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Presented by:
Mrs.Mamta Verulkar
Vimaltai Tidke convent & Jr.
College, Beltarodi ,
Nagpur
05/08/15 1India's Contribution to Geometry
• Mathematicsisthescienceof
structure, orderandrelationthat
developsgraduallyfrombasic
practiceofcounting, measuringand
describingtheshapesofobjects.
05/08/15 2India's Contribution to Geometry
Geometrical Proof
• AGeometricalproofinvolveswriting
reasoned, logicalexplanationsthatuse
definitions, axioms, postulatesand
previouslyprovedtheoremstoarrive
ataconclusionaboutageometric
statement.
05/08/15 3India's Contribution to Geometry
SquareRootof2
• Theoldestvalueofsquarerootof2is
obtainedfromBabyloniantimes
(1600B.C.)
• Thesexagesimalequivalentvalueof
squarerootof2asgiveninthe
Sulvasutra is1.24, 51, 10, 37
05/08/15 4India's Contribution to Geometry
Proof
• Construct ABC such that
• AB = BC = n
• AC = m, B =
•
Supposemandnareintegers
• Let us assume that m:n is the ratio
A
B C
m
n
05/08/15 5India's Contribution to Geometry
n
∆
∠ 90
0
n
m
AB
AC
=
2=
n
m
Draw arcs of lengths m
and n with centre A
AB = AD
AC = AE
• BAC and DAE
coincide
• ABC ADE
byS.A.S. test
05/08/15 India's Contribution to Geometry 6
∠ ∠
∆ ≅
EBF =
05/08/15 India's Contribution to Geometry 7
0
90∠
(Rightangle)
∠ BEF= =
ishalfofrightangle
BEFisrightisosceles
triangle.
∆
0
45
Hence BE = m– n
BF = m – n
• DF = m – n
• Then FDC isalso right
Isoscelestriangle
• FC = n – ( m – n )
= 2n - m
05/08/15 India's Contribution to Geometry 8
⇒
∆ EBF ∆ CDF≅
ABC & FDC are similar
05/08/15 India's Contribution to Geometry 9
∆ ∆
FC
AC
FD
AB
=
FD
FC
AB
AC
=
nm
mn
n
m
−
−
=
2
But triangle ABC is biggerin
size than FDC
i.e m > 2n-m
n > m-n
But assumption is is in
lowest form.
i.e is also in
lowest form
05/08/15 India's Contribution to Geometry 10
n
m
nm
mn
−
−2
• i.e m and n have common integral
factors but it contradicts the
assumption m:n is in lowest form.
• Therefore cannot be expressed
in form (rational form) hence
is irrational.
05/08/15 India's Contribution to Geometry 11
2
n
m
2
DuttaB.B. (1932)
Considertwosquareswhosesidesareofunit
length.
Dividethesecondsquareintothree
equalstripesI, IIanIII.
05/08/15 12India's Contribution to Geometry
III III
• Sub-dividethelast
stripintothree
smallsquaresIII1,
III2, III3 of
sides⅓rd
each.
05/08/15 India's Contribution to Geometry 01
III2
III1
III3
III III
III1
• ThenonplacingI,
IIandIII1
about
thefirstsquareS
inthepositionI’,
II’ andIII1
’ anew
squareisformed.
05/08/15 India's Contribution to Geometry 01
I
II III1
Nowdivideeachofthe
portionsIII2
andIII3
into
fourequalstripes.
05/08/15 15India's Contribution to Geometry
Arrange four partsof III2,III3
asshown in
figure
• Introducing asmall square
at thesouth-east corner, a
largesquarewill beformed.
05/08/15 India's Contribution to Geometry 16






++
)4)(3(
1
3
1
1
Each side of which
is equal to
05/08/15 India's Contribution to Geometry 17
In order to get theequivalent
arearemovetwo thin stripes,
say of width x from either side
of thesquare.
05/08/15 India's Contribution to Geometry 18
2
2
)4)(3(
1
)4)(3(
1
3
1
12 





=−





++ xx
Neglecting x2
as to small
2
)4)(3(
1
12
17
2 





=





x
)34)(4)(3(
1
2
1
17
12
)4)(3(
1
2
=

















=x
2
6271.41421568
)34)(4)(3(
1
)4)(3(
1
3
1
1
=
=
−++=Side of a square
after subtracting x
• In decimal form this value equals
1.41421568627
Today’s value of is
1.41421356237,a difference of
0.000021239
05/08/15 India's Contribution to Geometry 19
2
References
• Book by Saraswati Amma.
• Ganit Kaumidi (Mathematics book)
• www.cut-the-knot.orgproofs/sq_root.shtml
• httphttp://www.math.ubc.ca/~
• http://www.math.ubc.ca/~cass/courses/m309
-0l1a/kong/sulbasutra_geometry.htm
• Guided by Professor DR.Anant Vyawhare.
05/08/15 India's Contribution to Geometry 01
05/08/15 India's Contribution to Geometry 21

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Mamta verulkar[1]

Editor's Notes

  1. Mamta R. Verulkar
  2. Geometry
  3. By Pythagorus theorem
  4. BEF is half of right angle BEF is right isosceles triangle.
  5. CDF
  6. Side of a square after subtracting x