in India from the 6th century. The earliest certain use of zero as a *decimal* positional digit dates to the 5th century mention in the text Lokavibhaga. The glyph for the zero digit was written in the shape of a dot, and consequently called *bindu* ("dot"). The dot had been used in Greece during earlier ciphered numeral periods.

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.^{[29]}^{[30]}

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.

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 appears in the middle of 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).^{[31]} Zero is, however, even (see parity of zero).

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: = 0, for nonzero
*x*. But is undefined, because 0 has no multiplicative inverse (no real number multiplied by 0 produces 1), a consequence of the previous rule; see division by zero. - Exponentiation:
*x*^{0}=^{x}/_{x}= 1, except that the case*x*= 0 may be left undefined in some contexts; see Zero to the zero power. For all positive real*x*, 0^{x}= 0.

The expression, which may be obtained in an attempt to determine the limit of an expression of the form 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, 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.

- 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.

- A zero of a function
*f*is a point*x*in the domain of the function such that

- 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

- Several branches of mathematics have zero elements, which generalise either the property

The value zero plays a special role for many physical quantities. For some quantities, the zero level is naturally distinguished from all other levels, whereas for others it is more or less arbitrarily chosen. For example, on the Kelvin temperature scale, zero is the coldest possible temperature (negative temperatures exist but are not actually colder), whereas on the Celsius scale, zero is arbitrarily defined to be at the freezing point of water. Measuring sound intensity in decibels or phons, the zero level is arbitrarily set at a reference value—for example, at a value for the threshold of hearing. In physics, the zero-point energy is the lowest possible energy that a quantum mechanical physical system may possess and is the energy of the ground state of the system.

Zero has been proposed as the atomic number of the theoretical element tetraneutron. It has been shown that a cluster of four neutrons may be stable enough to be considered an atom in its own right. This would create an element with no protons and no charge on its nucleus.

As early as 1926, Professor Andreas von Antropoff coined the term neutronium for a conjectured form of matter made up of neutrons with no protons, which he placed as the chemical element of atomic number zero at the head of his new version of the periodic table. It was subsequently placed as a noble gas in the middle of several spiral representations of the periodic system for classifying the chemical elements.

The most common practice throughout human history has been to start counting at one, and this is the practice in early classic computer science programming languages such as Fortran and COBOL. However, in the late 1950s LISP introduced zero-based numbering for arrays while Algol 58 introduced completely flexible basing for array subscripts (allowing any positive, negative, or zero integer as base for array subscripts), and most subsequent programming languages adopted one or other of these positions. For example, the elements of an array are numbered starting from 0 in C, so that for an array of *n* items the sequence of array indices runs from 0 to . This permits an array element's location to be calculated by adding the index directly to address of the array, whereas 1 based languages precalculate the array's base address to be the position one element before the first.

There can be confusion between 0 and 1 based indexing, for example Java's JDBC indexes parameters from 1 although Java itself uses 0-based indexing.

In databases, it is possible for a field not to have a value. It is then said to have a null value. For numeric fields it is not the value zero. For text fields this is not blank nor the empty string. The presence of null values leads to three-valued logic. No longer is a condition either *true* or *false*, but it can be *undetermined*. Any computation including a null value delivers a null result. Asking for all records with value 0 or value not equal 0 will not yield all records, since the records with value null are excluded.

A null pointer is a pointer in a computer program that does not point to any object or function. In C, the integer constant 0 is converted into the null pointer at compile time when it appears in a pointer context, and so 0 is a standard way to refer to the null pointer in code. However, the internal representation of the null pointer may be any bit pattern (possibly different values for different data types).

In mathematics

-0=0=+0

- In some countries and some company phone networks, dialing 0 on a telephone places a call for operator assistance.
- DVDs that can be played in any region are sometimes referred to as being "region 0"
- Roulette wheels usually feature a "0" space (and sometimes also a "00" space), whose presence is ignored when calculating payoffs (thereby allowing the house to win in the long run).
- In Formula One, if the reigning World Champion no longer competes in Formula One in the year following their victory in the title race, 0 is given to one of the drivers of the team that the reigning champion won the title with. This happened in 1993 and 1994, with Damon Hill driving car 0, due to the reigning World Champion (Nigel Mansell and Alain Prost respectively) not competing in the championship.

- Grammatical number
- Number theory
- Peano axioms
- Zeroth (Zero as an ordinal number)

- Barrow, John D. (2001)
*The Book of Nothing*, Vintage. ISBN 0-09-928845-1. - Diehl, Richard A. (2004)
*The Olmecs: America's First Civilization*, Thames & Hudson, London. - Ifrah, Georges (2000)
*The Universal History of Numbers: From Prehistory to the Invention of the Computer*, Wiley. ISBN 0-471-39340-1. - Kaplan, Robert (2000)
*The Nothing That Is: A Natural History of Zero*, Oxford: Oxford University Press. - Seife, Charles (2000)
*Zero: The Biography of a Dangerous Idea*, Penguin USA (Paper). ISBN 0-14-029647-6. - Bourbaki, Nicolas (1998).
*Elements of the History of Mathematics*. Berlin, Heidelberg, and New York: Springer-Verlag. ISBN 3-540-64767-8. - Isaac Asimov (1978). Article "Nothing Counts" in
*Asimov on Numbers*. Pocket Books.

- A History of Zero
- Zero Saga
- The History of Algebra
- Edsger W. Dijkstra: Why numbering should start at zero, EWD831 (PDF of a handwritten manuscript)
- "My Hero Zero" Educational children's song in Schoolhouse Rock!

- Book: Principles of mathematics. 2. Bertrand. Russel. Forgotten Books. 1942. 1-440-05416-9. 125., Chapter 14, page 125
- Book: Catherine Soanes. Maurice Waite, Sara Hawker. The Oxford Dictionary, Thesaurus and Wordpower Guide. Hardback. 21 December 2007. 2nd. 2001. Oxford University Press. New York, United States. 978-0-19-860373-3.
- http://www.etymonline.com/index.php?search=aught&searchmode=none
*aught*at etymonline.com - Book: Number words and number symbols: a cultural history of numbers. Karl. Menninger. Courier Dover Publications. 1992. 0-486-27096-3. 401.
- Web site: "zero, n.". OED Online. December 2011. Oxford University Press. (accessed March 04, 2012).. http://www.webcitation.org/65yd7ur9u. 2012-03-06. no. 2012-03-04.
- http://www.oed.com/view/Entry/33155 "cipher | cypher, n.". OED Online. December 2011. Oxford University Press. (accessed March 04, 2012).
- http://www.merriam-webster.com/dictionary/zero Merriam Webster online Dictionary
- Kaplan, Robert. (2000).
*The Nothing That Is: A Natural History of Zero*. Oxford: Oxford University Press. - Bourbaki, Nicolas (1998).
*Elements of the History of Mathematics*. Berlin, Heidelberg, and New York: Springer-Verlag. 46. ISBN 3-540-64767-8. *Britannica Concise Encyclopedia*(2007), entry*algebra*- http://home.ica.net/~roymanju/Binary.htm Binary Numbers in Ancient India
- http://www.sju.edu/~rhall/Rhythms/Poets/arcadia.pdf Math for Poets and Drummers
- Robert Temple,
*The Genius of China, A place for zero*; ISBN 1-85375-292-4 *Aryabhatiya of Aryabhata*, translated by Walter Eugene Clark.- Ifrah, Georges (2000), p. 416.
- Bill Casselman (University of British Columbia), American Mathematical Society, "All for Nought"
- Ifrah, Georges (2000), p. 400.
- http://www.nownews.com/2008/11/03/142-2359146.htm 「零」字考與陳雲林會長來台
- http://www.math.sinica.edu.tw/math_media/pdf.php?m_file=ZDI2My8yNjMwNg
## 對中國傳統筆算之探討

- The original text, found in Chapter 8 of
*The Nine Chapters on the Mathematical Art*is 正負術曰: 同名相除，異名相益，正無入負之，負無入正之。其異名相除，同名相益，正無入正之，負無入負之。 The word*wúrù*(無入) used here, for which*zero*is the standard translation by mathematical historians, literally means: "no entry" or "null enters". The full Chinese text can be found at . - 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.
- Diehl, p. 186
- [Lemma (mathematics)|Lemma]
- Book: The historical roots of elementary mathematics. Lucas Nicolaas Hendrik. Bunt. Phillip S.. Jones. Jack D.. Bedient. Courier Dover Publications. 1988. 0-486-2556-3. 254–255., Extract of pages 254-255
- http://www.archive.org/details/ageoffaithahisto012288mbp Will Durant,
*'The Story of Civilization***, Volume 4,**The Age of Faith**, pp. 241.** - Book: Ifrah, Georges. The Universal History of Numbers: From Prehistory to the Invention of the Computer. Wiley. 2000. 0-471-39340-1.
- R. W.. Bemer. Towards standards for handwritten zero and oh: much ado about nothing (and a letter), or a partial dossier on distinguishing between handwritten zero and oh. Communications of the ACM. 10. 8. 1967. 513–518. 10.1145/363534.363563.
- http://books.google.com/books?id=A3cAAAAAMAAJ&printsec=frontcover&dq=brahmagupta
*Algebra with Arithmetic of Brahmagupta and Bhaskara* - Sigler, L.,
*Fibonacci's Liber Abaci*. English translation, Springer, 2003. - Grimm, R.E., "The Autobiography of Leonardo Pisano",
*Fibonacci Quarterly***11**/1 (February 1973), pp. 99–104. - Book: Reid. Constance. From zero to infinity: what makes numbers interesting. 1992. 4th. Mathematical Association of America. 23. 9780883855058.