M Rajagopala Rao

National Conference on Mathematics and

Astronomy of Bhaskara II

Sri Venkateswara Vedic University, Tirupati

18th-19th December 2014

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areas & volumes, compound interest, Vedic Binary

system)

• Level III (Vedic Algebra, Geometry and basic

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Advanced algebra, trigonometry, concepts of zero and

infinity)

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About the Author

M Rajagopala Rao, M.Sc.(Tech.)-Geophysics

Executive Director-Chief Knowledge

Officer (Retd.), ONGC

Principal Coach, Success With Self (SWIS)®

Academy, Hyderabad

ONGC Project Saraswati

Piligramage to Kailash-Mansarovar twice

Conducting classes at DAV schools in Hyderabad

• One of most famous Indian

mathematicians

• Born 1114 AD in Bijjada Bida

• Nicknamed Bhaskaracharya “Bhaskara

the Teacher”

• Studied Varahamihira and Brahmagupta

at Uijain

• Understood zero and negative

numbers

• Knew x^2 had 2 solutions *

• Had studied Pell’s equation and

other Diophantine problems

• First to declare a/0 = *

• First to declare + a =

• Wrote 6 works including

• Lilavati (mathematics)

• Bijaganita (algebra)

• Siddhantasiromani (Astronomy)

• Goladhyaya (Geometry of Spheres)

• Vasanabhasya (on Siddhantasiromani)

• Karanakutuhala (Astronomy)

• Vivarana

The ingenious method of expressing every

possible number using a set of ten symbols …

seems so simple … 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.

Laplace (1749-1827), the French

mathematician

• The Egyptian, Greek and Roman

number systems had no zeros

• Even though the Greek number

system was more sophisticated

than the Egyptian and Roman

systems, it was not the most

advanced.

• In the history of culture the discovery of zero will

always stand out as one of the greatest single

achievements of the human race.

-Tobias Danzig

• Without zero we would lack

• Calculus, financial accounting,

• the ability to make arithmetic

computations quickly

• and computers!

• Asthadhyayi – Panini (500 BCE) – Lopa – Null

Morphene

अदर्शनं लोपः (१:१:६०)• Pingala’s Chandassastra (300 BCE) – In Ch. VIII –

algorithm for positive integral power of 2 – Sunya– used as a marker

रूपे रू्न्यं द्वः रू्न्ये| (८.२९-३०)

• Indian Philosophy – Nyaya school – abhaava –

Baudhdha - Sunyavada

• In Vyasa Bhashya of Patanjali (100

BC)

यथैका रेखा शत स्थाने शतं दश स्थाने दश एक च एक स्थाने: Just as the same line (means) a

hundred in hundreds place, ten in tens

and one in one’s place

• Found near Peshavar, Pakisthan – of 200-

400CE – Important source of Mathematical

notation

• Three types of notations

• Fractions: one number below other – no

horizontal line

• –ve numbers: a small cross (+) to the right – probably simplified Devnagari ऋ

• Characters like यु, मू for operations +, √

abbreviations of words for the operations युतत, मूल

• Of 499 CE – Most comprehensive

astronomical mathematics

• Perfect presentation of number of

revolutions by planets – sure indication of

perfect knowledge of zero and place value

system

• So are algorithms for squareroot, cuberoot

• Sophisticated and ingenious number

representation – numbers of the order 1016

represented by single Character

In Shankara Bhashya (2.2.17) on

Brihat Samhita of Varaha Mihira(505-587CE)

यथा एकोऽपि सन ्देवदत्तः लोके स्वरूिम ्संबन्धिरूिं च अिेक्ष्य अनेक प्रत्ययभाग्भवतत – मनुष्यः, ब्राह्ममःः, श्रोत्रिय, वदाधय, बालः, युवा, स्थपवरः, पिता, िुिः, िौिः, भ्राता, जामाता इतत| यथा च एकपि सती रेखा (अङ्कः) स्थानाधयत्वेन तनपवशमाना एक-दश-शत-सहस्राददशब्दप्रत्ययभेदं अनुभवतत, तथा संबन्धिनोरेव ...Though Devdatta is only one person, due to own and

relational forms he becomes many – man, brahmin,

learned, generous, child, youth, old, father, son,

grandson, brother, son-in-law, just as one line (digit) due

to change of place is called one-ten-hundred-thousand and so on..

• By Brahmagupta - 628 CE – first work discussing operations with zero (शधूय िररक्रम), +ve and –ve (िन ऋः)

• Rules for संकलन िनयोिधनं ऋःंमःृयोः िनःधयोरधतरं समैकां खंरुःमैकां च िनमःृिनशधूययोः शधूययोः शधूयं • Pos + Pos = Pos, Neg + Neg = Neg

• Pos + Neg = Pos/Neg or zero when equal

• Pos + Zero = Pos, Neg + Zero = Neg

• Zero + Zero = Zero

• By Mahavira - 815-877 CE

• In multiplying as well dividing two

–ve or two +ve quantities result is

a +ve

• But it is –ve when one is +ve and

other –ve

Verses 50-52

• By Mahavira - 815-877 CE

• Operations with Zero

ताडितः खेन राशशः खं सोपवकारी हृतो युतः हीनोऽपि खवदाददः खं योगे कं योज्यरूिकं a x 0 = 0 x a = 0, a ÷ 0 = 0 ÷ a = 0

a + 0 = 0 + a = a, a – 0 = a, 0 – a = -a

a÷0 = 0 is not accepted in modern maths.

• A quantity divided by zero is called

fraction with Zero denominator (खहार) .

• In खहार no alteration, though many may

be inserted or extracted; as no change in

the infinite and immutable God when

worlds are created or destroyed, though

numerous orders of beings are absorbed

or put forth.

• Cannot find an answer, so it is

disallowed.

• 12÷6 = 2 as 6x2 = 12

• 12÷0 = p means 0xp = 12

• But no value would work for p because 0

times any number is 0.

• So division by zero doesn't work.

• 10/2=5 is 10 blocks, separated

into 5 groups of 2 each.

• 9/3=3 is 9 blocks, separated into 3

groups of 3 each.

• 5/1=5 is 5 blocks separated into 5

groups of 1 each.

• 5/0 = ? Into how many groups?

• 5/0 = ? how many groups of zero

could you separate 5 blocks?

• Any number of groups of zero

would never add up to five

since 0+0+0+0… = 0.

• It doesn't make sense since there

is not a good answer.

योगे खं क्षेिसमं वगाधदौ खं ख भान्जतो राशशः| खहरः स्यात ्खगःुः खं खगःुन्चचधत्यचच शषेपविौ||शूधये गःुके जात ेखंहारचचते िुनस्तदा राशशः|अपवकृत एव ज्ञेयस्थथवै खेनोतनतचच युतः||

• In addition sum equals the additive,

• in multiplication the result is zero

• If zero becomes a multiplier in the

numerator, the operation must be

postponed and if in further computation

zero appears in the denominator the

quantity must be retained as it is without any operation by 0/0.

• What is the number which when

multiplied by zero, being added to half of

itself, multiplied by three and divided by

zero amounts to sixty three

• Bhaskara worked it out as follows

0[x+(x/2)(3/0)] = 63 (3x/2)3 = 63 x = 14• Bhaskara declared that this kind of

calculation has great relevance in astronomy

• The difference between 0/0 and 1/0.

• Sequence A:

5*1 5*1/2 5*1/3 5*1/4 5*1/5----, ------, ------, ------, ------, ...1 1/2 1/3 1/4 1/5

• It approaches 0/0 as Numerator, Denomi-

nator both going to zero

• But every term is 5 so limit is 5 and 0/0=5

• Replace 5 by any number and you get

that as limit – so 0/0 can be anything

• Sequence B:

1 1 1 1 1---, -----, -----, -----, -----, ... Approaches 1/ 01 1/2 1/3 1/4 1/5

• But, It is 1,2,3,4,5 …. and approaches ∞

• So a/0 (a ≠ 0) is ± ∞

• Modern mathematics calls it indetermi-

nate

• Two types of zeros

• You have ₹100. you give ₹20 each to 5

people. You are left with ₹0

• You have a lump of gold. Every day you

give half of what you have to someone.

After a long time you will be left with

almost zero gold but not exactly zero.

This is called infinitesimal.

The hare and tortoise race

• Hare slept and Tortoise moved ahead by a distance p

• Hare woke up and covered the distance at a high speed V

• Hare took time t = p/V to cover the distance p

• Tortoise at slow speed v covered a small distance ∆p in time t

• Hare took a small time ∆t to cover distance ∆p.

• Tortoise travelled ∆(∆p) in time ∆t

• Hare took ∆(∆t) to cover ∆(∆p)

• Tortoise covered ∆(∆(∆p)) in ∆(∆t)

• And so on … ad infinitum

• How can the Hare ever cross the Tortoise mathematically?

• It can! Because the sum of an infinite series of this kind is

finite

Larger than the

largest number you

can think of

Hilbert Hotel

• You are working in the Hilbert Hotel, with the

reputation of never turning away any guest as it has

infinity number of rooms

• One day the hotel is full and a new guest has arrived.

In which room number will you accommodate her?

• Initially it may appear impossible but there is a

solution

• Request the guests through the public address

system to kindly cooperate in this extraordinary

situation and shift to the room having the next

number to their room number. Every one is

accommodated and the new guest moves into Room

No.1

• ईशावास्य उितनषद proclaims ब्रह्ममन ्thus:िूःधमदः िूःधशमदं िूःाधत ्िूःधमुदच्यते िूःधस्य िूःधमादाय िूःधमेवावशशष्यते• It is whole and this is also whole. The

whole comes out of the whole. Take the

whole out of the whole and the whole indeed remains

• Bhaskara’s खहार also is described in

similar terms

• Look at the following two series A & B

A 1 2 3 4 5 6 7 8 9 10 …

B 2 4 6 8 10 12 14 16 18 20 …

Clearly for every member of A there is one

corresponding member in B meaning their

cardinality is the same; both are infinity

But A has many members which are missing in

B meaning B is only a part of A

• Thus a part is equal to the whole

• A ‘point’ has no length, width and thickness

• A ‘line’ has length but no width and

thickness

• A line is made up of ‘infinite’ points

• That means ‘infinite’ zero lengths add up

to a finite length

• A ‘plane’ has length and breadth but no

height

• A plane is made up of infinite ‘lines’

• A plane is made up of infinite ‘points’ as

well

• The second infinity is more than the first

one

• There are ‘orders’ of infinities

• Jains (Surya Prajnapti – 400BC) have

classified numbers as

• Enumerable (संख्येय)• Unenumerable (असङ्ख्येय)• Infinite (अनधत)

• They mentioned different types of infinity

• One direction (एकतोनधतं), two directions

(द्पविानधतं), areal(देशपवस्तारानधतं), everywhere(सवधपवस्तारानधतं), eternal(शाचवतानधतं)

• Natural Numbers 1,2,3,4 …

• Natural to Integers (append -ve integers

and zero)

• Integers to rationals (append ratios of

integers),

• rationals to reals (appending limits of

convergent sequences),

• reals to complexes (appending the

square root of -1).

• Now, Let us apend +∞ and -∞ to the

set of complex numbers

• Define the operations + -

• ∞+r = r+ ∞ = ∞, -∞+r = r+(-∞),

• ∞+∞ = ∞, -∞+(-∞) = -∞, ∞-r = ∞,

-∞-r =-∞

• r-∞ = -∞, r-(-∞) = -∞, ∞-(-∞) = ∞, -∞-

∞= -∞

• Define the operations * ÷

• ∞*r=r*∞=∞, -∞*r=r*-∞=-∞ for r>0

• ∞*r=r*∞=-∞, -∞*r=-r*-∞=∞ for r<0

• ∞*∞=-∞*-∞=∞, -∞*∞=-∞*∞=-∞

• ∞/r=r/∞=∞, -∞/r=r/-∞=-∞ for r>0

• ∞/r=r/∞=-∞, -∞/r=-r/-∞=∞ for r<0

• ∞/∞=-∞/-∞=∞, -∞/∞=∞/-∞=-∞

• We get into trouble with

• ∞+(-∞), -∞+∞, ∞=-∞, -∞-∞, 0*∞, ∞*0,

0*(-∞), -∞*0, ∞/0, 0/∞, -∞/0, 0/(-∞), ∞/∞,

- ∞/∞, ∞/-∞, -∞/-∞

• These are ‘Indeterminate forms’

• So, infinity is a concept not a number

Zero and infinity are similar• कथोितनषद says

अःोरःीयान महतो महीयान (1.2.20)

The supreme is smaller than the smallest (infinitismal

zero) and larger than the largest (infinity)

• Every single cell in our body has the complete

genetic code which made us

• With holograms, each of the smaller parts still

contain a reflection of the complete, whole, 3-

dimensional image.

• perimeter of a closed curve increases, approaching

infinity as the length of the measuring rod

approaches zero.

How many corners does a circle have?

• Zero corners as the circle is smooth curve

• Draw a square such that the circle is super scribing it. There

are 4 corners

• Now replace the square by a hexagon (six corners), octagon

(eight corners) … and so on

• As the number of sides of the polygon increase, it would be

better approximating the circle. Number of corners also

increase

• Finally at zero length side an infinite sided polygon exactly

represents the circumference of the circle. It would have

infinite corners.

• That is, a circle which has zero corners also can be seen as

having infinite corners.

िूःधमदः िूःधशमदं िूःाधत ्िूःधमुदच्यते िूःधस्य िूःधमादाय िूःधमेवावशशष्यते