This lecture tries to clarify the big gap between the (finite) sequences we introduced in the last lecture, and "infinite" or "ongoing sequences" (we introduce the term "on-sequence") as are found in Sloane's Online Encyclopedia of Integer Sequences. We concentrate discussion on three such: the basic on-sequence of natural numbers, the on-sequence of prime numbers, and the on-sequence of Catalan numbers.
We lay out Euclid's well-known proof that given any finite list of primes we can find a prime not in it, and suggest some computational snags that perhaps we might worry about. The Catalan sequence of combinatorics fame has many possible definitions and formulas, illustrating the crucial distinction: finite sequences are specified by the elements of the sequence themselves, while "on-sequences'' can only be specified more indirectly, and generally non-uniquely.
Along the way we point out that the grammatical closeness of the terms finite and infinite suggest a dichotomy which is not really born out in experience.
Video Content:
00:00 Introduction
3:19 Course started with a "sequence"
6:59 More mundane concerns
8:45 Differences between finite and infinite sequences
12:13 New terminology
14:22 Sequence of prime numbers
18:19 Are there really "infinitely many" primes??
21:42 Fundamental Theorem of Arithmetic
25:27 Another definition of c(n)
28:22 Recursive definition
33:01 Difference between a finite sequence and an on-sequence
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Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of the lectures for various Playlists: great for review, study and summary.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/Norman_Wildberger
My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things.
Online courses will be developed at openlearning.com. The first one, already underway is Algebraic Calculus One at https://www.openlearning.com/courses/algebraic-calculus-one/ Please join us for an exciting new approach to one of mathematics' most important subjects!
If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at https://www.patreon.com/njwildberger Your support would be much appreciated.
***********************
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/Norman_Wildberger
My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things.
Online courses are being developed at openlearning.com. The first one, already underway, is Algebraic Calculus One at https://www.openlearning.com/courses/algebraic-calculus-one/ Please join us for an exciting new approach to one of mathematics' most important subjects!
If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at Patreon, https://www.patreon.com/njwildberger . Your support would be much appreciated.
Here are the Insights into Mathematics Playlists:
https://www.youtube.com/playlist?list=PL55C7C83781CF4316
https://www.youtube.com/playlist?list=PL3C58498718451C47
https://www.youtube.com/playlist?list=PL5A714C94D40392AB
https://www.youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
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https://www.youtube.com/playlist?list=PL8403C2F0C89B1333
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https://www.youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
https://www.youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
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