Surreal Numbers
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- Опубликовано: 11 июл 2024
- Introduction to the Surreal Number system including forms, ordering, negation, arithmetic operations with examples.
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Video on Ordinals, Cardinals and Transfinite Arithmetic
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• Ordinals, Cardinals an...
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Book: Surreal Numbers by D. Knuth
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www.amazon.com/Surreal-Number...
Nicely done. However, omega is not the smallest number after all the naturals, but rather the simplest. There are also omega - 1, omega - 2, ..., omega/2, ..., sqrt(omega), ... In fact, the "set" of infinite surreals less than omega is itself a proper class.
Good catch. You are absolutely correct.
In the video says is the smallest transfinite ordinal number. Which it is according to ordinals arithmetic. As a surreal, l agree with you, there are smaller, like omega-1 and so on
Ordinal addition is not commutative. The canonical example (given in the linked article) is that 1+ω=ω, but ω+1≠ω. Left subtraction is always possible. Given any ordinals α≤β, we can find an ordinal x such that α+x=β. This is called “left subtraction” because we’re figuratively asking for x=(−α)+β , i.e., adding the inverse on the left side, if such a thing made sense in the ordinals.
Right subtraction is not necessarily possible. That is, given ordinals α≤β, it might not be possible to find x+α=β. If we interpret ω−1
as “x such that 1+x=ω”, that’s ω itself.
But, if we want ω−1 to be “x such that x+1=ω”, it’s undefined. There’s no such ordinal.
I'm pinning this thread because there is an error in my characterization of omega *ω* and there are some good comments on this thread.
To clarify: Omega *ω* is not the smallest transfinite ordinal. It may be the first expressed through this generative process, but there are actually infinitely many expressions that come before it on the number line.
(Consider something like ω - ε.)
@egheitasean1 It's still the smallest transfinite ordinal, if we define ordinals to be surreals with no right options.
RIP John Conway 😭
wow, i'd never seen 1/3 expressed as the infinite sum of 1/4 + 1/16 + 1/64 + ... before. mind suitably blown.
evilotis01 ever seen floating point numbers?
It's actually easily proven intuitively just with dice.
Imagine you have 3 options you want to randomly choose from with equal likelihood, but you only have a 4-sided dice (a "d4" or tetrahedron).
You can assign numbers 1-3 on the dice to the 3 options (a, b, and c), and remaining number 4 as a re-roll.
Consider the probability space of all possible combinations of dice rolls.
What's the chance of getting an "a" with only one dice roll? Clearly, its 1/4. What about exactly two dice rolls? That would be the chance of a single re-roll, times the chance of the desired option ("a") on that second roll. That is, (1/4)*1/4), or (1/4)^2.
The chance with exactly 3 rolls is (1/4)(re-roll) * (1/4)(re-roll) * (1/4)("a"). In summary, the chance of getting an option with exactly any number of dice rolls "n" is the dice chance (1/4 in this case) to the nth power.
But the *total* probability of an option is the sum of _all_ working combinations of rolls. That is, getting an "a" on one, two, three, ..., or "n" rolls. That works out to: (1/4) + (1/4)^2 + (1/4)^3 + ...
But also remember, that there are only 3 possible options that must be equally likely, because this process has been completely symmetric. The chance of any option *must* be (1/3), so (1/3) must equal the infinite sum of (1/4)^n .
Not only that, but this argument can apply to any fraction of 1 divided by a natural number. (1/5) *must* equal the infinite sum of the powers of (1/6). That is what you would find by doing this with a six-sided dice and 5 options. One seventh must be the sum of the powers of one eighth. In general, 1/k = SUM{ from n=1 to infinity, of (k+1)^n }
But if you _really_ think about it, this is just a special case of the "Geometric Series." It says that infinite sum of the powers of any fraction "f" less than 1 equals: f/(1-f) . Plugging in 1/4 gets you 1/3.
Though in fact *_that_* is just a special case of the "Reimann Zeta-Function." And if you want to see something _really_ crazy, look up the "analytic extension" of the Reimman zeta-function.
(Spoilers: The sum of the natural numbers bizzarley corresponds to -1/12).
Though, to get back on topic. This method shows that you can construct a fair "dice-procedure" for any integer fraction probability so long as you have a two-sided dice (a coin). That's because you can model a d4 as two successive coin flips. Any 2^n sided dice can be modeled with n coin flips. And any betting chance can be fairly computed with a large enough dice and the correct assignment of re-rolls.
This is directly analogous to the fact that any real number can be represented as a sum of powers of 2. That is precisely what you're doing when you write numbers in base 2. And that's precisely how surreal numbers are able to represent non-diadic real numbers.
@@WWLinkMasterX well... shit. wow. thank you for the explanation. mind suitably blown (again!)
Actually every number 1/x = sum of 1/(x-1)^n over n
correction: 1/x = sum of 1/(x+1)^n over n
Amazing concept. That the surreal numbers include things like infinitesimals and still with the same arithmetic as the real numbers.
Thank you.
i don't get how the formula for the addition is x+y = (XL+y, x+YL | XR+Y, x+YD}, there's an addition there
It's a definition by recursion; every addition in the definition involves an earlier created number. Eventually, you'll get to 0, which has no options, and the process stops. An explicit example: 1+ 1 = {0|} + {0|} = {1+0,1+0|}; 1 + 0 = {0|} + {|} = {0+0|}; 0 + 0 = {|} + {|} = {|} = 0. Thus 0 + 0 = 0; 1 + 0 = {0+0|} = {0|} = 1; 1 + 1 = {1+0|} = {1|} = 2. A bit lengthy, but way shorter than Principia Mathematica.
The surreals are said to be well-founded, in that it is impossible to find an infinite sequence a0, a1, a2,..., where a1 is an option of a0, a2 is an option of a1, etc. (There are plenty of surreal numbers where you can construct such a sequence of any given finite length, but never one of infinite length.) This allows us to make definitions such as this one for addition, provided that we make sure to always involve earlier created numbers. For more information, look up transfinite recursion and transfinite induction, both of which are used extensively with the surreals.
Edit: It's not transfinite recursion and induction, but e-recursion and e-induction, where 'e' should be the 'is an element of' (epsilon) symbol, which my phone keyboard lacks.
Does anyone know whether the surreal numbers can represent i in their {...|...} notation?
It can't be {0|0}, {-1|1} or {-x|x}, because those can't be negated, but i can.
The definition of √ω as {0,1,2,3,...|ω,ω/2,ω/4,ω/8,...} suggests something like {0,1,2,3,...|-1,-1/2,-1/4,-1/8,...} or {-1,-1/2,-1/4,-1/8,...|0,1,2,3,...}, which on one hand makes sense (of course it wouldn't converge onto a single number), but on the other hand it seems impossible to derive recognizable complex numbers from these suggestions.
if you mean i as in the imaginary unit then no, however according to wikipedia there is an extension of the surreal numbers called the surcomplex numbers, although the surcomplex numbers aren't a proper class
naively, I would think you could just invent a new system where you "rotate" about the left and right "handed-ness" of this system.
Typo at 15:48 : the capital Y by itself in the second line, should be a lower-case y.
Do surreal numbers include the Gödel number of its non derivable sentence?
Great explanation of surreal numbers! What about the construction of arbitrary surreal numbers? I remember learning that there was a game involving drawing graphs to construct arbitrary ones.
Hackenbush? You can make any real value in an infinite Hackenbush position using the infinite binary expansion. You can also make other values, but the process is a bit more complicated.
The Surreal Numbers is a small subset of all games of the form:
{L | R}
Example: {0,0} isn't a number, but it IS a game. This pdf of some lecture notes does a good job of covering Conway's definitions and basic theorems:
www.cs.cmu.edu/afs/cs/academic/class/15859-s05/www/lecture-notes/comb-games-notes.pdf
@@egheitasean1 Thanks for the info! I thought all Hackenbush games were surreal numbers. Is there another name for the set of all Hackenbush games?
@@dcterr1 Hmm... I'm not sure about ALL Hackenbush games, just that certain subsets of games can be used to generate the Surreals and other types of numbers.
Example: nimbers... in a Hackenbush game, each connection to the ground can be considered a nim heap with a nimber value. They're interesting because they have their own arithmetic and all the separate connections to the ground line in a game of Hackenbush can be summed for a nimber of the game state.
I hope I helped a little, but maybe someone else can fully answer your question.
@@egheitasean1 This is all very interesting and intriguing to me! So if I understand you correctly, then a subset of the set of Hackenbush games generate the surreal numbers, which I assume are also a field extension of the real numbers. I wonder what field is generated by the set of all Hackenbush games and what kind of algebraic structure the set of Hackenbush games itself has! And is there a maximal field extension of the real numbers? Whatever it is, it needs to include the set of complex numbers, since this is also a field extension of the real numbers.
I'm not entirely sure if I grasped the concept but I have some questions:
1 could someone show me a number that it's purely surreal (it is not contained in any of the subset known)
2 is a surreal? (I don't know if I wrote it in the right way, it should be a number greater than all the surreal)
I am not an expert on this topic so I may be wrong but for your first question, {1,2,3,4,...|omega} = omega-1 I believe is only exists as a concept for surreal number. Other could include weird combinations of omega, i.e. sqrt(omega).
For your second question {No|} would definitely not be a surreal number as it is meant to be defined as a number larger than any surreal number. I am not entirely sure whether this concept is even well defined at all because to me it feels like something that is larger than anything that can be defined which would be a contradiction.
{No| } is not a surreal number because No is not a set.
@@jacobstern6343 {No | } actually has its own name! It's called On, and it's a special type of surreal form, called a "gap".
I still don't understand X_L•Y_L, X_R•Y_R, X_R•Y_L and X_L•Y_R
Are we multiply these set together?
Why the condition in the definition of inequality is so weird? that is why everywhere the write: "there is no x_L in X_L that Y ≤ x_L" but "Y ≤ x_L " means that "for all y in Y (y ≤ x_L)", so that means
"there is no x_L in X_L that (for all y in Y (y ≤ x_L) )" which is equivalent to "for all x_L in X_L there exists y in Y that it is not (y≤x_L) ) " which by linearity of the order reads
"for all x_L in X_L there exists y in Y that (x_L < y) ) " which is much simpler to comprehend .
where do I make a mistake ??
You're assuming that surreal numbers are necessarily ordered; that, for any surreal a and b, exactly one of a>b, a=b, a
What happens when you divide by zero in these?
the same like in every field - you can't -since divide by zero is multiply by the reciprocal of 0, but the reciprocal of zero is undefined (since 0 * any = 1 means 0 = 1 means FALSE)
and this is all just on the real axis. bruh imagine this with complex numbers, no, quarternions, no, infinite dimensions, no, UNCOUNTABLY INFINITE DIMENSIONS
There ARE actually surcomplex numbers.
@@ranjitsarkar3126 damn
we see infinity in surreal numbers
you sound completely out of breath in this video... gasping for air at the start of every sound bite.
additionally, there were several points in the video where you'd list a few ideas/elements/etc. with nothing related to them on the screen.
you really place a high cognitive load on the viewer when you present things this way.
regardless, thanks for talking about the surreal numbers. as someone who has taught about them in the past, it was nice to see a video that covered most of the details.
I had read through On Numbers And Games (Conway), and Surreal Numbers (Knuth). Being fundamentally a binary definition is interesting; and it's not taken far enough. The mechanical definition of the
UNREAL if u ask me ~~~~is there a purpose in all of this ~~~
the way you pronounce omega is just.. hilarious and weird.
Does anyone know whether the surreal numbers can represent *i* in their {...|...} notation?
It can't be {0|0}, {-1|1} or {-x|x}, because those can't be negated, but *i* can.
The definition of √ω as {0,1,2,3,...|ω,ω/2,ω/4,ω/8,...} suggests something like {0,1,2,3,...|-1,-1/2,-1/4,-1/8,...} or {-1,-1/2,-1/4,-1/8,...|0,1,2,3,...}, which on one hand makes sense (of course it wouldn't converge onto a single number), but on the other hand it seems impossible to derive recognizable complex numbers from these suggestions.
I don't think it's a great idea to ask the same question a whopping 9 times...
That's a good question. Conway considered all Forms *{...|...}* to be "games" while only those with certain characteristics were numbers. So *{0,0}* is not a number, but it is a game.
The Surreal Numbers is the "largest" class of ordinals taken as an extension of R. However you can get more types of numbers by considering higher dimensions of *R* like *R²* which isomorphic to *C* and generated by R[i]. Four dimensions gets you *i*, *j*, and *k* with the Quarternions. Eight gives the Octonions... etc. I made a video about this:
ruclips.net/video/A7vRtMyR9gE/видео.html
Does anyone know whether the surreal numbers can represent *i* in their {...|...} notation?
It can't be {0|0}, {-1|1} or {-x|x}, because those can't be negated, but *i* can.
The definition of √ω as {0,1,2,3,...|ω,ω/2,ω/4,ω/8,...} suggests something like {0,1,2,3,...|-1,-1/2,-1/4,-1/8,...} or {-1,-1/2,-1/4,-1/8,...|0,1,2,3,...}, which on one hand makes sense (of course it wouldn't converge onto a single number), but on the other hand it seems impossible to derive recognizable complex numbers from these suggestions.
Does anyone know whether the surreal numbers can represent *i* in their {...|...} notation?
It can't be {0|0}, {-1|1} or {-x|x}, because those can't be negated, but *i* can.
The definition of √ω as {0,1,2,3,...|ω,ω/2,ω/4,ω/8,...} suggests something like {0,1,2,3,...|-1,-1/2,-1/4,-1/8,...} or {-1,-1/2,-1/4,-1/8,...|0,1,2,3,...}, which on one hand makes sense (of course it wouldn't converge onto a single number), but on the other hand it seems impossible to derive recognizable complex numbers from these suggestions.
easy.... { | } + { | }i Just replace the real and imaginary numbers with surreal representations and leave the + and the i Dont make it harder than it actually is. Conway was all about keeping it simple and obvious.
Does anyone know whether the surreal numbers can represent *i* in their {...|...} notation?
It can't be {0|0}, {-1|1} or {-x|x}, because those can't be negated, but *i* can.
The definition of √ω as {0,1,2,3,...|ω,ω/2,ω/4,ω/8,...} suggests something like {0,1,2,3,...|-1,-1/2,-1/4,-1/8,...} or {-1,-1/2,-1/4,-1/8,...|0,1,2,3,...}, which on one hand makes sense (of course it wouldn't converge onto a single number), but on the other hand it seems impossible to derive recognizable complex numbers from these suggestions.
No, i is not a member of the surreals, you can however have surcomplex numbers a + b*I where a and b are surreals.
{0|0}, {-1|1} are members of a larger class called nimbers used in game theory, which again do not have I as an element.
Does anyone know whether the surreal numbers can represent *i* in their {...|...} notation?
It can't be {0|0}, {-1|1} or {-x|x}, because those can't be negated, but *i* can.
The definition of √ω as {0,1,2,3,...|ω,ω/2,ω/4,ω/8,...} suggests something like {0,1,2,3,...|-1,-1/2,-1/4,-1/8,...} or {-1,-1/2,-1/4,-1/8,...|0,1,2,3,...}, which on one hand makes sense (of course it wouldn't converge onto a single number), but on the other hand it seems impossible to derive recognizable complex numbers from these suggestions.
It seems that a complex surreal c = a + i b should have the form c= { {a_L | a_R}, {b_L| b_R} }, in particular i = { { | }, {0| } }.
Complexes are isomorphic to R^2: 1={1,0}, i = {0,1}, a+i b = {a , b}.
A potential development would somehow try express c as c= {c_L | c_R}, but we shall have to define suitable order for the complex elements of the surreals.
Does anyone know whether the surreal numbers can represent *i* in their {...|...} notation?
It can't be {0|0}, {-1|1} or {-x|x}, because those can't be negated, but *i* can.
The definition of √ω as {0,1,2,3,...|ω,ω/2,ω/4,ω/8,...} suggests something like {0,1,2,3,...|-1,-1/2,-1/4,-1/8,...} or {-1,-1/2,-1/4,-1/8,...|0,1,2,3,...}, which on one hand makes sense (of course it wouldn't converge onto a single number), but on the other hand it seems impossible to derive recognizable complex numbers from these suggestions.
The surreal numbers form an ordered field, and it’s pretty simple to prove that no ordered field contains i. So, the answer is no, i cannot be expressed in this way.
Does anyone know whether the surreal numbers can represent *i* in their {...|...} notation?
It can't be {0|0}, {-1|1} or {-x|x}, because those can't be negated, but *i* can.
The definition of √ω as {0,1,2,3,...|ω,ω/2,ω/4,ω/8,...} suggests something like {0,1,2,3,...|-1,-1/2,-1/4,-1/8,...} or {-1,-1/2,-1/4,-1/8,...|0,1,2,3,...}, which on one hand makes sense (of course it wouldn't converge onto a single number), but on the other hand it seems impossible to derive recognizable complex numbers from these suggestions.
I don't know for sure, but I suspect not. I think the surreals can be ordered, as positive or negative, obeying the usual arithmetical rules (negative times negative equals positive, etc), but the complex numbers cannot be. For instance, the usual laws of positive and negative arithmetic would suggest that if i.i=-1 then -i.i=+1, but it doesn't. It just equals -1 again.
However, you can obviously extend the surreal numbers to surreal complex numbers by adding i in.
@@donaldb1 Uhm... -i.i equals +1...
@@joelproko Oh yes, so it does. Haha. Um.
What I _meant to say_ is, if you can have positive or negative then,
1) For any number x either x or -x (but not both) should be greater than zero
2) the product of any two numbers greater than 0 should also be greater than zero
So, by 1) either i or -i should be greater than zero, in which case their square should be greater than zero, by 2). But they both square to -1. So they can't be positive or negative. Is that better?
Does anyone know whether the surreal numbers can represent *i* in their {...|...} notation?
It can't be {0|0}, {-1|1} or {-x|x}, because those can't be negated, but *i* can.
The definition of √ω as {0,1,2,3,...|ω,ω/2,ω/4,ω/8,...} suggests something like {0,1,2,3,...|-1,-1/2,-1/4,-1/8,...} or {-1,-1/2,-1/4,-1/8,...|0,1,2,3,...}, which on one hand makes sense (of course it wouldn't converge onto a single number), but on the other hand it seems impossible to derive recognizable complex numbers from these suggestions.
I think i cannot be expressed, since there is an order on the surreal numbers. If we could express i, we could express all complex numbers and then we had an order on the complex numbers that behaves well with addition and multiplication. But there is no such order on the complex numbers.