1 of 46

A Glimpse�of Infinity

Brahadeesh Sankarnarayanan

Research Scholar

Department of Mathematics

Indian Institute of Technology Bombay

2 of 46

Srinivasa Ramanujan�(1887–1920)�The Man Who Knew Infinity

Biography by Robert Kanigel (1991)
Biographical movie directed by Mathew Brown (2015)

3 of 46

Goddess Namagiri�Ramanujan’s family deity

“An equation for me has no meaning unless it expresses a thought of God.” – Ramanujan

4 of 46

G. H. Hardy�(1877–1947)�English mathematician

Number theory and mathematical analysis
“A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.” – Hardy

5 of 46

Number Theory

6 of 46

Infinite Continued Fractions

7 of 46

Infinite Continued Fractions: example

8 of 46

P. C. Mahalanobis�(1893–1972)�Founder of the Indian Statistical Institute (ISI)

Find the number x of a particular house on a street where the houses are numbered 1,2,3,…,n.
The house with number x has the property that the sum of the house numbers to the left of it is equal to the sum of the house numbers to the right of it.
Assume 50 < n < 500.

9 of 46

“...It was clear that the solution should obviously be a continued fraction; I then thought, which continued fraction? And the answer came to my mind.” – Ramanujan

P. C. Mahalanobis�(1893–1972)�Founder of the Indian Statistical Institute (ISI)

10 of 46

Infinite Nested Radicals

11 of 46

Infinite Nested Radicals

12 of 46

Real Analysis

13 of 46

Infinite Series

14 of 46

Infinite Series: Geometric series

15 of 46

Infinite Series: Ramanujan’s π formula

16 of 46

Infinite Series (revisited):�Grandi’s series�Guido Grandi (1671–1742)�Italian monk, philosopher and mathematician

1 – 1 + 1 – 1 + 1 – 1 + ··· = ?

(1 – 1) + (1 – 1) + (1 – 1) + ··· = 0

1 – (1 – 1) – (1 – 1) – (1 – 1) - ··· = 1

17 of 46

Infinite Series (revisited): Alternating series�Bernhard Riemann (1826–1866)�German mathematician

18 of 46

Infinite Series (revisited): Alternating series

19 of 46

Geometry

20 of 46

Georg Cantor�(1845–1918)�German mathematician�Father of modern set theory

21 of 46

The Cantor set

22 of 46

23 of 46

24 of 46

Helge von Koch�(1870–1924)�Norwegian mathematician

25 of 46

The Koch snowflake

26 of 46

27 of 46

28 of 46

Wacław Sierpiński�(1882–1969)�Polish mathematician

29 of 46

The Sierpiński triangle

30 of 46

31 of 46

32 of 46

33 of 46

Benoit Mandelbrot�(1924–2010)�Polish-born French–American mathematician

34 of 46

The Mandelbrot set

35 of 46

36 of 46

37 of 46

38 of 46

Pierre François Verhulst�(1804–1849)�Belgian mathematician

39 of 46

The Logistic Map

40 of 46

41 of 46

42 of 46

A puzzle to take home

43 of 46

44 of 46

The staircase paradox

45 of 46

46 of 46

Thank you!