2. EARLY LIFE
• Brahmagupta was born in 598 CE according to his own
statement.
• He lived in Bhillamala in Gurjaradesa (modern Bhinmal in
Rajasthan, India) during the reign of the Chavda dynasty ruler,
Vyagrahamukha.
• His father Jisnugupta was an Astrologer in the city of Bhinmal
(Rajasthan).
• He was a Hindu by religion, in particular, a Shaivite.
• He lived and worked there for a good part of his life.
3. EDUCATION
• Bhillamala wasthe capitalof the Gurjaradesa,the second- largest kingdom of Western India, comprising
southern Rajasthanand northern Gujarat in modern-day India.
• It was also a centre of learning for mathematicsand astronomy.
• Brahmaguptabecame an astronomer of the Brahmapaksha school, one of the four major schools of
Indianastronomy during this period.
• He studied the five traditional Siddhantason Indianastronomy as well as the work of other astronomers
includingAryabhata-I,Latadeva,Pradyumna,Varahamihira,Simha, Srisena, Vijayanandin and
Vishnuchandra
4. WORK
• Brahmaguptawas the first to give
rules to compute with zero.
• Brahmasphutasiddhanta,
composed in 628 CE.
• Khandakhadyaka,composed in
665 CE.
• Grahanarkajnana,(ascribed in one
manuscript)
5. SOLUTIONOF GENERAL LINEAREQUATION
The difference between rupas, when inverted and divided by the difference of the [coefficients] of the
[unknowns], is the unknown in the equation.The rupas are [subtracted on the side] below that from which
the square and the unknown are to be subtracted.
when the equationis bx + c = dx + c,
where b,c,d & e are constant.
• then it's solution is equivalentto- X = (e-c )/(b-d)
• note:-उन्होंने b,d को अज्ञात और c,e को rupas से संबोधित ककया था।
6.
7. ZERO
Brahmaguptagavethe first rules for dealingwith zero as a number...
When zero is added to a number or subtracted from a number,the number remains unchanged.A number multipliedbyzero
becomes zero.
...and for dealingwith positive(fortune)and negative(debt)numbers
A debt minus zero is a debt.
A fortune minus zero is a fortune.
Zero minus zero is a zero.
A debt subtracted fromzero is a fortune.A fortune subtractedfromzero is a debt.
The product ofzero multiplied bya debt or fortune is zero.
The product ofzero multiplied byzero is zero.
The product orquotient oftwo fortunes is one fortune.The product orquotient oftwo debts is one fortune.
The product or quotient ofa debt and a fortune is a debt.The product or quotient ofa fortune and a debt is a debt.
8. LITTLE ERRORS
• Almost 500 years later, in the 12th Century, another Indianmathematician,BhaskaraII, showed that the
answer should be infinity, not zero (on the Brahmaguptaestablishedthe basic mathematicalrules for
dealing with zero (1+0=1; 1-0 = 1; and 1 x0 = 0), althoughhis understandingof divisionby zero was
incomplete(he thought that 1 +0=0).
• grounds that 1 can be divided intoan infinitenumber of pieces of size zero), an answer that was
considered correct for centuries.
• However, this logic does not explain why 2÷0, 7÷0, etc., should also be zero - the modern view is that a
number dividedby zero is actually"undefined"(i.e. it doesn't make sense).
9. SOLUTION OF QUADRATICEQUATION
• He further gave two equivalentsolutionsto the general quadraticequation
• 18.44. Diminish by the middle [number] the square-root of the rupas multipliedby four times the
square and increased by the square of the middle [number]; divide the remainder by twice the square.
[The result is] the middle [number]. 18.45. Whatever is the square-root of the rupas multipliedby the
square [and] increased by the square of half the unknown, diminishedthat by half the unknown [and]
divide [the remainder] by its square. [The result is] the unknown.
10. SQUARES AND CUBES
• Brahmaguptathen goes on to give the sum of the squares and cubes of the first n integers.
• 12.20. The sum of the squares is that [sum] multipliedby twice the [number of] step[s] increased by
one [and] dividedby three. The sum of the cubes is the square of that [sum] Piles of these with identical
balls.
• Here Brahmaguptafound the result in terms of the sum of the first n integers, rather than in terms of n
as is the modern practice.
11. BRAHMAGUPTA’SFORMULA
• Brahmagupta's most famous result in geometryis his formula forcyclic quadrilaterals.Given the lengths of the sides of any
cyclic quadrilateral,Brahmagupta gavean approximate and an exact formula for the figure's area,
• 12.21. The approximate area is the product ofthe halves of the sums of the sides and opposite sides ofa triangle
• and a quadrilateral.The accurate [area]is the square root from
• the product ofthe halves of the sums of the sides diminished by[each] side of the quadrilateral.
• So given the lengths p,q, r and s of a cyclic quadrilateral,
• the approximate area is (p+q)/2.(q+s)/2while, lettingt = (p+q+r+s)/2,
• the exact area is
• √(t-p)(t-q)(t-r)(t − s).
• Although Brahmagupta does not explicitlystate thatthese quadrilaterals are cyclic,
• it is apparent fromhis rules that this is the case. Heron's formula is a special case of this
• formula and it can be derived by settingone of the sides equal to zero.
12. Brahmaguptacontinues,
• 12.23.The square-rootof the sum of the two products of the
sides and opposite sides of a non-unequal quadrilateralis
the diagonal. The square of the diagonal is diminished by
the square of half the sum of the base and the top; the
square-root is the perpendicular [altitudes].
• So, in a "non-unequal" cyclic quadrilateral (thatis, an
isosceles trapezoid),the length of each
diagonal is (pr+ qs)^(1/2).
14. PI
• In verse 40, he gives values of л,
• 12.40. The diameterand the square of the radius[each] multipliedby 3 are [respectively] the practical
circumference and the area [of a circle]. The accurate [values] are the square-roots from the squares of
those two multipliedby ten.[22]
• So Brahmaguptauses 3 as a "practical"valueof л, and 10≈ 3.1622... as an "accurate"value of л, with an
error less than 1%.
15. INTERPOLATIONFORMULA
• In 665 Brahmaguptadevised and used a special case of the Newton-Stirlinginterpolation formulaof the
second-order to interpolatenew values of the sine function from other values alreadytabulated.[34]
The formula gives an estimate for the valueof a function fat a valuea + xh of its argument (with h> 0
and -1 ≤ x ≤ 1) when its value is already known at a -h, a and a + h.
• The formula for the estimate is:
16.
17. ACCURATE SINETABLES
2.2-5. The sines: The Progenitors, twins; Ursa Major, twins, the Vedas;
the gods, fires, six; flavors, dice, the gods; the moon, five, the sky, the
moon; the moon, arrows, suns [...] [32]
Here Brahmaguptauses names of objects to represent the digits of
place-valuenumerals, as was common with numerical data in Sanskrit
treatises. Progenitorsrepresents the 14 Progenitors("Manu") in Indian
cosmology or 14, "twins" means 2, "Ursa Major"represents the seven
stars of Ursa Majoror 7, "Vedas" refers to the 4 Vedas or 4, dice
represents the number of sides of traditional diceand so on.
18. BRAHMAGUPTAAS ASTRONOMER
• Later, Brahmaguptamoved to Ujjaini, Avanti,which was also a
major centre for astronomy in centralIndia.
• At the age of 67, he composed his next well-known work
Khanda-khadyaka,a practicalmanual of Indian astronomyin the
karanacategory meant to be used by students.
19. • Calculatedthe length of a year is 365 days6 hours 12 minutes 9 seconds which is very close to today's
calculation.
• Brahmaguptatalked about 'gravity.
• To quote him, 'Bodies fall towardsthe earth as it is in the nature of the earth to attract bodies, just as it
is in the nature of water to flow."
• Proved that the Earth is a sphere and calculatedits circumference to be around36,000 km (modern
40,075 km).
20. MOSTCRITICALLYACCLAIMED – CHAKRAVALA
METHOD
• This method is used to solve indeterminatequadraticequationsof form:
• x² = Ny²+1
• He found out pairs of solutionsfor several values of N, very easily.
• One problem was unsolved since then, and was notoriouslydifficult
x² = 61y²+1
• Jayadeva (9thcentury) and Bhaskara (12th Century) found out the minimal Integral solution-
• x=1766319049 & y=226153980
• Solved in Europe not before 1657, when Fermat proposed this as a challenge
• Lagrange gave a general method but it's computationwas way more difficult than the Chakravala
method!
21. COMMENTARIES BY FOREIGN MATHEMATICIANS ON
CHAKRAVALA METHOD:
• "The method represents a best approximationalgorithmof minimallength that, owing to several
minimizationproperties, with minimaleffort and avoidinglarge numbers automaticallyproduces the
best solutionsto the equation.The chakravala methodanticipatedthe Europeanmethods by more than
a thousand years. But no European performances in the whole field of algebra at a time much laterthan
Bhaskara's, nay nearly equal up to our times, equalledthe marvellous complexity and ingenuityof
chakravala."
-Selenius
• "the finest thing achievedin the theory of numbers before Lagrange."
-Hermann Hankel
22. HALL OF FAME
• Bhaskara II named him as "Ganitachakra Chudamani"Meaning-'a brightest star in the galaxy of
mathematicians'
• Huge influx in Arabianmathematicswas adoptedfrom works of Brahmaguptaby translatinghis works.
• (e.g. Sindhindis an Arab translation.of Brahmasputasiddhanta).
