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 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 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:

Cantor’s array for reals…

Construct the following number 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: 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?