Transcript Abu Ja'far Muhammad ibn Musa Al

Arabic Mathematics, Indian
Mathematics and zero
Al-Khwarizmi
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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.
al-Khwarizmi
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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.
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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.
al-Khwarizmi
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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.
al-Khwarizmi
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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, "aljabr" 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).
al-Khwarizmi
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Al-Khwarizmi then shows how to solve the
six standard types of equations.
He uses both algebraic methods of solution
and geometric methods.
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How can we solve a quadratic quation
geometrically?
For example to solve the equation
x2 + 10 x = 39 he writes
al-Khwarizmi
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... 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.
al-Khwarizmi
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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.
al-Khwarizmi
al-Khwarizmi and Euclid
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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 Euclid’s Elements.
We know that he could have been, perhaps it
is even fair to say "should have been",
familiar with Euclid’s work.
al-Khwarizmi and Euclid
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In al-Rashid's reign, while al-Khwarizmi was still
young, al-Hajjaj had translated Euclid’s 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 Euclid’s
"Elements".
Al-Khwarizmi
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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
Al-Khwarizmi
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(a+bx)(c+dx)
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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
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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...
Al-Khwarizmi
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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 alKhwarizmi 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
al-Khwarizmi
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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.
al-Khwarizmi
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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.
al-Khwarizmi
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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.
al-Khwarizmi
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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.
al-Khwarizmi
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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.
al-Khwarizmi
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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.
al-Khwarizmi
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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 Ptolemy’s work too:It is certain that Ptolemy’s 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.
Al-Khwarizmi
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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 Ptolemy’s 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.
Al-Khwarizmi
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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 Ptolemy’s
data.
al-Khwarizmi
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A number of minor works were written by alKhwarizmi 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.
astrolabe
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An astrolabe (Greek: ἁστρολάβον
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.
Arabic mathematics
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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.
Arabic mathematics
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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.
Arabic mathematics
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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.
Arabic mathematics
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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.
Arabic mathematics
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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".
Arabic mathematics
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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".
Arabic mathematics
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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.
Arabic mathematics
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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
Arabic mathematics
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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 Euclid’s Elements by al-Hajjaj, were made
during al-Rashid's reign.
Arabic mathematics
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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
Arabic mathematics
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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.
Arabic mathematics
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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.
Arabic mathematics
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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
Arabic mathematics
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Al-Khawarizmi’s 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.
Arabic mathematics
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Let us follow the development of algebra for a
moment and look at Al-Khawarizmi’s successors.
About forty years after Al-Khawarizmi’s 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
Arabic mathematics
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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 = xm+n.
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.
Arabic mathematics
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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.
Indian Mathematics
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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.
Indian Mathematics
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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:-
Laplace: Indian mathematics
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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
Indian mathematics
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We shall look briefly at the Indian development of the placevalue 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.
Indian mathematics
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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.
Indian numerals
Indian numerals
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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.
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So there is much debate
Brahmi numerals
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... 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.
Brahmi’s numbers
Brahmi numerals
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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.
Brahmi numerals
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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.
Gupta numerals
Nagari numerals
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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.
Nagari numerals
Indian mathematics
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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.
Indian mathematics
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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 placevalue system was in use in India by the end of the
6th century.
Origins of zero
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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.
Origins of zero
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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.
Origins of zero
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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.
Origins of zero
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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 śūnya to refer to zero or void.
History of zero
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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
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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.
History of zero
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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.
The Olmec
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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 Mesoamerica’s
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.
History of zero
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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.
History of zero
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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.
History of zero
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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
History of zero
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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.
History of zero
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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).
History of zero
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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.
History of zero
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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.
History of zero
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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.
History of zero
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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.
History of zero
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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.