vm1
TRANSCRIPT
Ved
ic M
athe
mat
ics
M. K
rish
nam
oort
hy (
moo
rthy
)
emai
l ad
dres
s: m
oort
hy@
cs.r
pi.e
du
UR
L:
http
://w
ww
.cs.
rpi.e
du/~
moo
rthy
/vm
Ref
eren
ces
� 1)
Ved
ic M
athe
mat
ics
or S
ixte
en S
impl
eM
athe
mat
ical
For
mul
ae fr
om th
e V
edas
(For
one
-line
ans
wer
s to
all
mat
hem
atic
alpr
oble
ms)
, Mot
ilal B
anar
sida
ss P
ubl
ishe
rs,
Del
hi, 1
965.
ISB
N 8
1-20
8-01
64-4
� 2) H
isto
ry o
f H
ind
u M
athe
mat
ics
- A
sou
rce
Boo
k Pa
rts
1&2
B. D
atta
and
A. N
. Sin
gh,
Asi
a Pu
blis
hing
Hou
se, B
omba
y, 1
935.
Out
line
of th
e C
ours
e
� Intr
oduc
tion
and
Mul
tipl
icat
ion
Sut
ra(c
hapt
er 3
) 2/
2/97
� Pra
ctic
al A
ppli
catio
ns o
f M
ulti
plic
atio
nan
d D
ivis
ion
Sutr
a (c
hapt
er 4
) 2/
9/97
� Arg
umen
tal D
ivis
ion
(cha
pter
1)
2/16
/97
� Rec
apitu
latio
n a
nd
Mor
e ex
erci
ses
2/2
3/97
� Fact
oriz
atio
n o
f Si
mpl
e Q
uadr
atic
s (c
hapt
er7)
3/2
/97
Sum
mar
y of
Cou
rse
� Con
side
rs th
e o
rigi
n of
diff
eren
t met
hod
sus
ed in
Ved
ic M
athe
mat
ics
� Lea
rn S
utra
s to
kno
w h
ow to
do
calc
ulat
ions
in a
dif
fere
nt (
fast
er/m
ore
effi
cien
t man
ner)
.
� Use
fuln
ess o
f th
ese
met
hod
s in
aca
lcul
ator
/com
pute
r ag
e.
Abo
ut S
tude
nts
� Fin
d o
ut a
ny r
elev
ant b
ackg
rou
nd a
ndin
tere
st o
f th
e au
dien
ce
� Wha
t do
they
und
erst
and
by v
edic
mat
hem
atic
s?
� Why
one
shou
ld d
o m
athe
mat
ics
in th
e20
th/2
1st c
entu
ry?
� Can
you
nam
e a
fam
ous
20th
cen
tury
Ind
ian
Mat
hem
atic
ian?
Age
nda
for
Tod
ay
� Bri
ef H
isto
ry V
edas
, Ved
anga
s an
dU
pave
das
� Why
did
the
vedi
c pe
ople
do
mat
hem
atic
s?
� Som
e of
the
mat
hem
atic
al i
nven
tions
that
wer
e at
trib
uted
to In
dia.
-
� Fam
ous
Indi
an m
athe
mat
icia
ns o
f th
eby
gon
e a
nd th
eir
wri
ting
s
� Sutr
as fo
r M
ulti
plic
atio
n a
nd E
xam
ples
Ove
rvie
w
Let
us
say
that
you
wan
t to
mul
tiply
1) 1
257
* 1
001
2) 7
77 *
99
9
3) 1
13
* 1
08
HIS
TO
RY
Exp
lain
how
all
the
indi
vidu
al t
opic
s fi
tto
geth
er
Per
iod
of V
edic
Mat
hem
atic
s
Thi
s is
a b
ook
by
Shan
kara
char
ya (1
884
-19
60)
of
Puri
. It w
as m
eant
to b
e th
e fi
rst o
f16
vol
umes
cov
erin
g a
spec
ts o
fm
athe
mat
ics
by
expl
aini
ng th
e us
e of
16
sutr
as fr
om th
e V
edas
. All
of t
hese
vol
umes
wer
e lo
st. B
efor
e h
e di
es, h
e re
wro
te V
ol. 1
from
his
mem
ory.
Thi
s vo
lum
e co
vers
qui
ckm
etho
ds fo
r do
ing
ari
thm
etic
.
Oth
er C
laim
s
Som
e ot
hers
cla
im th
at th
is b
ook
is a
bout
Hin
du M
athe
mat
ics
betw
een
Ary
abha
ttaan
d B
hask
ara
- th
at is
rou
ghly
bet
wee
n 5
00an
d 1
100
AD
.
Cha
ndog
ya U
pani
shad
- St
ory
abo
ut N
arad
aan
d S
age
San
atku
mar
a.
Ved
as, V
edan
gas
and
Upa
ngas
Four
teen
-fol
d V
edic
Kno
wle
dge:
1) 4
Ved
as (R
g, Y
ajur
, Sam
a an
d A
thar
va)
-or
igin
ally
ora
l in
natu
re.
2) 6
Ved
anga
s (p
hone
tics,
gra
mm
ar,
etym
olo
gy, m
etro
nom
y(ch
anda
s),
Ast
ron
omy
and
Ast
rolo
gy,
and
Kal
pa)
4) 4
Upa
veda
s (A
naly
sis,
Lo
gic,
Pur
anic
Lit
erat
ure)
an
d D
arm
a Sa
stra
Upa
veda
s
4 U
pave
das
(Ayu
rved
a, G
and
harv
aved
a,D
han
urve
da a
nd
Sth
apat
yave
da)
Ved
as re
fer
to a
bod
y of
kno
wle
dge
that
reve
als
diff
eren
t mea
ns a
nd
end
s av
aila
ble
to th
e h
uman
bei
ng.
Ved
ic M
athe
mat
ics
Mos
t im
por
tant
sou
rces
for
Ved
icM
athe
mat
ics
are
fou
nd in
Ast
rono
my
&A
stro
logy
and
Kal
pa (r
ules
for
ritu
als
and
cere
mo
nies
).
Evi
denc
e is
usu
ally
in th
e fo
rm o
f su
tras
(ape
culia
r fo
rm o
f w
ritin
g w
hich
is v
ery
conc
ise
and
ofte
n u
ses
a po
etic
sty
le to
capt
ure
the
ess
ence
of
arg
umen
t or
resu
lt.
Use
s of
Mat
hem
atic
s
� Kal
pasu
tras
gav
e d
irec
tions
for
con
duct
ing
sacr
ific
ial f
ires
at d
iffe
rent
tim
es o
f th
e y
ear.
So th
ey n
eed
to c
alcu
late
the
plan
etar
ypo
siti
ons
whi
ch n
eede
d m
athe
mat
ics.
The
yal
so n
eed
to c
onst
ruct
cal
enda
rs/p
anch
anga
sfo
r lu
nar
as w
ell
sola
r m
ont
hs. A
stro
logy
also
nee
ded
cons
ider
able
mat
hem
atic
alca
lcul
atio
ns.
Sut
ras
� Con
dens
atio
n of
sut
ras
so th
at m
inim
alw
riti
ng m
ater
ial i
s us
ed. T
his
was
the
form
in w
hich
the
cont
ents
of
Bra
hman
as w
ere
pres
erve
d. T
his
was
late
r us
ed b
y va
riou
sph
iloso
phic
al a
nd s
cien
tific
sch
ool
s su
ch a
sst
atec
raft
(ar
thas
asas
tra)
. A c
onc
ise
Sans
krit
stat
emen
t tha
t giv
es a
rule
or
prin
cipl
e to
be
follo
wed
, for
exa
mpl
e, in
gra
mm
ar, i
n rit
ual
or in
mat
hem
atic
s
Exa
mpl
es
Sutr
a ca
n b
e th
ough
t of
Sans
krit
equi
vale
nt o
fso
met
hin
g li
ke: “
i” b
efor
e “e
” ex
cept
aft
er“c
”.
Exa
mpl
e: N
ikhi
lam
Nav
atas
hcar
amam
Das
hata
h
All
from
nin
e an
d th
e la
st f
rom
10.
Use
s of
that
sut
ra
Mul
tipl
icat
ion
8 x
7 x
98
8 -
27
-3
9 -
18
-2
----
----
----
----
-
7
25
6
Mor
e E
xam
ples
7 x
9
9 x
9
12 x
13
14 x
17
19 x
19
Why
doe
s it
wor
k?
(x-a
) *
(x-b
) =
x^2
-x(a
+b)
+ a
b
=
x(x
-a-b
) +
ab
In 8
x 9
x-a
= 8
, x-b
=9
and
x is
take
n to
be
10.
Upa
sutr
a
The
re is
a u
pasu
tra
(sub
sutr
a) o
f on
e w
ord
anur
upye
na w
hich
mea
ns “
Pro
port
iona
tely
”.In
oth
er w
ords
, one
can
take
any
bas
e th
at is
suita
ble.
Bef
ore
I ex
plai
n w
hat i
s m
eant
by
base
:
91 -
9
89 -
11
80 9
9.
Bas
e
The
bas
e w
e ch
oos
e in
the
prev
ious
exa
mpl
eis
100
. ( o
r x
in th
e fo
rmul
a is
10
0).
Bas
es c
an b
e un
ders
tood
from
the
pos
ition
alnu
mbe
r sy
stem
(aga
in th
is sy
stem
isbe
lieve
d to
be
the
inve
ntio
n o
f In
dian
s.
Nor
mal
ly, n
umbe
rs a
re re
pres
ente
d in
bas
e 1
0no
tatio
n. C
ompu
ters
use
bin
ary
num
bers
or
num
bers
repr
esen
ted
in b
ase
2.
Exa
mpl
es
56
645
-521
1
11
54
447
-320
9
09
----
----
----
----
-
---
----
----
--
60/2
24
21 1
544
0 9
9
302
4
Sum
mar
y
� His
tory
of
Ved
ic M
athe
mat
ics
� Mul
tipl
icat
ion
usi
ng s
utra
s
Whe
re to
get
mor
e in
form
atio
n
� Do
mor
e ex
ampl
es.
� Lo
ok a
t the
Inte
rnet
und
er a
ltav
ista
, sea
rch
with
ved
ic-m
athe
mat
ics
� Go
to lo
cal l
ibra
ry/B
arne
s N
oble
s/B
orde
rsre
ad a
boo
k tit
led
“T
he M
an W
ho K
new
Infi
nity
”
Fee
dbac
k
Ple
ase
give
me
feed
bac
k fr
om b
oth
par
ents
and
stud
ents
. If
som
ethi
ng g
oes
abov
e yo
urhe
ad, p
leas
e d
o no
t wor
ry. O
nce
you
hav
ean
ade
qua
te b
ackg
roun
d, r
ead
the
mat
eria
lag
ain.
It t
ook
me
man
y at
tem
pts
toun
ders
tand
the
mat
eria
l.