Transcript fract2
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: f ( x) 0 if x is rational 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?