0.999... Repeating Is Equal To 1, But Something Like It Is Not (Introduction To The Surreal Numbers)
4,043
by Super User, 10 years ago
Some of the comments appear to be based solely on reading the title or the thumbnail--so I've changed the title to more accurately reflect the contents. The surreal numbers contain the real numbers, and 0.9 repeating is equal to 1 in both. However, we think about 0.9 repeating conceptually as some number that is infinitely smaller than 1. There is such a number in the surreal numbers, and it is not equal to 1!
Technical Points
The surreal numbers are a pretty complicated logical system. One point to make is that the surreal numbers contain the real numbers, and so 0.999.. = 1 since this is true for the real numbers.
However, there is a psychology about 0.999... repeating as a number that is infinitesmally smaller than 1. The surreal number {0, 1/2, 3/4, 7/8,... | 1} is infinitesmally smaller than 1 and would satisfy our psychological idea of 0.999.. repeating. I believe {0, 9/10, 99/100, 999/1000,...| 1} would also represent the same surreal number; however, I have not been able to verify that so I did not include it in the video.
When fact-checking this video, I came across a number of conflicting arguments that seemed valid.
http://math.stackexchange.com/questions/926581/in-the-surreal-numbers-is-it-fair-to-say-0-9-repeating-is-not-equal-to-1
http://forums.xkcd.com/viewtopic.php?f=17&t=76752
http://en.wikipedia.org/wiki/0.999...#Hackenbush
I also did my best to work through John Conway's book that introduces surreal numbers called "On Numbers and Games."
http://en.wikipedia.org/wiki/On_Numbers_and_Games
I welcome thoughtful comments about the surreal numbers.
If you like my videos, you can support me at Patreon: http://www.patreon.com/mindyourdecisions
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http://goo.gl/BDlEkB
Here's a more detailed description of each book...
"The Joy of Game Theory" shows how you can use math to out-think your competition. (rated 4.2/5 stars on 17 reviews) http://amzn.to/1uQvA20
"Math Puzzles Volume 1" features classic brain teasers and riddles with complete solutions for problems in counting, geometry, probability, and game theory. Volume 1 is rated 4.6/5 stars on 9 reviews. http://amzn.to/1GhUUSH
"Math Puzzles Volume 2" is a sequel book with more great problems. http://amzn.to/1NKbyCs
"Math Puzzles Volume 3" is the third in the series. http://amzn.to/1NKbGlp
"40 Paradoxes in Logic, Probability, and Game Theory" contains thought-provoking and counter-intuitive results. (rated 4.9/5 stars on 7 reviews) http://amzn.to/1LOCI4U
"The Best Mental Math Tricks" teaches how you can look like a math genius by solving problems in your head http://amzn.to/18maAdo
"Multiply Numbers By Drawing Lines" This book is a reference guide for my video that has over 1 million views on a geometric method to multiply numbers. http://amzn.to/XRm7M4
Technical Points
The surreal numbers are a pretty complicated logical system. One point to make is that the surreal numbers contain the real numbers, and so 0.999.. = 1 since this is true for the real numbers.
However, there is a psychology about 0.999... repeating as a number that is infinitesmally smaller than 1. The surreal number {0, 1/2, 3/4, 7/8,... | 1} is infinitesmally smaller than 1 and would satisfy our psychological idea of 0.999.. repeating. I believe {0, 9/10, 99/100, 999/1000,...| 1} would also represent the same surreal number; however, I have not been able to verify that so I did not include it in the video.
When fact-checking this video, I came across a number of conflicting arguments that seemed valid.
http://math.stackexchange.com/questions/926581/in-the-surreal-numbers-is-it-fair-to-say-0-9-repeating-is-not-equal-to-1
http://forums.xkcd.com/viewtopic.php?f=17&t=76752
http://en.wikipedia.org/wiki/0.999...#Hackenbush
I also did my best to work through John Conway's book that introduces surreal numbers called "On Numbers and Games."
http://en.wikipedia.org/wiki/On_Numbers_and_Games
I welcome thoughtful comments about the surreal numbers.
If you like my videos, you can support me at Patreon: http://www.patreon.com/mindyourdecisions
Connect on social media. I update each site when I have a new video or blog post, so you can follow me on whichever method is most convenient for you.
My Blog: http://mindyourdecisions.com/blog/
Twitter: http://twitter.com/preshtalwalkar
Facebook: https://www.facebook.com/pages/Mind-Your-Decisions/168446714965
Google+: https://plus.google.com/108336608566588374147/posts
Pinterest: https://www.pinterest.com/preshtalwalkar/
Tumblr: http://preshtalwalkar.tumblr.com/
Instagram: https://instagram.com/preshtalwalkar/
Patreon: http://www.patreon.com/mindyourdecisions
Newsletter (sent about 2 times a year): http://eepurl.com/KvS0r
My Books
Here's a listing of all my books
http://goo.gl/BDlEkB
Here's a more detailed description of each book...
"The Joy of Game Theory" shows how you can use math to out-think your competition. (rated 4.2/5 stars on 17 reviews) http://amzn.to/1uQvA20
"Math Puzzles Volume 1" features classic brain teasers and riddles with complete solutions for problems in counting, geometry, probability, and game theory. Volume 1 is rated 4.6/5 stars on 9 reviews. http://amzn.to/1GhUUSH
"Math Puzzles Volume 2" is a sequel book with more great problems. http://amzn.to/1NKbyCs
"Math Puzzles Volume 3" is the third in the series. http://amzn.to/1NKbGlp
"40 Paradoxes in Logic, Probability, and Game Theory" contains thought-provoking and counter-intuitive results. (rated 4.9/5 stars on 7 reviews) http://amzn.to/1LOCI4U
"The Best Mental Math Tricks" teaches how you can look like a math genius by solving problems in your head http://amzn.to/18maAdo
"Multiply Numbers By Drawing Lines" This book is a reference guide for my video that has over 1 million views on a geometric method to multiply numbers. http://amzn.to/XRm7M4
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Super User uploaded a new media, 0.999... Repeating Is Equal To 1, But Something Like It Is Not (Introduction To The Surreal Numbers)
10 years ago
