HN Theater has aggregated all Hacker News stories and comments that mention Insights into Mathematics's video "Inconvenient truths about sqrt(2) | Real numbers and limits Math Foundations 80 | N J Wildberger".
Youtube Summary
This video begins a discussion on the role of irrationality in mathematics, starting with the "square root of 2". The difficulties with this concept go back to the ancient Greeks, as the Pythagoreans realized that the side and diagonal of a square were incommensurable. The Greeks realized that there was no rational number whose square was exactly two, a result which historically appeared in Euclid. In the modern age this idea that there were "irrational numbers" that could be incorporated into the Hindu-Arabic decimal number system was introduced by Stevin in 1585.
There are these days three approaches to "sqrt(2)": an applied one dealing with approximations, an algebraic one involving a finite field extension of the rational numbers, and an analytic one which attempts to apply the square root algorithm to assign to sqrt(2) an infinite decimal. It is this last approach which does not work, leading to serious logical problems with modern analysis.
This lecture is part of the MathFoundations series, which tries to lay out proper foundations for mathematics, and will not shy away from discussing the serious logical difficulties entwined in modern pure mathematics. For those interested in this topic, be sure to watch the further videos in this series which go into considerable detail about the various problems with real numbers as infinite decimals, equivalence classes of Cauchy sequences, and Dedekind cuts. None of them work.
************************ 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.
*********************** Here are all the Insights into Mathematics Playlists:
Elementary Mathematics (K-6) Explained: https://www.youtube.com/playlist? list=PL8403C2F0C89B1333 Year 9 Maths: https://www.youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK Ancient Mathematics: https://www.youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e Wild West Banking: https://www.youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH Sociology and Pure Mathematics: https://www.youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua Old Babylonian Mathematics (with Daniel Mansfield): https://www.youtube.com/playlist? list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov Math History: https://www.youtube.com/playlist?list=PL55C7C83781CF4316 Wild Trig: Intro to Rational Trigonometry: https://www.youtube.com/playlist?list=PL3C58498718451C47 MathFoundations: https://www.youtube.com/playlist?list=PL5A714C94D40392AB Wild Linear Algebra: https://www.youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp Famous Math Problems: https://www.youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf Probability and Statistics: An Introduction: https://www.youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse Boole's Logic and Circuit Analysis: https://www.youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb Universal Hyperbolic Geometry: https://www.youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI Differential Geometry: https://www.youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP Algebraic Topology: https://www.youtube.com/playlist?list=PL6763F57A61FE6FE8 Math Seminars: https://www.youtube.com/playlist?list=PLBF39AFBBC3FB30AF ************************
And here are the Wild Egg Maths Playlists:
Triangle Centres: https://www.youtube.com/watch?v=iLBGXDSUohM&list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM Six: An elementary course in pure mathematics: https://www.youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS Algebraic Calculus One: https://www.youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB Algebraic Calculus Two: https://www.youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG- м
I only "believe" in the computable numbers - that can be defined by an algorithm at arbitrary precision.
That includes all rationals along with irrational that we have a way to approximate by algorithms - like sqrt(2) and PI. It does not include the incalculable numbers like Omega.
Being linked to algorithms, such numbers are only countable many, so they cannot contain "most" of irrational and therefore they lack most of "Real numbers".
You might find Norman J Wildebergers Youtube videos on the subject interesting: http://www.youtube.com/watch?v=REeaT2mWj6Y He argues quite convincingly that real numbers like pi shouldn't be seen as numbers and that infinity does not exist. It's very fascinating reasoning to listen to even if you don't agree with all of it.
HN Theater is an independent project and is not operated by
Y Combinator or any of the video hosting platforms linked to on this
site.
Lorem ipsum dolor sit amet,
consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et
dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation
ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor
in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla
pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui
officia deserunt mollit anim id est laborum.