Surreal Numbers (writing the first book) - Numberphile
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- เผยแพร่เมื่อ 26 มิ.ย. 2016
- Donald Knuth coined the term "Surreal Numbers" and wrote the first book about them after lunch with the man who devised them, John Conway.
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"My wife did visit me a couple times that week. Because we always wondered what it would be like to have an affair in a hotel room." The man, the legend Donald Knuth.
nitowa1 wonder if there was any LaTeX involved in those visits
average numberphile enjoyer
Kids on the playground: "Yeah, well I'm infinity plus one!"
Grade school: "No such thing."
First year calculus: "We have to be careful not to treat infinity as a number, but it's useful for some calculations."
Mathematicians on a napkin over lunch: "The kid is onto something."
lol
You have to know the rules before you can break them.
@@jacobshirley3457 I disagree strongly in the case of mathematics. All you need to do is have respect for the concept of rules and consistency. That is, it's okay to break the rules if you can still find a way for things to be consistent, and once you have done so, you have found something truly marvelous.
and now we know that at least some math is completely absurd and should be discarded. There is no such thing as infinity +1 as by definition infinity is infinitely greater than the largest known number and adding one still yields infinity.
@@kennethgee2004 That's only with cardinal numbers. In ordinals, there is a family of infinities. There literally is infinity + 1. Aleph Null and such. There are also smaller and larger infinities. The whole numbers (1, 2, 3, 4, ... ∞) is a smaller infinity than [0, 1], or the real numbers that are in between 0 and 1 are larger than all the whole numbers, it's a larger infinity. 0, ... 0.00001 ... 0.0001 ... 0.3, ..... 0.6, ... 0.99 ... 0.99999999 ... 0.999999999999999999 ... 1.
It's always fun to see a legendary genius speaking as a normal guy in a one-on-one interview.
Some quick googling turns up "The theory of well-quasi-ordering: A frequently discovered concept", which cites a French paper in footnote 9 as the origin of the name surordinal, almost certainly the one Knuth refers to. Someone want to tell him?
'In 1967, writing in French, Pierre Jullien [9] generalized the concept of an ordinal slightly to what he called a “surordinal.” He announced several results, among which is the statement that the class of “surordinaux” are wqo.9. P. JULLIEN, Theorie des relations. - Sur la comparaison des types d’ordres disperses, C. R. Acad. Sci. Paris, Se’r. A 264 (1967), 594-595.'
Cool
Name awesome! I hope he got this information
Is this the man who invented the arrow notation?
yes
how does one write square roots? or ANY non-[in between |or extended from] function?
Also the man who invented TeX?
Yes, and he also wrote The Art of Computer Programming.
Actually, he’s still writing it.
Wrote it in 6 days and rested on the 7th
Is Donald Kunth god?
He's surreal.
Conway is according to the book
He created Life.
Edit: He *could have* created Life, if he were Conway.
*****
D'oh! That's right.
I'm being oppressed by facts!
No; the rhetoric was successful the first time, why not try it again?
He was just trying to be interesting. IMO, dragging religious connotations into this just fouls it up.
Fifty years behind on his writing? Wow, he procrastinates worse than I do.
I would worship him, but I'm watching TH-cam
+Callum Munro I'll do it after I finished watching TH-cam XD
Because it obviously wasn't obvious?
+Callum Munro I'll worship him tomorrow! Or maybe later, since I have many important TH-cam videos to watch…
+true reality writing ", obviously" IS a valid way to symbolise sarcasm.
Kappa makes as much sense as "obviously" - language is just what it is. I think "obviously" isn't as obvious at marking sarcasm as other methods tho.
BTW: what does kappa actually mean?
"There's no reason for us to think that the universe obeys the laws of real numbers."
I'm starting to come to the conclusion that a large part of Mathematics is really just a branch of Philosophy.
Most systems of thought are defined as being a specific branch of philosophy (e.g Science) . You would be correct
I think along similar lines too. If philosophy is putting forward an argument by making a premise and initial statements and following them to a logical conclusion with words, then maths could well be described as philosophy codified in numbers.
Will Fairweather
I didn't mean philosophy in the classical sense where everything is a philosophy. I meant in the existential "what does it all mean?" kind of philosophy. You knew what I meant. You're just trying to be contrary.
+TheDarxide23 You're asking a worldview question -- this has more to do with starting points (or initial premises) than logical conclusions. To arrive at viable premises, the best tools we have (as far as I know) are to evaluate existing belief systems for (1) correspondence to reality and (2) internal consistency.
The biggest difference between math and philosophy is that philosophers almost always disagree on premises--what we are accepting as true, what we are taking as definitions. Mathematicians rarely disagree on this. They will when something is first introduce, and over the years the best way of thinking about thing will be established. This is a HUGE difference between philosophy and mathematics. Math only need to accept the definitions that makes the math work out the way WE want. We don't actually need it to be "true" in the sense philosophers do. This makes mathematics much less restrictive.
This guy is a legend. One of the few names I remember from times when I studied CS in the 90-s. And now for the first time I actually see him talking ;)
Get Donald Knuth to make a computerphile video.
get donald kunth to make any video
Yeah totally. Could talk about the failure of itanium for example. I think it's something he weighed in on before. Well anything really. He's one of the all time greats for sure.
Get Donald Knuth.
University/Graduate Mathematics : Numberphile :: Art of Programming : (???)
You can tell by the way he talks that this is someone with ALOOOOOOOOOT going on in his mind :O
He's just old. Not a bad thing as much as an inevitable thing, if you live long enough.
@@jacobshirley3457 Try getting old and tell me again that it's not so bad.
@@megamaser that's not what he meant
@@megamaser It's better than the alternative
@@iank2615 only up to a point
Man, right off the bat: I love this guy's creativity!
So in other words the book was the program Knuth wrote to help him remember (rediscover, via the help of two abstract entities in the universe he created) the proofs Conway provided. Sounds about right.
I can't thank you enough Numberphile for getting Donald tell the story live on camera. This is pure gold, and future generations will thank you for this :).
This guy reminds me of my father who passed away last year, meekness & gentleness covering a awe inspiring intellect!
Is this the guy who invented TeX?
yes
Yes, also invented Knuth Up Arrow Notation
@@hamiltonianpathondodecahed5236 This is a difference in opinion, and some people tend to believe mathematics is inventyed, others believe it is discovered. Personally, I tend to be on the more "discovered" side of thinking. However, notation is not a fundamental fact of the universe. There are multiple ways to represent the operations which can be represented in Knuth Up-Arrow Notation, and there is nothing inherent about using arrows to represent them. Your argument is like saying I "discovered" the word "florbifod" despite just making it up just now.
@@hamiltonianpathondodecahed5236 It's a bit of both. We invent axioms/definitions then discover facts about them.
Arnav kumar Sinha Well, notation isn’t discovered since it’s pretty arbitrary if you think about it. It’s invented ti describe concept that are discovered (e.g. hyperoperators). Still, Knuth’s a genius. No doubt about that ;)
watching this was sad he has so much to say but cant say it....
Who should explain it?
stefanozurich Like a stutter? I do the same thing but, for me, it's more like diarrhea of the mouth, constipation of the brain.
frtard Cool, who should explain it though? Who can explain it better?
He tries so hard
#SAVETHENUMBERS
How simple and humble was JC in referring to his own death.
Now, killed by the virus of COVID19 it seems to me a synchronicity.
Something surreal.
To infinity, and beyond John Conway!
Nino Meloni I didn’t know! I’m so sorry to hear this.
Fake. Comorbidity.
@@nintendo9231889 Monster group
Thanks for the TeX, Don!
Man we got to bring 70s math back. Dialogues, games, fantastic stuff
Do the series of Millenium Problems.
You mean gender studies? :)
+Gazeth Sonica Huh?
+Gazeth Sonica I meant the Millennium Problems (7) set by Clay Institute of Mathematics. One has been solved.
Not gonna pretend i knew, but in was under the assumption you probably did mean something related to math. :)
+Gazeth Sonica I think you heard of the Riemann Hypothesis. That is one of them, desperately waiting for someone to solve it.
His choppy speech is driving me insane. I'm trying to listen for his words but it's making me crazy.
I think he is getting old and you slow down a bit and the idea he was trying to explain is a complicated one.
Get over yourself
He sounds like that on 20-year-old lecture videos. Not as pronounced as here, though.
His finished thoughts are great! He just has the habit of speaking before his brain has fully finished his thought.
Pretty sure he is halting a stutter. Pretty well too I would say.
lol this is so brilliant... “the mistakes that I would make, the characters in the story would make those same mistakes”
Bradford Folkens I loved that :)
The voice of this man make me cry
Never trust anyone who sounds to confident in himself. Such a person has not, and will never begin to grasp the universe.
Only a very self-confident person would say that.
Really, i didnt finish the video because of that
Bronze To Challenger ?
Why?
I really enjoyed this, seems like such a genuine dude as well. Will have to give his book a read
It seems so much a theoretical and outlandish concept, but actually I just recently used these to simplify an algorithm significantly. Basically I stripped all of border handling from the algorithm by only looking at the numbers between the reals (where no border could be) instead of the reals themselves. It went from about 250 LOC to 30 LOC (plus about 80 for obvious but many-cases back-and-forth transformation to ordinary reals). It is now also much easier to read and understand. It wasn't even a particularly sophisticated algorithm, it just simplified some filter expressions for number data.
I have wanted this video for ages. Thank you!
Probably one the best videos by Numberphile.
The main application of Conway's version of surreal numbers is in game theory, where the advantage to a player of some well-balanced games is more than 0 but smaller than any positive real number. The books "Winning Ways" by Conway, Guy, and Berlecamp give details.
I feeeeeeeel like they're just making stuff up XD
Math is really the study of systems of rules and their logical consequences. Mathematicians are basically allowed to make up whatever rules they want as long as the rules don't contradict themselves and they have interesting consequences.
I recommend the video "Mathematics: Measuring x Laziness^2" by Zogg from Betelgeuse.
Mathematics is a series of games, laying some groundwork and seeing what structure we can build on top of it. It just so happens that so many of our games become these magnificent descriptions of the real world and enhance our scientific understanding of the universe immensely.
I totally agree with you guys haha I just found it funny
No... Surreal numbers are a byproduct of studying mathematics via model theory. Model theory isn't just some random rules that were created to create the surreal numbers, maths isn't done like that. You don't just decide that you want surreal numbers and find some convenient rules that imply their existence, model theory is a way of studying the fundamental structures of all mathematical universes, and the insights of studying mathematics at this level of generality, among other achievements, uncovered the existence of surreal numbers naturally.
There is a whole area of mathematics, that features in most numberphile video not to do with number theory, done by mathematicians who do indeed just make rules up, but they do so to find interesting and challenging problems, not to create whole new theories of mathematics; mathematical theories come from studying first known 'natural' mathematical objects, then older theories that study general classes of these objects. Theory building is bottom up, not top down. The bottom however, as I've just explained, is not arbitrarily decided upon as you are suggesting.
I've been wanting you to do a video on surreals since your interviews with Conway. Excellent interview!
That feeling, when you get back from a convention and are happy that you got a signature from someone you admire and the internets adds new people to that list immediately...
...and on the seventh day he rested..
knuth's stutters make me feel less insecure about my own :) since i stutter on words too because my mouth cant keep up as fast as im thinking
Fascinating , immediately purchased this book ! John Conway is a real genius !
1:02 - probably it was Pierre Jullien, „Sur la comparaison des types d’ordres dispersés“ from 1967
RIP John Conway. Your works shall be remembered!
Sad days
It sounds like the Genesis of math.
"In the beginning there was nothing,
therefore there was one thing one void,
Thus there was two things, one and the void, zero."
The book cover is quite pretty!
Though I was also expecting some fancy math.
The very existence of nothing means there exists one thing.
It's beautiful.
The 7 day hotel room story alludes to the bible even further (and for me, much more obviously).
Nothing is not something, but the concept of nothing is not nothing.
This is how numbers are constructed from set theory. For the natural numbers it is called the Zermelo-Fraenkel set, after the guys who worked it out. Surreal numbers take a just slightly different approach, and suddenly ε is a number. I still can't believe it is really a thing. (But surreally it is.)
@@Woodside235 so are you Artrax from Das Dorf?
Wow, the legend himself!
Thank you, Numberphile :D
Seems like it took a lot of time and effort to edit this video. Thanks for making it flow well Brady!
I did my dissertation for a Masters on this - to this day still don't get it really.
I'm amazed. 3 years of being a Numberphile subscriber, and this is the first video that has made absolutely no sense to me. I thought at first that the guy was just crazy, but apparently this is the man who created arrow notation, among other things. So I'm assuming he's just speaking on a higher level of intelligence than I can comprehend.
No, he just doesn't understand his own concept.
Did you check on Wikipedia who the heck is this grandpa? Please do!
@@EGarrett01 its John Conways concept. And to understand it, you need to interprete the sides in the set as options for Alice and Bill playing a combinatorial game
@@gereonlanzerath9286 He's the interviewee, he's presenting the idea. It's 'his own' in this case similar to how a person who burns meat would screw up "his own" steak that he was serving.
@@EGarrett01 ok. Thought you were talking about the concept of surreal numbers
after watching Numberphile for so many years, it's still blows my mind, finding 'a new topic for me' that you already explored 6 years ago. Feels a bit like a worldline-spacetime-dilation ;-) THANK YOU
When he said that about dropping slithers into holes we did not know that they existed that reminded me of fractals and the problem of measuring the length of the coastline, because there you also zoom closer and closer and find more and more nooks and stuff.
Numbers are fractals in that way.
Same when he talked about creating universes from simple rules. That reminds me of fractal curves.
Am I the only one that caught the "because we had always wondered what it would be like to have an affair in a hotel room" @ 5:30 ?
Being so honest about this subject is really impressive ! Gogo Donald Knuth !
I find that very romantic to say.
"Book tells how mathematical research is done"
"He gave me a napkin with some notes defining numbers"
Sounds funny, mathematicians are nice guys (:
That was amazing! Thank you!!
I bought On Numbers and Games by Conway several years ago, and I have tried to read it a few times, but always get lost after a few chapters. I'm definitely going to have to find a copy of Knuth's book, I imagine after reading it, On Numbers and Games might finally make sense! This was a fantastic video, both technically informative and a great insight into the creative, artistic side of intellectual exploration that many people don't even know exists. Most people picture intellectual work as very austere and devoid of passion or art, and the truth couldn't be further from that!
I just invented a number that is bigger than zero but smaller than any surreal number and still it is as big as any infinity and it is also a negative number and if you divide it with zero you get the answer twenty six. I call it Zsrlinmber and to write it you use the letter H
Maan, I was keep getting 42 as the answer referring to an old research paper, idk it was in French, or maybe English, I can't recall. Thanks though
So you literally need to break mathematics in order to obtain the Zsrlinmber. I'm intrigued.
What happens when you divide H by plaid?
Emma May I have a hunch you get somewhere in or near the North Pole. Don't quote me on that though.
If you assume contradictory properties for the number, you can indeed prove anything - even that when you divide the number by 0, you get 26.
HAHAHAHA "...where Ibsen used to live, so I could get some of his VIBES..."
Vibesen.
I did not get the joke earlier, but thanks to you now I do ^_^
Beautiful interview.
This was a really fun book! Anyone who’s into recreational mathematics will enjoy it.
i actually loved how he speaks about how he created it, it's so cool
I think after watching this video, NJ Wiildberger's would be loosing his mind over surreal numbers lololol
you're still awesome, professor Wildberger ^_^
He wouldn't buy any of this hehe ^__^
He would have few things to say about it i would think.
+Blackadder 1620 he might as well try to refute that surreal numbers do not exist. Honestly, I might have forgotten the whole video if they actually have a formal definition what a surreal number is and if it is a subset of the complex. I admire the construction tho. Very creative.
+mdphdguy1 that I completely agree. I am curious to read his book.
+Johanus Gauss he really makes you think about the basics but, we have algebra and, it's hard to live without it even if some of the foundations aren't as strong as we would like. Give it another 100 years or so, im sure we'll know more.
I'm so glad I ran across this interview. I should read more into the surreal numbers. I've been dissuaded by those stuck in the reals, but now that I'm no longer in school or stuck in endless internet debates, I think I should review the surreals.
first time see him talking, une légende ... emotions ...
Have we had Don on Computerphile yet?
10:00 How inspiration works
he is as pure and this is as pure as art can ever be! he is a true artist.
Donald is pure enthusiasm, intelligence and inspiration.
8:33 These are smileys from an other planet.
5:26 slowly clicks away from the video...
Soo cool!! Your books are amazing!!
The concept seems rather similar to the one described in the video about "monads". I wonder what the technical differences are...
Can we get more explanation on what exactly is going on and why this is helpful. It seems like he's just replacing normal numbers with dots and lines and then he starts writing in normal fractional numbers
If you're interested in what's going on, the book "On Numbers and Games" by John Conway is arguably the best book.
It does not only cover the surreal numbers, but also games, which has the surreal numbers as subset (crazy, i know).
I agree with you that they should have gone much further.
He replaced normal numbers with dots and lines to represent how surreal numbers can be used to define any numbering system. While I do agree it's more convenient to use numbers already in place, the book is more of a proof of concept than anything else.
Gave me some perspective on what genius really is. Such an abstract thinker that he can't even keep up with himself sometimes! Great man and a fascinating mind.
Hearing Conway talking about his impending death at the end destroyed me.
Um, why don't you just take more of whatever you were on during that week?
Donald Knuth
Mr Trump! What an opportunity to ask you a question that has troubled me. Some psychologists, although they haven't met with you personally, have published a description of your behavior as that of an "extreme narcissist." This obviously a serious diagnosis. It means that you may be suffering from a near-psychotic condition, which means you may actually be delusional. My question to you is, have you personally ever sought the help of a psychiatrist or psychologist? If so, what was their diagnoses? If you haven't been to a psychiatrist or psychologist, don't you feel it is time for you to consult one? After all, you yourself ought to be sympathetic to the notion that the commander-in-chief of US military should not be led by a madman.
Of course, I realize I speaking to a wall. A narcissist, especially an extreme one, seldom ever acknowledges his faults.
@@tapolna lern sum gramer
I started this video after reading half of Knuths book on Surreal Numbers. It's really fun, funny and more detailed and so far, very easy to me, with little mathematical maturity. Recommended.
I love this guy! What a nice story.
Normal people: 70% water
Gamers: 70% salt
Mathematicians: 70% acid
explanation needed
Arnav kumar Sinha
People are ~70% water by volume
Gamers, or any competitive person tend to be upset when they lose, or "salty"
The last line is obvious, so I won't waste time explaining it.
Hope this helps!
Conky im pretty sure he means the drug acid
Conky I would say chemist
Hel lo oh...! dang I feel dense - thanks
What about cereal numbers? Are there any?
There’s 0 , the cheerio number.
I bought the book several years ago. I think it was mentioned in some other book(maybe some psychology book?) i was reading at the time. Very cool book.
Last week I tried generalising the construction of the natural numbers and arithmetic operations from zero and the successor function to ask what if the successor functions are not all the same (obviously this has all been done before, as is the rule with maths). The definition of "+" didn't work any more, but what I ended up with was a functional language. So, given that it's all been done before, is there a programming langauge defined up from this construction of surreal numbers?
The surreals are my favorite ordered field. They're very interesting, and as evidenced by the disbelief about their existence in the comments here they're highly counterintuitive.
Not trying to be rude, just curious, but does Mr. (Dr?) Knuth have a stutter?
No. Did you not listen?
He wants to speak a lot of things but obviously no one can. So much information wanting to be put out and the mouth can't handle it.
Wow, you registered it? It's a wonder your achievements will never even approach his.
Ah good old Don Knuth. Huge person in computer science... I still need to get and read his "Art of Computer Programming"
there is no number named infinity. Infinit means without end. it is not a number. only processes can be infinit in the sense that the proces has no definiit end.ตอบกลับ
Yes and no. If you're interested in the subject I recommend a video "How To Count Past Infinity" from the channel Vsauce.
That's why they use the lowercase Omega instead of the infinity symbol
Infinity may mean that somewhere else, but here it means .
Omega stands for infinity.
There is no real number named infinity. There is a surreal number named infinity.
At least that's what I gather from that video.
Just wondering if i understand this right, because im a layman.
1=0,99999_
but if you use surreal numbers then X can exist where
1>X>0,99999_
Would that be true?
Yup
Surreals are weird.
Well, I don't think that the notation 0.999... makes any sense here. And if 0.999...=1, then that's it, and there wouldn't be any X between, because then 0.999.. and 1 couldn't be equal anymore.
Zardo Dhieldor
I think that's the reason why X can only exist as a surreal number. What you describe is all still within real numbers.
I'm not saying that with intent of 'correcting' you btw. As i mentioned earlier I'm a layman and quite ignorant of this topic.
Gazeth Sonica
Not really. It just depends on what you mean by 0.999...!
1:42
"That's what almost everybody thought except Kanye"
I soon realised that he didn't say 'Kanye'.
Funnily enough, I remember going into a store during my teen years and asking for the new "Conway West" album 😆
Wow, that last comment from Conway has a new depth to it. RIP.
I own an H. Ipsen poem book, printed in 1875. Some of his work is amazing!
This man is "The Man"... As a programmer myself, he is one of my references, i admire him, I respect him, I wish I could have learn with him and had a chance to work with him...
Those generation had great brains, Knuth, Kernighan & Richie, Aho Sethi And Ullman, van Dam, Foley, etc.
When I read some post-2k kids who believe THEY are the digital generation and mock "elders" as if they knew nothing about "the net", it pisses me off. All this brain washing caused by hyped media, who only knows about facebook and twitter, confusing internet and www, etc...
I love the book reviews. He truly gives 1/ω shits. :)
So he at least cares a little bit.
Yup... the littlest bit.
When I was a kid I came up with the idea of 0.0 repeating 1 as the smallest number over 0, a few years ago I discovered from numberphile, that I was actually on to something, that number is an example of a surreal number!
did he break his brain thinking of this?
well, he broke his ability to write a letter…
edrudathec And to form words.
Are you that dense?
Luke M. Yes
Be a little more respectful you spoiled brat, he accomplished more in 6 days than you ever will in your entire life.
I would be frustrated as hell reading that book. It's Alice and Bob! Not Alice and Bill!
I was just about to say the same thing!
Why does he always put gullible in the description?
He didn't this time. It says "Glulible".
+Dogeasaurus Rex Oh yeah! I didn't notice!
Wow. I never noticed.
where?
New Erigion Its a prank bro
It's rare to get the chance to listen to genius
2:25 There exists an empty, a null set.
2:27 Each number has one after it.
It's the first two Russells' prepositions for his following reasonings in Principia Mathematica.
6:53 Set of all set that does not contain themselves (Russell's paradox).
Gentleman on the recording has recreated a set theory.
7:45 Ok, synthetic, formal languages.
11:15 Tape with whole numbers from a busy beaver POV.
I'd love to see a prove why 1/infinity is bigger than 0. Srsly.
Honestly. If 0.9 repeating is 1, then 1/infinity should be zero.
It depends on whether or not you accept infinitesimals.
is surreal number 0.9 repeating isn't 1
This follows directly from the definition of the order on surreal numbers, which can for example be found on Wikipedia.
This is the case for Real numbers as explained in the video, but Surreal numbers are far bigger, buy the boo!k or find it in the library!
0:08 Proof that Donald Knuth is God
One over a tiny sliver would be a great big sliver. Which makes me smile to put it like that.
he is as pure and this is as pure as art csn ever be! he is a true artist.
So if there are infinitely many surreals inbetween any 2 real numbers, wouldn't that mean there is also an infinite set of surreals inbetween each surreal? And then another infinite set of surreals inbetween each of the surreals that are between the surreals, and then ANOTHER infinite set ect ect ect and so on. But then again, there are also an infinite set of real numbers between any 2 real numbers, and then pick any 2 of THOSE real numbers and you'll find another infinite set of reals inbetween those 2 real numbers, and so on. So I honestly fail to see the difference between surreals and reals.
There are many sets of numbers with the properties you mention; just among subsets of real numbers, we have the rational numbers, the algebraic numbers, the transcendental numbers, the computable numbers, the dyadic rationals, and so on and so on. All of these quite different from each other, and the surreals are quite different from all of them.
Deedlit11 So whats the difference between an irratoinal number that is inbetween 2 real numbers and the surreal number that is inbetween 2 real numbers? Aren't they both essentially the same thing, just an infinitely long string of numbers? It really seems like a surreal is just an irrational number with a fancy name.
The surreal numbers are far vaster than the real numbers, they include things like ω, 1/ω, ω^ω, pi - sqrt(ω), and so on. Your argument seems to be that there is only one set that has the property that in between any two elements of the set there is another element of the set. But this is just something called a dense order, and there are many dense orders, such as the rationals, reals, and surreals. Consider the difference between the rationals and the reals, and how that refutes your argument.
Deedlit11 Ok, well let me ask you this. Is it possible to find an irrational number inbetween 2 surreal numbers? My problem is that my understanding of an irrational number is that it's just an infinite string of digits that don't repeat. My understand of a surreal number is that it's ALSO an infinite string of digits that don't repeat. Am I wrong on that?
Assuming that's roughly correct, then if you started writing down a surreal number, wouldn't it look exactly the same as if you started writing down an irrational number? And if so, my question is whats the difference? I don't mean to be annoying or anything, but I still really don't see any difference between an irrational number and surreal number. They really do seem to be exactly the same thing.
Here, I just thought of another way to express my point. Lets say I picked out a random number inbetween 1.4 and 1.5. I can easily tell whether it's a rational or an irrational number. But how would I be able to tell if that number is an irrational number or if it's a surreal number?
That depends. You can indeed find a pair of surreal numbers between which there are no real numbers: for example, 1/ω^2 and 1/sqrt(ω). Both numbers are more than 0 and less than any real number greater than 0.
To be honest, I've never been able to read much of Knuth's writings. It's quite annoying to be constantly interrupted by someone who thinks their jokes are really funny, while all you're trying to do is to learn something. That includes the TEXbook.
Absolutely brilliant
I was taught AP Calculus AB by his son. Needless to say, he was a great teacher.
this is marveoulos, but I find Mr. Knuth hard to listen to
reading his books as an alternative
but don't read his books with his voice, it's really hard