The document discusses the organization of Vedic mathematics courses aimed at enhancing mathematical understanding through four levels of instruction, ranging from basic arithmetic to advanced topics like calculus and trigonometry. It highlights the historical significance of zero in mathematics, elaborating on its conceptual foundation and mathematical operations involving zero, including division and its indeterminate nature. Additionally, the text explores various infinite concepts in mathematics, including sequences and different cardinalities of infinity, along with historical perspectives from ancient Indian mathematicians.

1. M Rajagopala Rao
National Conference on Mathematics and
Astronomy of Bhaskara II
Sri Venkateswara Vedic University, Tirupati
18th-19th December 2014

2. • SWIS ® (Success with Self) Trust organizes (Veda
Ganitam the Amazing mathematics) courses at schools
• Course comprises 4 levels
• Level I (Basic Arithmetic and important Vedic Maths
concepts)
• Level II (Advanced Arithmetic, Squares & Square roots,
Cubes& Cube roots, Divisibility, HCF&LCM, percentages,
areas & volumes, compound interest, Vedic Binary
system)
• Level III (Vedic Algebra, Geometry and basic
Trigonometry)
• Level IV (Advanced topics like Calculus, Combinatorics,
Advanced algebra, trigonometry, concepts of zero and
infinity)

3. Interested in organizing Vedic Maths course
at a school?
• SWIS can help you organize Vegam courses in schools
by providing
• Teachers’ Training
• Audio-visual material for classes
• Class room activities, games, quizzes
• Work Books
• Examinations
• Hand-holding in initial stages
• Contact successwithself@gmail.com
• Tel: +914027138854 or Mob: 8978864560 for details

4. 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

5. • One of most famous Indian
mathematicians
• Born 1114 AD in Bijjada Bida
• Nicknamed Bhaskaracharya “Bhaskara
the Teacher”
• Studied Varahamihira and Brahmagupta
at Uijain

6. • Understood zero and negative
numbers
• Knew x^2 had 2 solutions *
• Had studied Pell’s equation and
other Diophantine problems

7. • 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

8. 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

9. • 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.

10. • 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!

11. • 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 र्ून्यं द्वः र्ून्ये| (८.२९-३०)
• Indian Philosophy – Nyaya school – abhaava –
Baudhdha - Sunyavada

12. • 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

13. • 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

14. • 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

15. In Shankara Bhashya (2.2.17) on
Brihat Samhita of Varaha Mihira(505-587CE)
यथा एकोऽपि सन ् देवदत्तः लोके स्वरूिम् 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-सहस्रादद
शब्दप्रत्ययभेदं 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..

16. • By Brahmagupta - 628 CE – first work
discussing operations with zero (शूधय
िररक्रम), +ve and –ve (िन ऋः)
• Rules for संकलन
िनयोिधनं ऋःंमृःयोः िनःधयोरधतरं समैकां खं
रुःमैकां च िनमृःिनशूधययोः शूधययोः शूधयं
• Pos + Pos = Pos, Neg + Neg = Neg
• Pos + Neg = Pos/Neg or zero when equal
• Pos + Zero = Pos, Neg + Zero = Neg
• Zero + Zero = Zero

17. • 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

18. • 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.

19. • 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.

20. • 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.

21. • 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?

22. • 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.

23. योगे खं क्षेिसमं वगाधदौ खं ख भान्जतो
राशशः|
खहरः स्यात् खगुःः खं खगुःन्चचधत्यचच
शेषपविौ||
शूधये गुःके जाते खंहारचचेत िुनस्तदा
राशशः|
अपवकृ त एव ज्ञेयस्थथैव खेनोतनतचच युतः||

24. • 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.

25. • 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

26. • 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

27. • Sequence B:
1 1 1 1 1
---, -----, -----, -----, -----, ... Approaches 1/ 0
1 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

28. • 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.

29. 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

30. Larger than the
largest number you
can think of

31. 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

32. • ईशावास्य उितनषद proclaims ब्रह्ममन् thus:
िूःधमदः िूःधशमदं िूःाधत् िूः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

33. • 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

34. • 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

35. • 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

36. • 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)

37. • 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).

38. • Now, Let us apend +∞ and -∞ to the
set of complex numbers
• Define the operations + -
• ∞+r = r+ ∞ = ∞, -∞+r = r+(-∞),
• ∞+∞ = ∞, -∞+(-∞) = -∞, ∞-r = ∞,
-∞-r =-∞
• r-∞ = -∞, r-(-∞) = -∞, ∞-(-∞) = ∞, -∞-
∞= -∞

39. • 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
• ∞/∞=-∞/-∞=∞, -∞/∞=∞/-∞=-∞

40. • 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

41. 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.

42. 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.

43. िूःधमदः िूःधशमदं िूःाधत् िूःधमुदच्यते
िूःधस्य िूःधमादाय िूःधमेवावशशष्यते