Presented by:

Mrs.Mamta Verulkar

Vimaltai Tidke convent & Jr. College, Beltarodi ,

Nagpur

04/18/23 1India's Contribution to Geometry

• Mathematics is the science of structure, order and relation that develops gradually from basic practice of counting, measuring and describing the shapes of objects.

04/18/23 2India's Contribution to Geometry

Geometrical Proof

• A Geometrical proof involves writing reasoned, logical explanations that use definitions, axioms, postulates and previously proved theorems to arrive at a conclusion about a geometric statement.

04/18/23 3India's Contribution to Geometry

Square Root of 2

• The oldest value of square root of 2 is obtained from Babylonian times (1600 B.C.)

• The sexagesimal equivalent value of square root of 2 as given in the Sulvasutra is 1.24, 51, 10, 37

04/18/23 4India's Contribution to Geometry

Proof• Construct ABC such that

• AB = BC = n • AC = m, B =

•

Suppose m and n are integers

• Let us assume that m:n is the ratio in its lowest term.

A

B C

m

n

04/18/23 5India's Contribution to Geometry

n

900

n

m

AB

AC

2n

m

Draw arcs of lengths m and n with centre A

AB = AD AC = AE

• BAC and DAE coincide• ABC ADE by S.A.S. test

04/18/23 India's Contribution to Geometry 6

EBF =

04/18/23 India's Contribution to Geometry 7

090(Right angle)

BEF = = is half of right angle

BEF is right isosceles triangle.

045

Hence BE = m – n

BF = m – n

• DF = m – n• Then FDC is also right

Isosceles triangle

• FC = n – ( m – n )

= 2n - m

04/18/23 India's Contribution to Geometry 8

EBF CDF

ABC & FDC are similar

04/18/23 India's Contribution to Geometry 9

FC

AC

FD

AB

FD

FC

AB

AC

nm

mn

n

m

2

But triangle ABC is bigger in 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

04/18/23 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.

04/18/23 India's Contribution to Geometry 11

2n

m

2

Dutta B.B. (1932)Consider two squares whose sides are of unit length.

Divide the second square into three equal stripes I, II an III.

04/18/23 12India's Contribution to Geometry

III III

• Sub-divide the last

strip into three small squares III1, III2, III3 of

sides ⅓rd each.

04/18/23 India's Contribution to Geometry 01

III2

III1

III3

III III

III1

• Then on placing I,

II and III1 about

the first square S

in the position I’,

II’ and III1’ a new

square is formed.

04/18/23 India's Contribution to Geometry 01

I

II III1

Now divide each of the portions III2 and III3 into four equal stripes.

04/18/23 15India's Contribution to Geometry

Arrange four parts of III2,III3 as shown in figure

• Introducing a small square at the south-east corner, a large square will be formed.

04/18/23 India's Contribution to Geometry 16

)4)(3(

1

3

11

Each side of which is equal to

04/18/23 India's Contribution to Geometry 17

In order to get the

equivalent area remove

two thin stripes , say of

width x from either side

of the square.

04/18/23 India's Contribution to Geometry 18

22

)4)(3(

1

)4)(3(

1

3

112

xx

Neglecting x2 as to small

2

)4)(3(

1

12

172

x

)34)(4)(3(

1

2

1

17

12

)4)(3(

12

x

2

6271.41421568

)34)(4)(3(

1

)4)(3(

1

3

11

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

04/18/23 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.

04/18/23 India's Contribution to Geometry 01

04/18/23 India's Contribution to Geometry 21