Some Interesting Facts, Myths and History of Mathematics
Some Interesting Facts, Myths and History of Mathematics
Some Interesting Facts, Myths and History of Mathematics
ABSTRACT : This paper deals with primary concepts and fallacies of mathematics which many a times
students and even teachers ignore. Also this paper comprises of history of mathematical symbols, notations and
methods of calculating time. I have also included some ancient techniques of solving mathematical real time
problems. This paper is a confluence of various traditional mathematical techniques and their implementation
in modern mathematics.
I.
INTRODUCTION
I have heard my father saying that Mathematics is the only genuine subject as it does not change with boundary
of countries. It is lucrative just because of its simplicity. Galileo once said, Mathematics is the language with
which God wrote the Universe. He was precise in calling mathematics a language, because like any dialect,
mathematics has its own rubrics, formulas, and nuances. In precise, the symbols used in mathematics are quite
unique to its field and are profoundly engrained in history. The following will give an ephemeral history of
some of the greatest well-known symbols employed by mathematics. Categorized by discipline within the
subject, each section has its own interesting subculture surrounding it. Arithmetic is the most rudimentary part
of mathematics and covers addition, subtraction, multiplication, and the division of numbers. One category of
numbers are the integers, -n,-3,-2,-1,0,1,2,3,n , where we say that n is in .The capital letter Z is written to
represent integers and comes from the German word, Zahlen, meaning numbers. Two vital operations in
mathematics, addition, +, and subtraction, -, credit the use of their symbols to fourteenth and fifteenth century
mathematicians. Nicole d' Oresme, a Frenchman who lived from 1323-1382, used the + symbol to abbreviate
the Latin et, meaning and, in his AlgorismusProportionum.
The fourteenth century Dutch mathematician Giel Vander Hoecke, used the plus and minus signs in his
Eensonderlingheboeck in dye edelconsteArithmetica and the Brit Robert Recorde used the same symbols in his
1557 publication, The Whetstone of Witte (Washington State Mathematics Council). The division and
multiplication signs have correspondingly fascinating past. The symbol for division,, called an obelus, was
first used in 1659, by the Swiss mathematician Johann Heinrich Rahn in his work entitled TeutscheAlgebr. The
symbol was later presented to London when the English mathematician Thomas Brancker deciphered Rahns
work (Cajori, A History of Mathematics, 140). Descartes, who lived in the primary part of the 1600s, turned the
German Cossits into the square root symbol that we now have, by knocking a bar over it .The symbol
meaning infinity, was first presented by Oughtreds student, John Wallis, in his 1655 book De
SectionibusConicus .It is theorized that Wallis borrowed the symbol from the Romans, which meant 1,000 (A
History of Mathematical Notations, 44). Preceding this, Aristotle (384-322 BC) is noted for saying three things
about infinity: i) the infinite exists in nature and can be identified only in terms of quantity, ii) if infinity exists it
must be defined, and iii) infinity do not exist in realism. From these three statements Aristotle came to the
conclusion that mathematicians had no use for infinity. This idea was later refuted and the German
mathematician, Georg Cantor, who lived from 1845-1918, is quoted as saying; I experience true pleasure in
conceiving infinity as I have, and I throw myself into itAnd when I come back down toward finiteness, I see
with equal clarity and beauty the two concepts [of ordinal numbers(first, second, third etc.) and cardinal
numbers (one, two, three etc.)] once more becoming one and converging in the concept of finite integer. Cantor
not only acknowledged infinity, but used aleph, the first letter of the Hebrew alphabet, as its symbol. Cantor
referred to it as transfinite. Another interesting fact is that Euler, while accepting the concept of infinity did
not use the familiar symbol, but instead he wrote a sideways s.
1.1 Intersection and union
The notations and were used by Giuseppe Peano (1858-1932) for intersection and union in 1888
in Calcologeometrico secondo l'Ausdehnungslehre di H. Grassmann (Cajori vol. 2, page 298); the logical part
of this work with this notations is ed. in: Peano, operescelte, 2, Rom 1958, p. 3-19.Peano also shaped the large
notations for general intersection and union of more than two classes in 1908.
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II.
One mendacious belief which present day mathematicians are nourishing is that Null Set is denoted by Greek
Notation (Phi). Common notations for the empty set include "{}", "", and " ". The latter two notations were
introduced by the Bourbaki group (specifically Andr Weil) in 1939, inspired by the letter in the Norwegian
and Danish alphabets (and not related in any way to the Greek letter which is read as Phi). Phi is the 21st
letter of the Greek alphabet and it does not denote an Empty Set. Null, void, empty and vacuous are
synonymous and interchangeable. and are not same ( is Danish Symbol while is a Greek Symbol).
One thing which is noticeable is that {} is an example of singleton set as this is a set of an element
which is Null. When speaking of the sum of the elements of a finite set, one is inevitably led to the convention
that the sum of the elements of the empty set is zero. The reason for this is that zero is the identity element for
addition. Similary, the product of the elements of the empty set should be considered to be one (see empty
product), since one is the identity element for multiplication. A disarrangement of a set is a permutation of the
set that leaves no element in the same position. The empty set is a disarrangement of itself as no element can be
found that retains its original position.
III. GREATEST COMMON DIVISOR AND LEAST COMMON MULTIPLE OF NEGATIVE NUMBERS
The least common multiple (also called the lowest common multiple or smallest common multiple) of
two integers a and b, usually denoted by LCM(a, b), is the smallest positive integer that is divisible by
both a and b. Since division of integers by zero is undefined, this definition has meaning only if a and b are both
different from zero. However, some authors define LCM(a,0) as 0 for all a, which is the result of taking the
LCM to be the least upper bound in the lattice of divisibility.
We know that LCM(2,3)=6. Let us find LCM(2,-3), LCM(-2,3) and LCM(-2,-3). A nave mathematician will
come with answer -6.But Least common multiple means that multiple must be minimum and if we look on
number line -12<-6 and -18<-12. So, we will keep on moving on negative side of number line and we will never
come with a solution. So question arises what should be appropriate answer.
We may define
.
The least common multiple can be defined generally over commutative rings as follows: Let a and b be
elements of a commutative ring R. A common multiple of a and b is an element m of R such that
both a and b divide m (i.e. there exist elements x and y of R such that ax = m and by = m). A least common
multiple of a and b is a common multiple that is minimal in the sense that for any other common
multiple n of a and b, m divides n. The LCM of more than two integers is also well-defined: it is the smallest
positive integer that is divisible by each of them.
LCM (2, -3) = LCM (-2, 3) = LCM (-2, -3) = LCM (2,3) = 6.
In general, two elements in a commutative ring can have no least common multiple or more than one. However,
any two least common multiples of the same pair of elements are associates. In a unique factorization domain,
any two elements have a least common multiple. In a principal ideal domain, the least common multiple
of a and b can be characterised as a generator of the intersection of the ideals generated by a and b (the
intersection of a collection of ideals is always an ideal).The Greatest Common Factor (HCF) or Greatest
Common Divisor (GCD) of two non-zero integers is the largest positive integer that divides both numbers
without remainder.(The negative number sign may be ignored as divisibility is not affected).The Highest
Common Factor (HCF) or Greatest Common Divisor (GCD) of two non-zero integers is the largest positive
integer that divides both numbers without remainder.(The negative number sign may be ignored as divisibility is
not affected)
HCF(x,y)=HCF(x,-y)=HCF(-x,-y)=HCF(-x,y) x,y N
More generalized definition will be
HCF(a,b)=HCF(|a|,|b| )a,b z.
Computer Code for GCD
#include <stdio.h>
int main()
{
int a, b;
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IV.
REMAINDER
Different programming languages have adopted different conventions: Pascal showed the result of
the mod operation positive, but does not allow d to be negative or zero (so, a = (a div d )*d + a mod d is not
always valid). C99 chooses the remainder with the same sign as the dividend a. (Before C99, the C language
allowed other choices) Perl, Python (only modern versions), and Common Lisp choose the remainder with the
same
sign
as
the
divisor d. Haskell and Scheme offer
two
functions, remainder and modulo
PL/I has mod and rem, while Fortran has mod and modulo; in each case, the former agrees in sign with the
dividend, and the latter with the divisor. When a and d are floating-point numbers, with d non-zero, a can be
divided by d without remainder, with the quotient being another floating-point number. If the quotient is
constrained to being an integer, however, the concept of remainder is still necessary. It can be proved that there
exists a unique integer quotient q and a unique floating-point remainder r such that a = qd + r with 0 r < |d|.
Code snippet of remainder using C is stated below.
int mod(int a, int b)
{int r = a % b;
return r <0 ? r + b : r;}
V.
Versine
versin()=1-cos()
Vercosine vercosin()=1+cos()
Coversine coversin()=1-sin()
Covercosine covercosine()=1+sin()
Haversine haversin()=versin()/2
Havercosine havercosin()=vercosin()/2
Hacoversine hacoversin()=coversin()/2
Hacovercosine
hacovercosin()=covercosin()/2
Exsecant
exsec()=sec()-1
Excosecant excsc()=csc()-1
[]
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ParenthesesParentheses are used for two purposes: (1) to control the order of operations in an expression, and (2) to supply
parameters to a constructor or method.
VII.
Why
Any number raised to power zero is one except zero. We can take an example and demonstrate.
In order to calculate
we may write
, (i.e. It is also an indeterminate form)
VIII.
During the Vedic period (1500500 BCE), driven by geometric construction of the fire altars and astronomy, the
use of a numerical system and of elementary mathematical operations developed in northern India. Hindu
cosmology required the mastery of very huge numbers such as the kalpa (the lifetime of the universe) said to be
4,320,000,000 years and the "orbit of the heaven" said to be 18,712,069,200,000,000 yojanas. Numbers were
expressed using a "named place-value notation", using names for the powers of 10: dasa, shatha, sahasra,
ayuta, niyuta, prayuta, arbuda, nyarbuda, samudra, madhya, anta, parardha etc., the last of these being the
name for a trillion. For example, the number 26432 was expressed as "2 ayuta 6 sahasra 4 shatha 3 dasa 2. In
the Buddhist text Lalitavistara, the Buddha is said to have narrated a scheme of numbers up to 10^53.The form
of numerals in Ashoka's inscriptions in the Brahmi script (middle of the third century BCE) involved separate
signs for the numbers 1 to 9, 10 to 90, 100 and 1000. A multiple of 100 or 1000 was represented by a
modification (or "enciphering" of the sign for the number using the sign for the multiplier number. Such
enciphered numerals directly represented the named place-value numerals used verbally. They continued to be
used in inscriptions till the end of the 9th century.
In his seminal text of 499 CE, Aryabhata devised a novel positional number system, using Sanskrit consonants
for small numbers and vowels for powers of 10. Using the system, numbers up to a billion could be expressed
using short phrases, e.g., khyu-gh representing the number 4,320,000. The system did not catch on because it
produced quite unpronouncable phrases, but it might driven home the principle of positional number system
(called dasa-gunottara, exponents of 10) to later mathematicians. A more elegant katapayadi scheme was
devised in later centuries representing a place-value system including zero.
The place value system, however, developed later. The Brahmi numerals have been found in engravings in
caves and on coins in regions near Pune, Mumbai, and Uttar Pradesh. These numerals (with minor deviations)
were in use over a long time span up to the fourth century.
While the numerals in texts and inscriptions used a named place-value notation, a more efficient notation might
have been employed in calculations, possibly from the 1st century CE. Computations were carried out on clay
tablets covered with a thin layer of sand, giving rise to the term dhuli-karana ("sand-work") for higher
computation. Karl Menninger believes that, in such computations, they must have dispensed with the enciphered
numerals and written down just sequences of digits to represent the numbers. A zero would have been
represented as a "missing place," such as a dot. The single manuscript with worked examples available to us, the
Bakhshali manuscript (thought to be a copy of an original written in fourth to seventh century CE), uses a place
value system with a dot to denote the zero. The dot was called the shunya-sthna, "empty-place." The same
symbol was also used in algebraic expressions for the unknown (as in the canonical x in modern algebra).
However, the date of the Bakhshali manuscript is subject to considerable debate.Textual references to a placevalue system are seen from the 1st century CE onwards. The Buddhist philosopher Vasubandhu in the 1st
century says "when [the same] clay counting-piece is in the place of units, it is denoted as one, when in
hundreds, one hundred." A commentary on Patanjali's Yoga Sutras from the 5th century reads, "Just as a line in
the hundreds place [means] a hundred, in the tens place ten, and one in the ones place, so one and the same
woman is called mother, daughter and sister."
A system called bhta-sankhya ("object numbers" or "concrete numbers") was employed for representing
numerals in Sanskrit verses, by using a concept representing a digit to stand for the digit itself. The Jain text
entitled
the
Lokavibhaga,
dated
458
CE,
mentions
the
objecit
fiednumeral"
panchabhyahkhalushunyebhyahparamdvesaptachambaramekamtrini cha rupamcha" meaning, "five voids, then
two and seven, the sky, one and three and the form", i.e., the number 13107200000. Such objectified numbers
were used extensively from the 6th century onwards, especially after Varahamihira (c. 575 CE). Zero is
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VIII.
Excavations at Harappa, Mohenjo-daro and other sites of the Indus Valley Civilisation have uncovered evidence
of the use of "practical mathematics". The people of the IVC manufactured bricks whose dimensions were in the
proportion 4:2:1, considered favourable for the stability of a brick structure. They used a standardised system of
weights based on the ratios: 1/20, 1/10, 1/5, 1/2, 1, 2, 5, 10, 20, 50, 100, 200, and 500, with the unit weight
equaling approximately 28 grams (and approximately equal to the English ounce or Greek uncia). They massproduced weights in regular geometrical shapes, which included hexahedra, barrels, cones, and cylinders,
thereby demonstrating knowledge of basic geometry.
The inhabitants of Indus civilisation also tried to standardise measurement of length to a high degree of
accuracy. They designed a rulerthe Mohenjo-daro rulerwhose unit of length (approximately 1.32 inches or
3.4 centimetres) was divided into ten equal parts. Bricks manufactured in ancient Mohenjo-daro often had
dimensions that were integral multiples of this unit of length.
SAMHITAS AND BRAHMANAS
The religious texts of the Vedic Period provide evidence for the use of large numbers. By the time of
the Yajurvedasahit- (1200900 BCE), numbers as high as 1012 were being included in the texts. For example,
the mantra (sacrificial formula) at the end of the annahoma ("food-oblation rite") performed during
the avamedha, and uttered just before-, during-, and just after sunrise, invokes powers of ten from a hundred to
a trillion.
Hail to ata ("hundred," 102), hail to sahasra ("thousand," 103), hail to ayuta ("ten thousand," 104), hail
to niyuta ("hundred thousand," 105), hail to prayuta ("million,"106), hail to arbuda ("ten million," 107), hail
to nyarbuda ("hundred million," 108), hail to samudra ("billion," 109, literally "ocean"), hail to madhya ("ten
billion," 1010, literally "middle"), hail to anta ("hundred billion," 1011,lit., "end"), hail to parrdha ("one
trillion," 1012 lit., "beyond parts"), hail to the dawn (uas), hail to the twilight (vyui), hail to the one which is
going to rise (udeyat), hail to the one which is rising (udyat), hail to the one which has just risen (udita), hail
to svarga (the heaven), hail to martya (the world), hail to all.
The solution to partial fraction was known to the Rigvedic People as states in the purushSukta (RV 10.90.4):
With three-fourths Purua went up: one-fourth of him again was here.
The SatapathaBrahmana (ca. 7th century BCE) contains rules for ritual geometric constructions that are similar
to the Sulba Sutras.
ULBASTRAS
The ulbaStras (literally, "Aphorisms of the Chords" in Vedic Sanskrit) (c. 700400 BCE) list rules for the
construction of sacrificial fire altars.Most mathematical problems considered in the ulbaStras spring from "a
single theological requirement," that of constructing fire altars which have different shapes but occupy the same
area. The altars were required to be constructed of five layers of burnt brick, with the further condition that each
layer consist of 200 bricks and that no two adjacent layers have congruent arrangements of bricks.
According to (Hayashi 2005, p. 363), the ulbaStras contain "the earliest extant verbal expression of
the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians."
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109
30
12
=
3.1104
Trillion
Years
(1
year
of
Brahma)
3.1104 1012 50 = 155.52 Trillion Years (50 years of Brahma)
(6 71 4320000 ) + 7 1.728 10^6 = 1852416000 years elapsed in first six Manvataras, and SandhiKalas in the current
Kalpa
27 4320000 = 116640000 years elapsed in first 27 Mahayugas of the current Manvantara
1.728 10^6 + 1.296 10^6 + 864000 = 3888000 years elapsed in current Mahayuga
3102 + 2016 = 5118 years elapsed in current Kaliyuga.
So the total time elapsed since current Brahma is
155520000000000 + 1852416000 + 116640000 + 3888000 + 5115 = 155,521,972,949,117 years (one hundred fifty-five
trillion, five hundred twenty-one billion, nine hundred seventy-two million, nine hundred forty-nine thousand, one hundred
seventeen years) as of 2016 AD
The current Kali Yuga began at midnight 17 February / 18 February in 3102 BCE in the proleptic Julian calendar. As per the
information above about Yuga periods, only 5,118 years are passed out of 432,000 years of current Kali Yuga, and hence
another 426,882 years are left to complete this 28th Kali Yuga of VaivaswathaManvantara.
IX.
There are plethoras of ancient techniques which provide quick and accurate solutions for various types of problems. Some of
the basic methods are listed in the Table.
SUTRAS
MEANING
EKADHIKINA PURVENA
PARAAVARTYA YOJAYET
SHUNYAM SAAMYASAMUCCAYE
(ANURUPYE) SHUNYAMANYAT
SANKALANA-VYAVAKALANABHYAM
PURANAPURANABYHAM
CHALANA-KALANABYHAM
YAAVADUNAM
VYASHTISAMANSTIH
SHESANYANKENA CHARAMENA
SOPAANTYADVAYAMANTYAM
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GUNITASAMUCHYAH
GUNAKASAMUCHYAH
Table 1
X.
Abbreviations deriving from Latin terms and phrases can be troublesome for us non-Latin speakers. Heres the long and
short of the most common short forms adopted into English from the classical language:
10.1 e.g.-This abbreviation of exempli gratia (for example) is not only often left bereft of its periods (or styled eg.), its
also frequently confused for a similar abbreviation youll find below. Use e.g. (followed by a comma) to signal sample
examples.
10.2 etc.-This sloppily formed abbreviation of et cetera (and so forth) is often misspelled ect., perhaps because were
accustomed to words in which c precedes t, but not vice versa. (Curiously, Merriam-Webster spells out etcetera as such as a
noun, but at the end of an incomplete list, retain the two-word form, or translate it.) A comma should precede it.Refrain from
using etc. in an e.g. list; the abbreviations are essentially redundant, and note that etc. is also redundant in a phrase that
includes including.
10.3 et al.-This abbreviation of et alia (and others), used almost exclusively to substitute for the names of all but the
primary author in a reference to a multiauthor publication or article but occasionally applied in other contexts, should have
no period after et, because that word in particular is not an abbreviation.Also, unlike as in the case of etc., refrain from
preceding it with a comma, presumably because only one name precedes it. Fun fact: We use a form of the second word in
this term alias to mean otherwise known as (adverb) or an assumed name (noun).
10.4 i.e.-This abbreviation of id est (that is) is, like e.g., is frequently erroneously styled without periods (or as ie.). It,
followed by a comma, precedes a clarification, as opposed to examples, which e.g. serves to introduce.
10.5 fl.-This abbreviation of flourit (flourished) is used in association with a reference to a persons heyday, often in lieu
of a range of years denoting the persons life span.
10.6 N.B.-This abbreviation for nota bene (note well), easily replaced by the imperative note, is usually styled with
uppercase letters and followed by a colon.
10.7 per cent.-This British English abbreviation of per centum (for each one hundred) is now often (and in the United
States always) spelled percent, as one word and without the period.
10.8 re-This abbreviation, short for in re (in the matter of) and often followed by a colon, is often assumed to be an
abbreviation for reply, especially in email message headers.
10.9 viz.-This abbreviation of videlicet (namely), unlike e.g., precedes an appositive list one preceded by a reference to
a class that the list completely constitutes: Each symbol represents one of the four elements, viz. earth, air, fire, and water.
Note the absence of a following comma.
10.10 vs.-This abbreviation of versus (against) is further abbreviated to v. in legal usage. Otherwise, the word is usually
spelled out except in informal writing or in a jocular play on names of boxing or wrestling matches or titles of schlocky
science fiction movies. (In this title bout of Greed vs. Honesty, the underdog never stood a chance.)
XI.
CONCLUSIONS
We need to be little bit more frugal to deal with symbols, notations and primary concepts. Mathematics is not only a subject
but a language of modern computing. Fallacies are present everywhere like we find exceptions in Chemistry and bugs in
Computer Science. Confluence of Vedic Mathematics and modern computer programming will embellish and uplift
techniques of modern computing.
XII.
[1].
[2].
[3].
[4].
[5].
REFERENCES
Weil, Andr (1992), The Apprenticeship of a Mathematician, Springer, p. 114, ISBN 9783764326500.
Halmos, Paul (1950), Measure Theory. New York: Van Nostrand. pp. vi. The symbol is used throughout the entire book in place
of such phrases as "Q.E.D." or "This completes the proof of the theorem" to signal the end of a proof.
Kenneth E. Iverson (1962), A Programming Language, Wiley, retrieved 20 April 2016
Keller, Agathe (2006), Expounding the Mathematical Seed. Vol. 1: The Translation: A Translation of Bhaskara I on the
Mathematical Chapter of the Aryabhatiya, Basel, Boston, and Berlin: BirkhuserVerlag, 172 pages, ISBN 3-7643-7291-5.
Keller, Agathe (2006), Expounding the Mathematical Seed. Vol. 2: The Supplements: A Translation of Bhaskara I on the
Mathematical Chapter of the Aryabhatiya, Basel, Boston, and Berlin: BirkhuserVerlag, 206 pages, ISBN 3-7643-7292-3.
www.ijmsi.org
66 | Page
[25].
[26].
[27].
[28].
[29].
[30].
[31].
[32].
[33].
[34].
[35].
[36].
[37].
[38].
[39].
[40].
[41].
[42].
Neugebauer, Otto; Pingree (eds.), David (1970), The Pacasiddhntik of Varhamihira, New edition with translation and
commentary, (2 Vols.), Copenhagen.
Pingree, David (ed) (1978), The Yavanajtaka of Sphujidhvaja, edited, translated and commented by D. Pingree, Cambridge,
MA: Harvard Oriental Series 48 (2 vols.).
Sarma, K. V. (ed) (1976), ryabhaya of ryabhaa with the commentary of SryadevaYajvan, critically edited with Introduction
and Appendices, New Delhi: Indian National Science Academy.
Sen, S. N.; Bag (eds.), A. K. (1983), Theulbastras of Baudhyana, pastamba, Ktyyana and Mnava, with Text, English
Translation and Commentary, New Delhi: Indian National Science Academy.
Shukla, K. S. (ed) (1976), ryabhaya of ryabhaa with the commentary of Bhskara I and Somevara, critically edited with
Introduction, English Translation, Notes, Comments and Indexes, New Delhi: Indian National Science Academy.
Shukla, K. S. (ed) (1988), ryabhaya of ryabhaa, critically edited with Introduction, English Translation, Notes, Comments and
Indexes, in collaboration with K.V. Sarma, New Delhi: Indian National Science Academy.
Bourbaki, Nicolas (1998), Elements of the History of Mathematics, Berlin, Heidelberg, and New York: Springer-Verlag, 301
pages, ISBN 3-540-64767-8.
Boyer, C. B.; Merzback (fwd. by Isaac Asimov), U. C. (1991), History of Mathematics, New York: John Wiley and Sons, 736
pages, ISBN 0-471-54397-7.
Bressoud, David (2002), "Was Calculus Invented in India?", The College Mathematics Journal (Math. Assoc. Amer.), 33 (1): 2
13, doi:10.2307/1558972,JSTOR 1558972.
Bronkhorst, Johannes (2001), "Panini and Euclid: Reflections on Indian Geometry", Journal of Indian Philosophy, Springer
Netherlands, 29 (12): 4380,doi:10.1023/A:1017506118885.
Burnett, Charles (2006), "The Semantics of Indian Numerals in Arabic, Greek and Latin", Journal of Indian Philosophy, SpringerNetherlands, 34 (12): 1530,doi:10.1007/s10781-005-8153-z.
Burton, David M. (1997), The History of Mathematics: An Introduction, The McGraw-Hill Companies, Inc., pp. 193220.
Cooke, Roger (2005), The History of Mathematics: A Brief Course, New York: Wiley-Interscience, 632 pages, ISBN 0-471-44459-6.
Dani, S. G. (25 July 2003), "On the Pythagorean triples in the ulvastras" (PDF), Current Science, 85 (2): 219224.
Datta, Bibhutibhusan (Dec 1931), "Early Literary Evidence of the Use of the Zero in India", The American Mathematical
Monthly, 38 (10): 566572, doi:10.2307/2301384,JSTOR 2301384.
Datta, Bibhutibhusan; Singh, Avadesh Narayan (1962), History of Hindu Mathematics : A source book, Bombay: Asia Publishing
House.
De Young, Gregg (1995), "Euclidean Geometry in the Mathematical Tradition of Islamic India", HistoriaMathematica, 22 (2): 138
153, doi:10.1006/hmat.1995.1014.
Encyclopaedia Britannica (Kim Plofker) (2007), "mathematics, South Asian", Encyclopdia Britannica Online: 112, retrieved 18
May 2007.
Filliozat, Pierre-Sylvain (2004), "Ancient Sanskrit Mathematics: An Oral Tradition and a Written Literature", in Chemla, Karine;
Cohen, Robert S.; Renn, Jrgen; et al., History of Science, History of Text (Boston Series in the Philosophy of Science), Dordrecht:
Springer Netherlands, 254 pages, pp. 137157, pp. 360375, ISBN 978-1-4020-2320-0.
Fowler,
David (1996),
"Binomial
Coefficient
Function", The
American
Mathematical
Monthly, 103 (1):
1
17, doi:10.2307/2975209, JSTOR 2975209.
Hayashi, Takao (1995), The Bakhshali Manuscript, An ancient Indian mathematical treatise, Groningen: Egbert Forsten, 596
pages, ISBN 90-6980-087-X.
Hayashi, Takao (1997), "Aryabhata's Rule and Table of Sine-Differences", HistoriaMathematica, 24 (4): 396
406, doi:10.1006/hmat.1997.2160.
Hayashi, Takao (2003), "Indian Mathematics", in Grattan-Guinness, Ivor, Companion Encyclopedia of the History and Philosophy
of the Mathematical Sciences, 1, pp. 118130, Baltimore, MD: The Johns Hopkins University Press, 976 pages, ISBN 0-8018-73967.
Hayashi, Takao (2005), "Indian Mathematics", in Flood, Gavin, The Blackwell Companion to Hinduism, Oxford: Basil Blackwell,
616 pages, pp. 360375, pp. 360375,ISBN 978-1-4051-3251-0.
Henderson, David W. (2000), "Square roots in the Sulba Sutras", in Gorini, Catherine A., Geometry at Work: Papers in Applied
Geometry, 53, pp. 3945, Washington DC:Mathematical Association of America Notes, 236 pages, pp. 3945, ISBN 0-88385-164-4.
Joseph, G. G. (2000), The Crest of the Peacock: The Non-European Roots of Mathematics, Princeton, NJ: Princeton University
Press, 416 pages, ISBN 0-691-00659-8.
Katz, Victor J. (1995), "Ideas of Calculus in Islam and India", Mathematics Magazine (Math. Assoc. Amer.), 68 (3): 163
174, doi:10.2307/2691411.
Katz, Victor J., ed. (2007), The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton, NJ:
Princeton University Press, 685 pages, pp 385514, ISBN 0-691-11485-4.
Keller, Agathe (2005), "Making diagrams speak, in Bhskara I's commentary on the Aryabhaya", HistoriaMathematica, 32 (3):
275302, doi:10.1016/j.hm.2004.09.001.
Kichenassamy, Satynad (2006), "Baudhyana's rule for the quadrature of the circle", HistoriaMathematica, 33 (2): 149
183, doi:10.1016/j.hm.2005.05.001.
Pingree, David (1971), "On the Greek Origin of the Indian Planetary Model Employing a Double Epicycle", Journal of Historical
Astronomy, 2 (1): 8085.
Pingree, David (1973), "The Mesopotamian Origin of Early Indian Mathematical Astronomy", Journal of Historical
Astronomy, 4 (1): 112,doi:10.1177/002182867300400102.
Pingree, David; Staal, Frits (1988), "Reviewed Work(s): The Fidelity of Oral Tradition and the Origins of Science by Frits
Staal", Journal of the American Oriental Society, 108(4): 637638, doi:10.2307/603154, JSTOR 603154.
Pingree,
David (1992),
"Hellenophilia
versus
the
History
of
Science", Isis, 83 (4):
554
563, Bibcode:1992Isis...83..554P, doi:10.1086/356288, JSTOR 234257
Pingree, David (2003), "The logic of non-Western science: mathematical discoveries in medieval India", Daedalus, 132 (4): 45
54, doi:10.1162/001152603771338779.
Plofker, Kim (1996), "An Example of the Secant Method of Iterative Approximation in a Fifteenth-Century Sanskrit
Text", HistoriaMathematica, 23 (3): 246256,doi:10.1006/hmat.1996.0026.
Plofker, Kim (2001), "The "Error" in the Indian "Taylor Series Approximation" to the Sine", HistoriaMathematica, 28 (4): 283
295, doi:10.1006/hmat.2001.2331.
www.ijmsi.org
67 | Page
[46].
[47].
[48].
[49].
[50].
[51].
[52].
[53].
[54].
[55].
[56].
[57].
[58].
[59].
[60].
[61].
[62].
[63].
[64].
[65].
[66].
[67].
[68].
[69].
[70].
[71].
Plofker, K. (2007), "Mathematics of India", in Katz, Victor J., The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A
Sourcebook, Princeton, NJ: Princeton University Press, 685 pages, pp 385514, pp. 385514, ISBN 0-691-11485-4.
Plofker, Kim (2009), Mathematics in India: 500 BCE1800 CE, Princeton, NJ: Princeton University Press. Pp. 384., ISBN 0-69112067-6.
Price, John F. (2000), "Applied geometry of the Sulba Sutras" (PDF), in Gorini, Catherine A., Geometry at Work: Papers in
Applied Geometry, 53, pp. 4658, Washington DC: Mathematical Association of America Notes, 236 pages, pp. 4658, ISBN 088385-164-4.
Roy, Ranjan (1990), "Discovery of the Series Formula for by Leibniz, Gregory, and Nilakantha", Mathematics Magazine (Math.
Assoc. Amer.), 63 (5): 291306,doi:10.2307/2690896.
Singh, A. N. (1936), "On the Use of Series in Hindu Mathematics", Osiris, 1 (1): 606628, doi:10.1086/368443, JSTOR 301627
Staal, Frits (1986), The Fidelity of Oral Tradition and the Origins of Science, Mededelingen der KoninklijkeNederlandseAkademie
von Wetenschappen, Afd. Letterkunde, NS 49, 8. Amsterdam: North Holland Publishing Company, 40 pages.
Staal, Frits (1995), "The Sanskrit of science", Journal of Indian Philosophy, Springer Netherlands, 23 (1): 73
127, doi:10.1007/BF01062067.
L. Berggren (2004), J. M. Borwein and P. B. Borwein, Pi: a Source Book, Springer-Verlag, New York, third edition.
David M. Burton (2003), The History of Mathematics: An Introduction, McGraw-Hill, New York.
Tobias Dantzig and Joseph Mazur (2007), Number: The Language of Science, Plume, New York.
Howard Eves, An Introduction to the History of Mathematics, Holt, Rinehart and Winston, New York, 1990.
Gupta,R.C. (1983), Spread and triumph of Indian numerals, Indian Journal of Historical Science, vol. 18, pg. 23-38, available
at Online article.
Georges Ifrah (2000), The Universal History of Numbers: From Prehistory to the Invention of the Computer, translated from
French by David Vellos, E. F. Harding, Sophie Wood and Ian Monk, John Wiley and Sons, New York.
Victor J. Katz (1998), A History of Mathematics: An Introduction, Addison Wesley, New York.
RevielNetz and William Noel (2007), The Archimedes Codex, Da Capo Press.
Josephine Marchant (2008), Decoding the Heavens: Solving the Mystery of the Worlds First Computer, Arrow Books, New York.
John Stillwell (2002), Mathematics and Its History, Springer, New York.
Dirk J. Struik (1987), A Concise History of Mathematics, Dover, New York, 1987.
Staal,
Frits (1999),
"Greek
and
Vedic
Geometry", Journal
of
Indian
Philosophy, 27 (12):
105
127, doi:10.1023/A:1004364417713.
Staal, Frits (2001), "Squares and oblongs in the Veda", Journal of Indian Philosophy, Springer Netherlands, 29 (12): 256
272, doi:10.1023/A:1017527129520.
Staal, Frits (2006), "Artificial Languages Across Sciences and Civilisations", Journal of Indian Philosophy, Springer
Netherlands, 34 (1): 89141, doi:10.1007/s10781-005-8189-0.
Stillwell, John (2004), Berlin and New York: Mathematics and its History (2 ed.), Springer, 568 pages, ISBN 0-387-95336-1.
Thibaut, George (1984) [1875], Mathematics in the Making in Ancient India: reprints of 'On the Sulvasutras' and
'BaudhyayanaSulva-sutra', Calcutta and Delhi: K. P. Bagchi and Company (orig. Journal of Asiatic Society of Bengal), 133 pages.
van der Waerden, B. L. (1983), Geometry and Algebra in Ancient Civilisations, Berlin and New York: Springer, 223 pages, ISBN 0387-12159-5
van der Waerden, B. L. (1988), "On the Romaka-Siddhnta", Archive for History of Exact Sciences, 38 (1): 1
11, doi:10.1007/BF00329976
van der Waerden, B. L. (1988), "Reconstruction of a Greek table of chords", Archive for History of Exact Sciences, 38 (1): 23
38, doi:10.1007/BF00329978
Van Nooten, B. (1993), "Binary numbers in Indian antiquity", Journal of Indian Philosophy, Springer Netherlands, 21 (1): 31
50, doi:10.1007/BF01092744
Whish, Charles (1835), "On the Hind Quadrature of the Circle, and the infinite Series of the proportion of the circumference to the
diameter exhibited in the four S'stras, the TantraSangraham, YuctiBhsh, CaranaPadhati, and Sadratnamla", Transactions of
the Royal Asiatic Society of Great Britain and Ireland, 3 (3): 509523,doi:10.1017/S0950473700001221, JSTOR 25581775
Yano, Michio (2006), "Oral and Written Transmission of the Exact Sciences in Sanskrit", Journal of Indian Philosophy, Springer
Netherlands, 34 (12): 143160,doi:10.1007/s10781-005-8175-6
www.ijmsi.org
68 | Page