Numbering Systems
and Infinity
Moreno Marzolla
Dip. di Informatica—Scienza e Ingegneria (DISI)
Università di Bologna
http://www.moreno.marzolla.name/
Copyright © 2014, Moreno Marzolla, Università di Bologna, Italy
(http://www.moreno.marzolla.name/teaching/CS2013/)
This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License
(CC-BY-SA). To view a copy of this license, visit http://creativecommons.org/licenses/bysa/3.0/ or send a letter to Creative Commons, 543 Howard Street, 5th Floor, San
Francisco, California, 94105, USA.
Image credits: the Web; The Computational Beauty of Nature website
http://mitpress.mit.edu/sites/default/files/titles/content/cbnhtml/home.html
Complex Systems
2
Summary
●
Infinity plays an important role in several topics we will
address (e.g., fractals), therefore it is appropriate that
we give a closer look at it
Complex Systems
3
Zeno's Paradox
●
●
●
Achille runs 1 Km in 1 min
The Tortoise runs 1 / 2 Km in
1 min
The Tortoise is given 1Km
advantage
–
–
–
●
After 1 min, the Tortoise is
1 / 2 Km ahead
After 1 / 2 min the Tortoise is
1 / 4 Km ahead
After 1 / 4 min the Tortoise is
1 / 8 Km ahead
The Tortoise is always ahead
of Achille !?
Complex Systems
4
Enter Infinity
●
To reach the Tortoise, Achille needs to run for
1 1 1
1+ + + +…
2 4 8
...
1/8
1
1/2
1/16
1/4
●
We now know that this sum of infinite elements is
indeed finite!
∞
1
∑i=0 a = 1−a , if 0<a<1
i
Complex Systems
5
Counting numbers: naturals
●
There are infinitely many natural numbers (1, 2, 3, …)
–
–
1
●
Even naturals (2, 4, 6, …) are a proper subset of naturals...
...so there seem to be less even numbers than natural
numbers, right?
2
3
4
5
6
7
8
9
10
11
12
...
Wrong!!
–
There are as many naturals as there are even naturals
1
2
3
4
5
6
2
4
6
8
10
12
Complex Systems
6
The Extraordinary Hotel
●
The Extraordinary Hotel or the Thousand and First
Journey of Ion the Quiet
–
●
Attributed to Stanislaw Lem, but probably written by the
russian mathematician Naum Yakovlevich Vilenkin
You are the director of an hotel with (countably)
infinitely many rooms {1, 2, 3, ...}
Complex Systems
7
The Extraordinary Hotel
●
●
The hotel is fully booked, i.e., all rooms are already
occupied by a guest
A new guest arrives. What should you do?
Hotel (infinitely many rooms, fully booked)
g
1
2
3
4
5
6
7
8
...
New guest
Complex Systems
8
The Extraordinary Hotel
●
●
Again, the hotel is fully booked.
Now, (countably) infinitely many new guests arrive at
the same time. What should you do?
Hotel (infinitely many rooms, fully booked)
... g4
g3 g2
g1
1
2
3
4
5
6
7
8
...
Infinitely many new guests
Complex Systems
9
The Extraordinary Hotel
●
●
●
Suppose that there are infinitely many hotels, each
with infinitely many rooms.
All hotels are fully booked, but all of them (except for
one) have to be closed down for restructuring. The
one hotel which is not to be closed is empty.
Therefore, you should try to fit infinitely many guests
from infinitely many hotels into a single (infinite) hotel.
What should you do?
Hotel (infinitely many rooms, empty)
...
g1,4 g1,3 g1,2 g1,1
...
g2,4 g2,3 g2,2 g2,1
...
g3.4 g3,3 g3,2 g3,1
1
2
3
Complex Systems
4
5
6
7
8
...
10
Counting numbers: rationals
●
●
(Positive) rational numbers
are those of the form p / q,
with p, q naturals
The set of rational
numbers if countable, and
therefore we can define a
one-to-one mapping from
the set of naturals
–
Therefore, there are as
many natural numbers as
there are rationals
1/1
1/2
2/2
1/3
2/3
3/3
1/4
2/4
3/4
4/4
1/5
2/5
3/5
4/5
5/5
1/6
2/6
3/6
4/6
5/6
6/6
...
Complex Systems
11
Rationals
●
There are infinitely many rationals between any two
rational numbers (p / q) < (r / s) (p, q, r, s integers)
–
●
Proof: (p/q + r/s) / 2 = (sp + rq) / 2qs is a rational number
between p/q and r/s
A single rational number can be used to encode the
whole content of the RAM of your PC!
–
Proof: the content of the memory of your PC is just a very
long but finite sequence of 0 and 1, e.g.,
0110100100101010. Write the sequence as
0.0110100100101010 and you have a rational number
encoding the whole value.
Complex Systems
12
Counting numbers: irrationals
●
Irrational numbers are
those who can not be
expressed as p / q
–
●
They have infinite,
nonperiodic decimal
representation
The set of irrationals is
not countable
–
Therefore, there are more
irrational numbers than
rational (and natural)
numbers
0.
x11
x12
x13
x14
x15
0.
x21
x22
x23
x24
x25
0.
x31
x32
x33
x34
x35
0.
x41
x42
x43
x44
x45
0.
...
Complex Systems
13
Counting numbers: reals
●
Consider a closed interval I = [a, b], and the open halfinfinite interval J = [0, +∞). Are there more real
numbers in J than I?
Complex Systems
14
Counting numbers: reals
●
●
Consider a closed interval I = [a, b], and the open halfinfinite interval J = [0, +∞). Are there more real
numbers in J than I?
Answer: both contain the same (infinite) quantity of
real numbers
Complex Systems
15
Conclusions
●
●
●
●
●
Natural numbers occupy fixed and equally-spaced
positions on the real line
There are “as many” natural numbers as there are
rational numbers
Yet, between any two naturals there are infinitely many
rationals
There are “more” real numbers than rational (or
natural) numbers
There are “as many” real numbers in a (finite) segment
than in the whole real line!
Complex Systems
16