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Infinity and
Beyond!
A prelude to Infinite Sequences and
Series (Chp 10)
Infinity and Fractals…
• Fractals are self-similar objects whose
overall geometric form and structure repeat
at various scales they provide us with a
“glimpse” into the wonderful way in which
nature and mathematics meet.
• Fractals often arise when investigating
numerical solutions of differential (and other
equations).
… go to XAOS
Paradoxes of Infinity
• Zeno
– Motion is impossible
– Achilles and the
tortoise
– Math prof version
Sizes of Infinity…
• How can you decide if two sets are the
same size?
• How many fractions are there between
0 and 1?
• Which is bigger – the set of counting
numbers or the et of fractions?
Cantor (and the concept of countable and
uncountable sets)
• In the 1870’s Cantor
began his great work on
the theory of sets and in
so doing startled the
mathematical world with
fundamental discoveries
concerning the nature of
infinity.
• Cantor developed the idea
of countable and
uncountable sets…
Why the number of Rationals is the
same as the number of Naturals
1/1
2 /1
3/1
4 /1
1/ 2
2/2
3/ 2
4/2
1/ 3
2/3
3/ 3
4/3
1/ 4...
2 / 4...
3/ 4...
4 / 4...
Since the rationals can be put in a 1:1 relation with the natural
numbers they are a countable set and the size of the set of
rationals is the same as the naturals – these “infinities” are the
same size!
The Reals are NOT countable…
• Cantor come to the following
remarkable conclusion. He showed
that one cannot “count” the reals. To
see this consider how you would
answer the question: “Do the real
numbers form a countable set?”
• What are the answers you could give?
Reductio ad Absurdum
• Let’s assume that we CAN make a 1:1
relationship between the reals and the
natural numbers.
• Any real number can be written as a
decimal expansion:
1/ 3  0.3333333...
2  1.41421356237309505...
Cantor’s array for reals…
1  0.a1a2 a3a4 a5 ...
2  0.b1b2b3b4b5 ...
3  0.c1c2c3c4c5 ...
4  0.d1d 2 d 3d 4 d 5 ...

Construct the following number
0. 1 2 3 4 5 ...
 1  a1 ,  2  b2 ,  3  c3
• This number can’t be in the table of
reals, therefore the original assumption
is false!
Cantor’s Unsettling Conclusion…
• The infinity of real numbers is bigger
than the infinity of integers!
Some Infinites are bigger than
other Infinities!
A challenge…
• Early in the course we encountered the
following function:
0
if x is rational
f ( x)  1 if x is irrational
The Riemann Sum definition of the didn’t
work! What do you think is the area under
this curve between x = 0 and x = 1 (and
why)?
The Koch Snowflake and Infinite
Sequences…
What is a Koch Snowflake?
• How “long” is a section of the Koch
Snowflake between x = 0 and x = 1?
• Anything else odd about this?
– What “dimension” is it?
– Can you differentiate it?