Digital Design
(CS F215)
Harikrishnan N B
Assistant Professor
Department of Computer Science and Information Systems
BITS Pilani K K Birla Goa Campus
Lecture 2:
05 August 2025
Agenda
Number System
Solve Base Conversion Problems
Connection Between Base Conversion and Chaos Theory :)
Question 1: Convert the decimal no. (105)10 into its binary
(base 2)
Question 1: Convert the decimal no. (105)10 into its trinary
(base 3)
Question 1: Convert the decimal no. (105)10 into its octal
(base 8)
Question 2: Convert the decimal no. (0.625)10 into its
binary (base 2)
Question 2: Convert the decimal no. (0.625)10 into its octal
(base 8)
Question 2: Convert (743)8 into its decimal (base 10)
Question 2: Convert (432)5 = (?)7
Question 2: Convert (101)7 = (?)3
Convert (101101)2 = (?)10
Most commonly used Number System
Decimal
Binary
Octal
Hexadecimal
Decimal, Binary, Octal Hexadecimal equivalence number
representation
Convert (1101001.111)2 = (??)8
Convert (1101001.111)2 = (??)16
Connection Between Base Conversion & Chaos Theory
Consider the following algorithm
X = 0.625
Store [2x]
x
2x-[2x]
Can I extend this to any base ‘r’
X = 0.625
Store [rx]
x
rx-[rx]
How does Tr (x) = rx-[rx] looks like?
Let us say r = 2
What is Chaos?
Isaac Newton, 17th Century
Pierre-Simon, marquis
de Laplace
Laplace’s Demon
According to Laplace’s world view, the universe is completely
deterministic, there is no uncertainty and unpredictability.
Gösta
Mittag-Leffler
(1885)
King Oscar II of
Sweden and
Norway
The birth of chaos theory goes back to
Mittag-Leffler’s idea of organizing a
mathematics international prize
competition honouring the 60th
birthday of King Oscar II of Sweden
and Norway. The competition was
announced in 1885 and winning prize
consisted of a gold medal and 2,500
Swedish kronor.
Henri Poincaré
(1889)
N Body Problem
For a system of arbitrarily many mass points that attract each other
according to Newton’s law, assuming that no two points ever collide,
find a series expansion of the coordinates of each point in known
functions of time converging uniformly for any period of time. The
Prize Jury (1885)
Three body problem
Implications:
Is the solar system
stable?
Will the sun rise
tomorrow?
Henri Poincaré
Birth of chaos
This was the birth of chaos theory, however the term
‘chaos’ had to wait another 85 years to be coined. This
was after the discovery of sensitive dependence on initial
conditions by Edward Lorenz in 1961
Edward Lorenz Experiment
Butterfly Effect
This sensitive dependence on initial conditions making the long-term
prediction impossible is popularly known as the Butterfly Effect - a
butterfly flapping its wings in Bengaluru leading to unpredictable
changes in Delhi weather. Later in 1975, James A. Yorke named this
phenomena as chaos theory.
Deterministic Chaos
A dynamical system D (D is T: X —>X, where T is a map from set X onto itself) is chaotic if it satisfies the
following properties:
1.
2.
3.
The periodic points in D are dense.
D is topologically transitive.
D is sensitive to initial conditions.
Logistic Map
31
Resources
1)
2)
3)
Images - Google
Read this: https://www.mittag-leffler.se/about-us/history/prize-competition/
Doubling Map:
https://marksmath.org/classes/Spring2017ComplexDynamics/text/subsectiondoubling_map.html