0.999... Repeating Is Equal To 1, But Something Like It Is Not (Introduction To The Surreal Numbers)
2,677
by Super User, 8 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
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.
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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)
8 years ago
