Arabic Mathematics, Indian Mathematics and zero

1
Arabic Mathematics, Indian Mathematics and zero
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Al-Khwarizmi

• Born about 780 in Baghdad (now in Iraq)Died
  about 850 CE
• We know few details of Abu Ja'far Muhammad ibn
  Musa al-Khwarizmi's life. perhaps we should call
  him Al for short
• One unfortunate effect of this lack of knowledge
  seems to be the temptation to make guesses based
  on very little evidence.

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al-Khwarizmi

• Having introduced the natural numbers,
  al-Khwarizmi introduces the main topic of the
  first section of his book, namely the solution of
  equations.
• His equations are linear or quadratic and are
  composed of units, roots and squares.
• Known now as the solution by radicals
• For example, to al-Khwarizmi a unit was a number,
  a root was x, and a square was x2.
• However, although we shall use the now familiar
  algebraic notation in this presentation to help
  us understand the notions, Al-Khwarizmi's
  mathematics is done entirely in words with no
  symbols being used.

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al-Khwarizmi

• He first reduces an equation (linear or
  quadratic) to one of six standard forms
• 1. Squares equal to roots.2. Squares equal to
  numbers.3. Roots equal to numbers.4. Squares
  and roots equal to numbers
• e.g. x2 10 x 39.5. Squares and
  numbers equal to roots
• e.g. x2 21 10 x.6. Roots and numbers
  equal to squares
• e.g. 3 x 4 x2.

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al-Khwarizmi

• The reduction is carried out using the two
  operations of al-jabr and al-muqabala.
• Here "al-jabr" means "completion" and is the
  process of removing negative terms from an
  equation. It is where we get the word algebra
  from Restoration and equivalence
• For example, using one of al-Khwarizmi's own
  examples, "al-jabr" transforms x2 40 x - 4 x2
  into 5 x2 40 x.
• The term "al-muqabala" means "balancing" and is
  the process of reducing positive terms of the
  same power when they occur on both sides of an
  equation.
• For example, two applications of "al-muqabala"
  reduces
• 50 3x x2 29 10 x to 21 x2 7 x (one
  application to deal with the numbers and a second
  to deal with the roots).

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al-Khwarizmi

• Al-Khwarizmi then shows how to solve the six
  standard types of equations.
• He uses both algebraic methods of solution and
  geometric methods.
• How can we solve a quadratic quation
  geometrically?
• For example to solve the equation
• x2 10 x 39 he writes

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al-Khwarizmi

• ... a square and 10 roots are equal to 39 units.
  The question therefore in this type of equation
  is about as follows what is the square which
  combined with ten of its roots will give a sum
  total of 39? The manner of solving this type of
  equation is to take one-half of the roots just
  mentioned. Now the roots in the problem before us
  are 10. Therefore take 5, which multiplied by
  itself gives 25, an amount which you add to 39
  giving 64. Having taken then the square root of
  this which is 8, subtract from it half the roots,
  5 leaving 3. The number three therefore
  represents one root of this square, which itself,
  of course is 9. Nine therefore gives the square.

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al-Khwarizmi

• The geometric proof by completing the square
  follows.
• Al-Khwarizmi starts with a square of side x,
  which therefore represents x2 (see Figure to
  follow).
• To the square we must add 10x and this is done by
  adding four rectangles each of breadth 10/4 and
  length x to the square (see Figure again).
• The figure has area x2 10 x which is equal to
  39. We now complete the square by adding the four
  little squares each of area 5/2 5/2 25/4.
• Hence the outside square in the Figure has area
• 4 25/4 39 25 39 64. The side of the
  square is therefore 8. But the side is of length
  5/2 x 5/2 so x 5 8, giving x 3.

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al-Khwarizmi
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al-Khwarizmi and Euclid

• These geometrical proofs are a matter of
  disagreement between experts.
• The question, which seems not to have an easy
  answer, is whether al-Khwarizmi was familiar with
  Euclids Elements.
• We know that he could have been, perhaps it is
  even fair to say "should have been", familiar
  with Euclids work.

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al-Khwarizmi and Euclid

• In al-Rashid's reign, while al-Khwarizmi was
  still young, al-Hajjaj had translated Euclids
  Elements into Arabic and al-Hajjaj was one of
  al-Khwarizmi's colleagues in the House of Wisdom.
  
• This would support the comments of a mathematical
  historian-
• ... in his introductory section al-Khwarizmi uses
  geometrical figures to explain equations, which
  surely argues for a familiarity with Book II of
  Euclids "Elements".

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Al-Khwarizmi

• Al-Khwarizmi continues his study of algebra in
  Hisab al-jabr w'al-muqabala by examining how the
  laws of arithmetic extend to an arithmetic for
  his algebraic objects.
• For example he shows how to multiply out
  expressions such as

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Al-Khwarizmi

• (abx)(cdx)
• although again please remember that al-Khwarizmi
  uses only words to describe his expressions, and
  no symbols are used.
• Rashed (a historian of mathematics) sees a
  remarkable depth and novelty in these
  calculations by al-Khwarizmi which appear to us,
  when examined from a modern perspective, as
  relatively elementary.

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Al-Khwarizmi's

• Al-Khwarizmi's concept of algebra can now be
  grasped with greater precision it concerns the
  theory of linear and quadratic equations with a
  single unknown, and the elementary arithmetic of
  relative binomials and trinomials. ... The
  solution had to be general and calculable at the
  same time and in a mathematical fashion, that is,
  geometrically founded. ... The restriction of
  degree, as well as that of the number of
  unsophisticated terms, is instantly explained.
  From its true emergence, algebra can be seen as a
  theory of equations solved by means of radicals,
  and of algebraic calculations on related
  expressions...

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Al-Khwarizmi

• Al-Khwarizmi's algebra is regarded as the
  foundation and cornerstone of the sciences.
• In a sense, al-Khwarizmi is more entitled to be
  called "the father of algebra" than Diophantus
  because al-Khwarizmi is the first to teach
  algebra in an elementary form and for its own
  sake, Diophantus himself is primarily concerned
  with the theory of numbers and the integer
  solution to equations

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al-Khwarizmi

• The next part of al-Khwarizmi's Algebra consists
  of applications and worked examples.
• He then goes on to look at rules for finding the
  area of figures such as the circle and also
  finding the volume of solids such as the sphere,
  cone, and pyramid.
• This section on mensuration certainly has more in
  common with Hindu and Hebrew texts than it does
  with any Greek work.

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al-Khwarizmi

• The final part of the book deals with the
  complicated Islamic rules for inheritance but
  require little from the earlier algebra beyond
  solving linear equations.

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al-Khwarizmi

• Al-Khwarizmi also wrote a treatise on
  Hindu-Arabic numerals.
• The Arabic text is lost but a Latin translation,
  Algoritmi de numero Indorum in English
  Al-Khwarizmi on the Hindu Art of Reckoning gave
  rise to the word algorithm deriving from his name
  in the title.
• Unfortunately the Latin translation (which has
  been translated into English) is known to be much
  changed from al-Khwarizmi's original text (of
  which even the title is unknown).
• The work describes the Hindu place-value system
  of numerals based on 1, 2, 3, 4, 5, 6, 7, 8, 9,
  and 0.

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al-Khwarizmi

• The first use of zero as a place holder in
  positional base notation was probably due to
  al-Khwarizmi in this work.
• Methods for arithmetical calculation are given,
  and a method to find square roots is known to
  have been in the Arabic original although it is
  missing from the Latin version.

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al-Khwarizmi

• The Indian text on which al-Khwarizmi based his
  treatise was one which had been given to the
  court in Baghdad around 770 as a gift from an
  Indian political mission.
• There are two versions of al-Khwarizmi's work
  which he wrote in Arabic but both are lost.
• In the tenth century al-Majriti made a critical
  revision of the shorter version and this was
  translated into Latin by Adelard.

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al-Khwarizmi

• There is also a Latin version of the longer
  version and both these Latin works have survived.
• The main topics covered by al-Khwarizmi in the
  Sindhind zij are calendars calculating true
  positions of the sun, moon and planets, tables of
  sines and tangents spherical astronomy
  astrological tables parallax and eclipse
  calculations and visibility of the moon.

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al-Khwarizmi

• Although his astronomical work is based on that
  of the Indians, and most of the values from which
  he constructed his tables came from Hindu
  astronomers, al-Khwarizmi must have been
  influenced by Ptolemys work too-
• It is certain that Ptolemys tables, in their
  revision by Theon of Alexandria, were already
  known to some Islamic astronomers and it is
  highly likely that they influenced, directly or
  through intermediaries, the form in which
  Al-Khwarizmi's tables were cast.

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Al-Khwarizmi

• Al-Khwarizmi wrote a major work on geography
  which give latitudes and longitudes for 2402
  localities as a basis for a world map.
• The book, which is based on Ptolemys Geography,
  lists with latitudes and longitudes, cities,
  mountains, seas, islands, geographical regions,
  and rivers.
• The manuscript does include maps which on the
  whole are more accurate than those of Ptolemy.

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Al-Khwarizmi

• In particular it is clear that where more local
  knowledge was available to al-Khwarizmi such as
  the regions of Islam, Africa and the Far East
  then his work is considerably more accurate than
  that of Ptolemy, but for Europe al-Khwarizmi
  seems to have used Ptolemys data.

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al-Khwarizmi

• A number of minor works were written by
  al-Khwarizmi on topics such as the astrolabe, on
  which he wrote two works, on the sundial, and on
  the Jewish calendar.
• He also wrote a political history containing
  horoscopes of prominent persons.

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astrolabe

• An astrolabe (Greek ?st????ß?? astrolabon
  'star-taker') is a historical astronomical
  instrument used by classical astronomers,
  navigators, and astrologers.
• Its many uses include locating and predicting the
  positions of the Sun, Moon, planets, and stars
  determining local time given local latitude and
  vice-versa and surveying.

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Arabic mathematics

• Recent research paints a new picture of the debt
  that we owe to Arabic/Islamic mathematics.
• Certainly many of the ideas which were previously
  thought to have been brilliant new conceptions
  due to European mathematicians of the sixteenth,
  seventeenth and eighteenth centuries are now
  known to have been developed by Arabic/Islamic
  mathematicians around four centuries earlier.

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Arabic mathematics

• In many respects the mathematics studied today is
  far closer in style to that of the Arabic/Islamic
  contribution than to that of the Greeks.
• There is a widely held view that, after a
  brilliant period for mathematics when the Greeks
  laid the foundations for modern mathematics,
  there was a period of stagnation before the
  Europeans took over where the Greeks left off at
  the beginning of the sixteenth century.

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Arabic mathematics

• The common perception of the period of 1000 years
  or so between the ancient Greeks and the European
  Renaissance is that little happened in the world
  of mathematics except that some Arabic
  translations of Greek texts were made which
  preserved the Greek learning so that it was
  available to the Europeans at the beginning of
  the sixteenth century.

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Arabic mathematics

• That such views should be generally held is of no
  surprise.
• Many leading historians of mathematics have
  contributed to the perception by either omitting
  any mention of Arabic/Islamic mathematics in the
  historical development of the subject or with
  statements such as that made by Duhem (historian)
  
• ... Arabic science only reproduced the teachings
  received from Greek science.

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Arabic mathematics

• Before proceeding it is worth trying to define
  the period that we are covering and give an
  overall description to cover the mathematicians
  who contributed.
• The period covered is easy to describe it
  stretches from the end of the eighth century to
  about the middle of the fifteenth century.
• Giving a description to cover the mathematicians
  who contributed, however, is much harder. There
  are works on "Islamic mathematics", detailing
  "Muslim contribution to mathematics".

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Arabic mathematics

• Other authors try the description "Arabic
  mathematics.
• However, certainly not all the mathematicians we
  included were Muslims some were Jews, some
  Christians, some of other faiths.
• Nor were all these mathematicians Arabs, but for
  convenience we will call our topic "Arab
  mathematics".

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Arabic mathematics

• The regions from which the "Arab mathematicians"
  came was centred on Iran/Iraq but varied with
  military conquest during the period.
• At its greatest extent it stretched to the west
  through Turkey and North Africa to include most
  of Spain, and to the east as far as the borders
  of China.

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Arabic mathematics

• The background to the mathematical developments
  which began in Baghdad around 800 AD is not well
  understood.
• Certainly there was an important influence which
  came from the Hindu mathematicians whose earlier
  development of the decimal system and numerals
  was important.
• There began a remarkable period of mathematical
  progress with al-Khwarizmis work and the
  translations of Greek texts

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Arabic mathematics

• This period begins under the Caliph Harun
  al-Rashid, the fifth Caliph of the Abbasid
  dynasty, whose reign began in 786.
• He encouraged scholarship and the first
  translations of Greek texts into Arabic, such as
  Euclids Elements by al-Hajjaj, were made during
  al-Rashid's reign.

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Arabic mathematics

• The next Caliph, al-Ma'mun, encouraged learning
  even more strongly than his father al-Rashid, and
  he set up the House of Wisdom in Baghdad which
  became the centre for both the work of
  translating and of of research.
• Al-Kindi (born 801) and the three Banu Musa
  brothers worked there

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Arabic mathematics

• One should emphasise that the translations into
  Arabic at this time were made by scientists and
  mathematicians such as those previously named
  above, not by language experts ignorant of
  mathematics, and the need for the translations
  was stimulated by the most advanced research of
  the time.
• It is important to realise that the translating
  was not done for its own sake, but was done as
  part of the current research effort.

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Arabic mathematics

• Most of the important Greek mathematical texts
  were translated and list of these exist
• Algebra was a unifying theory which allowed
  rational numbers, irrational numbers, geometrical
  magnitudes, etc., to all be treated as "algebraic
  objects".
• It gave mathematics a whole new development path
  so much broader in concept to that which had
  existed before, and provided a vehicle for future
  development of the subject.

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Arabic mathematics

• Another important aspect of the introduction of
  algebraic ideas was that it allowed mathematics
  to be applied to itself in a way which had not
  happened before. One commentary states

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Arabic mathematics

• Al-Khawarizmis successors undertook a systematic
  application of arithmetic to algebra, algebra to
  arithmetic, both to trigonometry, algebra to the
  Euclidean theory of numbers, algebra to geometry,
  and geometry to algebra. This was how the
  creation of polynomial algebra, combinatorial
  analysis, numerical analysis, the numerical
  solution of equations, the new elementary theory
  of numbers, and the geometric construction of
  equations arose.

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Arabic mathematics

• Let us follow the development of algebra for a
  moment and look at Al-Khawarizmis successors.
• About forty years after Al-Khawarizmis is the
  work of al-Mahani (born 820), who conceived the
  idea of reducing geometrical problems such as
  duplicating the cube to problems in algebra.
• Abu Kamil (born 850) forms an important link in
  the development of algebra between Al-Khawarizmi
  and al-Karaji

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Arabic mathematics

• Despite not using symbols, but writing powers of
  x in words, he had begun to understand what we
  would write in symbols as xn.xm xmn.
• Let us remark that symbols did not appear in
  Arabic mathematics until much later.
• Ibn al-Banna and al-Qalasadi used symbols in the
  15th century and, although we do not know exactly
  when their use began, we know that symbols were
  used at least a century before this.

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Arabic mathematics

• Omar Khayyam (born 1048) gave a complete
  classification of cubic equations with geometric
  solutions found by means of intersecting conic
  sections.
• Khayyam also wrote that he hoped to give a full
  description of the algebraic solution of cubic
  equations in a later work-
• If the opportunity arises and I can succeed, I
  shall give all these fourteen forms with all
  their branches and cases, and how to distinguish
  whatever is possible or impossible so that a
  paper, containing elements which are greatly
  useful in this art will be prepared.

44
Indian Mathematics

• It is without doubt that mathematics today owes a
  huge debt to the outstanding contributions made
  by Indian mathematicians over many hundreds of
  years.
• What is quite surprising is that there has been a
  reluctance to recognise this and one has to
  conclude that many famous historians of
  mathematics found what they expected to find, or
  perhaps even what they hoped to find, rather than
  to realise what was so clear in front of them.

45
Indian Mathematics

• We shall examine the contributions of Indian
  mathematics now, but before looking at this
  contribution in more detail we should say clearly
  that the "huge debt" is the beautiful number
  system invented by the Indians on which much of
  mathematical development has rested.
• Laplace put this with great clarity-

46
Laplace Indian mathematics

• The ingenious method of expressing every possible
  number using a set of ten symbols (each symbol
  having a place value and an absolute value)
  emerged in India. The idea seems so simple
  nowadays that its significance and profound
  importance is no longer appreciated. Its
  simplicity lies in the way it facilitated
  calculation and placed arithmetic foremost
  amongst useful inventions. the importance of this
  invention is more readily appreciated when one
  considers that it was beyond the two greatest men
  of Antiquity, Archimedes and Apollonius

47
Indian mathematics

• We shall look briefly at the Indian development
  of the place-value decimal system of numbers
  later.
• First, however, we go back to the first evidence
  of mathematics developing in India.
• Histories of Indian mathematics used to begin by
  describing the geometry contained in the
  Sulbasutras but research into the history of
  Indian mathematics has shown that the essentials
  of this geometry were older being contained in
  the altar constructions described in the Vedic
  mythology text the Shatapatha Brahmana and the
  Taittiriya Samhita.

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Indian mathematics

• Also it has been shown that the study of
  mathematical astronomy in India goes back to at
  least the third millennium BC and mathematics and
  geometry must have existed to support this study
  in these ancient times.

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Indian numerals
50
Indian numerals

• There is no problem in understanding the symbols
  for 1, 2, and 3. However the symbols for 4, ... ,
  9 appear to us to have no obvious link to the
  numbers they represent. There have been quite a
  number of theories put forward by historians over
  many years as to the origin of these numerals.
  Ifrah (historian) lists a number of the
  hypotheses which have been put forward.
• 1 The Brahmi numerals came from the Indus valley
  culture of around 2000 BC.
• 2 The Brahmi numerals came from Aramaean
  numerals.
• 3 The Brahmi numerals came from the Karoshthi
  alphabet.
• 4 The Brahmi numerals came from the Brahmi
  alphabet.
• 5 The Brahmi numerals came from an earlier
  alphabetic numeral system, possibly due to
  Panini.
• 6. The Brahmi numerals came from Egypt.
• So there is much debate

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Brahmi numerals

• ... the first nine Brahmi numerals constituted
  the vestiges of an old indigenous numerical
  notation, where the nine numerals were
  represented by the corresponding number of
  vertical lines ... To enable the numerals to be
  written rapidly, in order to save time, these
  groups of lines evolved in much the same manner
  as those of old Egyptian Pharonic numerals.
  Taking into account the kind of material that was
  written on in India over the centuries (tree bark
  or palm leaves) and the limitations of the tools
  used for writing (calamus or brush), the shape of
  the numerals became more and more complicated
  with the numerous ligatures, until the numerals
  no longer bore any resemblance to the original
  prototypes.

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Brahmis numbers
53
Brahmi numerals

• One might have hoped for evidence such as
  discovering numerals somewhere on this
  evolutionary path.
• However, it would appear that we will never find
  convincing proof for the origin of the Brahmi
  numerals.

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Brahmi numerals

• If we examine the route which led from the Brahmi
  numerals to our present symbols (and ignore the
  many other systems which evolved from the Brahmi
  numerals) then we next come to the Gupta symbols.
  
• The Gupta period is that during which the Gupta
  dynasty ruled over the Magadha state in
  North-eastern India, and this was from the early
  4th century AD to the late 6th century AD.
• The Gupta numerals developed from the Brahmi
  numerals and were spread over large areas by the
  Gupta empire as they conquered territory.

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Gupta numerals
56
Nagari numerals

• The Gupta numerals evolved into the Nagari
  numerals, sometimes called the Devanagari
  numerals.
• This form evolved from the Gupta numerals
  beginning around the 7th century AD and continued
  to develop from the 11th century onward.
• The name literally means the "writing of the
  gods" and it was the considered the most
  beautiful of all the forms which evolved.
  Comments include-
• What we the Arabs use for numerals is a
  selection of the best and most regular figures in
  India.

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Nagari numerals
58
Indian mathematics

• The oldest dated Indian document which contains a
  number written in the place-value form used today
  is a legal document dated 346 in the Chhedi
  calendar which translates to a date in our
  calendar of 594 AD.
• This document is a donation charter of Dadda III
  of Sankheda in the Bharukachcha region.
• The only problem with it is that some historians
  claim that the date has been added as a later
  forgery.

59
Indian mathematics

• Although it was not unusual for such charters to
  be modified at a later date so that the property
  to which they referred could be claimed by
  someone who was not the rightful owner, there
  seems no conceivable reason to forge the date on
  this document.
• Therefore, despite the doubts, we can be fairly
  sure that this document provides evidence that a
  place-value system was in use in India by the end
  of the 6th century.

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Origins of zero

• Early history
• By the middle of the 2nd millennium BCE, 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 BCE, 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 BCE), the scribe Bêl-bân-aplu wrote his zeros
  with three hooks, rather than two slanted wedges.

61
Origins of zero

• 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 (260), 3 and 180
  (360), 4 and 240 (460), looked the same because
  the larger numbers lacked a final sexagesimal
  placeholder.
• Only context could differentiate them.

62
Origins of zero

• 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 depend in large part on the
  uncertain interpretation of zero.

63
Origins of zero

• The concept of zero as a number and not merely a
  symbol for separation is attributed to India
  where by the 9th century CE practical
  calculations were carried out using zero, which
  was treated like any other number, even in case
  of division.
• The Indian scholar Pingala (circa 5th -2nd
  century BCE) 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.
• He and his contemporary Indian scholars used the
  Sanskrit word sunya to refer to zero or void.

64
History of zero

• 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
• A glyph is an element of writing
• 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 BCE.

65
History of zero

• Since the eight earliest Long Count dates appear
  outside the Maya homeland it is assumed that the
  use of zero in the Americas predated the Maya and
  was possibly the invention of the Olmecs.

66
The Olmec

• The Olmec were an ancient Pre-Columbian
  civilization living in the tropical lowlands
  lowlands of south-central Mexico, in what are
  roughly the modern-day states of Veracruz and
  Tabasco.
• The Olmec flourished during Mesoamericas
  Formative period, dating roughly from 1400 BCE to
  about 400 BCE. They were the first Mesoamerican
  civilization and laid many of the foundations for
  the civilizations that followed.

67
History of zero

• Many of the earliest Long Count dates were found
  within the Olmec heartland, although the Olmec
  civilization ended by the 4th century BCE,
  several centuries before the earliest known Long
  Count dates.

68
History of zero

• Although zero became an integral part of Maya
  numerals, it did not influence Old World numeral
  systems.
• The use of a blank on a counting board to
  represent 0 dated back in India to 4th century
  BCE.

69
History of zero

• In China counting rods were used for calculation
  since the 4th century BCE.
• Chinese mathematicians understood negative
  numbers and zero, though they had no symbol for
  the latter, until the work of the Song Dynasty
  mathematician Qin Jiushao in 1247 established a
  symbol for zero in China.
• The Nine chapters of the Mathematical Art, which
  was mainly composed in the 1st century CE, 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

70
History of zero

• By 130 CE, 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.

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History of zero

• 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).

72
History of zero

• Another zero was used in tables alongside Roman
  numerals by 525 (first known use by Dionsius
  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).
• An isolated use of the initial, N, was used in a
  table of Roman numerals by Bede or a colleague
  about 725, a zero symbol.

73
History of zero

• In 498 CE, Indian mathematician and astronomer
  Aryabhata stated that "Sthanam sthanam dasa
  gunam" or place to place in ten times in value,
  which may be the origin of the modern
  decimal-based place value notation.

74
History of zero

• 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
  CE.
• This text uses Sanskrit numeral words for the
  digits, with words such as the Sanskrit word for
  void for zero.

75
History of zero

• 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 in India, dated 876 CE.
• There are many documents on copper plates, with
  the same small o in them, dated back as far as
  the sixth century CE, but their authenticity may
  be doubted.

76
History of zero

• The Arabic numerals and the positional number
  system were introduced to the Islamic
  civilisation by Al-Khwarizmi.
• Al-Khwarizmi's book on arithmetic 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 Arabic numeral system was introduced to the
  Western world through Latin translations of his
  Arithmetic.