Page 56 - Ramanujan Yatra
P. 56
= 1 x 1 + Remainder √2 = 1 + Remainder
As Silver Rectangles repeat, their aspect ratios (L:B) is maintained. They are similar rectangles. Hence the ratio of Length (= 1 unit) and breath (= 2sq + remainder) is maintained for all.
The rectangles get smaller and smaller and smaller, but retain their aspect ratio to their mother rectangle.
So, we get an equation:
Remainder = 1
2 + remainder
Substituting this in the previous formula, we get
√2 = 1 + 1
2 + remainder
=1 + 1 1
2 + 2 + reaminder
=1 + 1 2+
This is continued fraction for √2. Let us see what we get when we converge fractions at every level:
3 = 1.5; 7 = 1.4; 17 = 1.4166 2 5 12
41 = 1.4137; 99 = 1.4142 .... 29 70
The last one √2 = 1.4142 .... is recognizable. This is the beauty of Continued Fraction.
VSS Sastry is a math communicator. He has worked in all areas of communication like puppetry, illustration, cartoons, short films, story-telling, modeling, etc. to deliver mathematics to masses.
11
2 + 2 + remainder
6— 1
6— 1
47
Ramanujan
YATRA
6— 1
6— 1
6— 1 6
