Surreal Numbers (Don Knuth Extra Footage) - Numberphile
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- เผยแพร่เมื่อ 27 มิ.ย. 2016
- Main video: • Surreal Numbers (writi...
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I love cereal numbers. They go great with milk!
Regarding Brady's question about 'how to get to infinity', I think Knuth kinda nailed it in his book. In "Surreal Numbers", Alice and Bill are the only two characters on an island going back and forth trying to understand these rules and constructing a number system based on them. When they run into the necessity of infinite sets to define the number "3", Knuth writes Conway's voice in as a god-like rumble of thunder in the sky foreshadowing the students' later realization. It isn't until the system begins to reveal itself that Bill and Alice are able to understand where things are headed and the necessity of infinity to account for the world around them that infinite sets and infinity itself is introduced.
It's so subtle that even the characters in the book mistake Conway's interjection as thunder, a harbinger of the upcoming monsoon season. The readers of the book are left with a puzzlement. Is this an actual editorial comment John Conway had of Knuth's book? Is this is the "JHWH Conway" character from the stone carving in the story? Is Conway speaking to the characters or to us?
It's an amazing storytelling device that forces us to become aware of a new dimension of reality. The story exists as a story : the reader as the reader. Those who contributed to the story are on one side : the reader's growing curiosity and understanding on the other. Alice and Bill shrink to slivers smaller than the smallest piece of our reality : while realization fills us and we look up from the book to stare beyond the reality we once knew.
Don't they need infinity for a third? In the surreal number system you need infinite numbers to get to 2/3. Normal 3 should just be {0,1,2 | }
Dr. D. Knuth is such a sweet man in these interviews. I can hardly believe he designed TeX.
He did?
@@zokalyx Yes, he did :)
He has reformed himself in his later years. He is no longer the monster he once was.
The st comment.
Yes, but i thought that would be inconsistent because conventionally we state the primary position as 1st rather than 0th.
+Rai Paribesh Not always. My house has a 0th floor (ground) and my cpu has a 0th core.
But you could sometimes start building something at 0?
The 0th day of the month, is simply the last day of the previous month.
+Rai Paribesh On Numbers and Games had two parts: the zeroth and the first.
:●>
PLEASE do more videos with this guy, maybe one on Knuth's up arrow notation?
I have no problem whatsoever with accepting any self-consistent number system as mathematically valid, completely regardless of whether there is any real-world analogue or use for it. Whether I spend much time contemplating it is another matter, but I will not hesitate to accept it for what it is.
I agree. But, that said, the surreal numbers do portray something outside their self contained system, namely games. Conway found them when looking into game theory.
Thanks for doing another video with Dr. Knuth. A true genius.
Very interesting interview!
Really love this way of looking at serial numbers
What?
*surreal
I just looked it up - they have the book in our library. :-)
I would like to see more videos about surreal numbers and about their relation with Robinson's non-standard analysis
I'm no expert, but I think non-standard analysis uses subsets of the class of surreals called hyperreal sets (or more precisely, hyperreal fields).
fantastic book. and knuth talking about Conway - could there be a better video?
Yes, a similar video twice as long.
I pondered over what the difference could be between "greater than or equal to" and "not less than", but then some fellow somewhere here pointed out that the empty set is not greater than or equal to any numbers on the left, and yet is also not less than those same numbers. And thinking about that, as a programmer, it makes sense, considering that there is a sort of equivalent for floating point: NaN.
NaN >= x is false, and at the same time, NaN < x is also false.
The difference is that you first need to prove that all numbers are comparable. This does happen to be true but is false in the larger world of games (what Knuth terms pseudonumbers). For example, the game {0|0}, termed 'star', is neither greater than, less than, nor equal to 0; it's incomparable with (or 'confused with') 0. In this more general arena, 'not greater than or equal to' is very definitely distinct from 'less than'.
There's a video (his only one, tragically) by Owen Maitzen on this subject titled 'HACKENBUSH: a window to a new world of math', which I cannot recommend highly enough.
legend
So what exactly is the difference between "greater than or equal to" and "not less than"?
Since he said that it makes the development of the system easier to use 'not less than' I would imagine that it's probably primarily a matter of how you prove things. To prove "greater than or equal to", you have to prove both that the case of equality and the case of greater-than are both valid, but with 'not less than', you only have to prove that the less-than condition produces a contradiction. But it might be more than just that since the surreal numbers deal so much in infinitesimals, I'm not entirely sure.
It caught my ear as well and I went searching. Apparently, it has to do with the law of trichotomy. The real numbers obey this law and every pair of real numbers, x and y, x is either less than, equal to, or greater than y. Other numbers do not necessarily obey this law, including the complex numbers. A fourth category arises of numbers being potentially incomparable.
All this is just from som quick reading. I don't claim to understand it and I might be getting it all squiggly. But thought it might still be welcome, in a blind leading blind sort of way. What a fascinating mathematics!
can you list your sources?
+Nikolaj Lepka nothing fancy. Just Wikipedia.
Thomas R. Jackson yeah I tried to Wikipedia it but I guess my search terms were too loose to get any tangible results
Hence why I'm asking :3
Is there a polish translation?
This seems like a notation variant of the hyperreals. What's the difference besides notation?
The hyperreals are a subfield of the surreals. Every hyperreal number is a surreal number, but there are many, many more surreal numbers than hyperreal numbers. To say that 0% of all surreal numbers are hyperreal numbers would be an understatement. The collection of surreal numbers is actually too big to be a set - it's a proper class.
But what would pi be in surreal notation?
First, you need to know binary...
Can anyone explain why 'greater than or equal" is not the same as "not less than"?
Of the top of my head I guess the difference might be relevant in the earlier stages of creation when one or other the sets involved is the empty set. You can't really say that the empty set is "greater than or equal to" anything, but you can say that it is "not less than" anything.
Handedness: the greater is on the right . Not less than can be the ordering, greater than or equal cheats the ordering. The logic holds only if the ordering holds. Thus the comment on proofs.
So, not less than is equivalent to the the truncation function, while greather than or equal is the floor function. Like taking the trace.
I feel like 1/infinity should really be 1/(2^infinity), at least the way that was defined in the video. {1/2,1/4,1/8,1/16,...|} and I think 1/infinity should be defined on day infinity plus 1 {1/1,1/2,1/3,1/4,...|}
Just a thought that popped into my head.
Woo
from what I understand, the surreal number statement implies infinitely small quantities. therefore the universe cannot work within this system because we have set limits such as a planks length
plancks length is not a set limit. i am two years late but i cannot let this stand. its the minimum MEASURABLE distance but you can move half a planck length and until quantized space is empirically proven its not able to be called a set limit.
By that logic, the universe can't work with real numbers either, or even all the rationals.
Okay.... So, isn't the concept of Smallest /largest present here even?
And isn't a differential pretty much the same?
And is half of a differential less than a differential?
Does anyone know if "pi" is a surreal number? It was hard enough to figure out if it was transcendental back in the day, but how can we prove that it has a countably infinite amount of digits? And for that matter, what about e or phi?
Yes, "pi" is a surreal number. Once you reach "omega" (which is defined as a number larger than every other positive number, causing it to be misunderstood as "infinity"), you start getting a whole bunch of new numbers, filling in gaps you weren't aware were there. Until you reach "omega", the only fractions achievable have a denominator of 2^n, but with "omega", you can get to stuff like 1/3 and "pi" very easily (easy in the sense of surreal numbers).
every number is a surreal number
frank franksen What about complex numbers
Ekaterina Nosenko They're called surcomplex numbers. They keep the form a+bi, but now a and b can be any surreal number.
Are we sure he wasn't in Amsterdam instead of Norway?
Maybe the real world is not discrete at all, but instead defined by the constructive interferences of waves which would make it more akin to octaves and harmonics.
Is this the arrow guy?
Yes
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What I find especially surreal is that the same axioms can have different interpretations.
His formal description of a number strictly greater than 0 but less than any positive fraction has no home in standard analysis. The limit concept is 'too strong' in the sense that it deletes the information about the inequalities presented. The limit of (1/n) is 0, after all.
To think of a such a surreal number is beyond analysis. The sequence of sets ( [0,1/n] ) 'approaches' {0}. I suppose [0,1/w] can be used to describe 'the set just before the limit set' but experience tells me that set won't be unique.
Like 1/w can be but then 1/w can also be .
I think what we have here is a model of the extended number line generated by the axioms. You have a way to generate all the fractions, you have a way to write any inequality and thus you have a way to refer to all the real numbers... and more.
Actually, it is not needed to represent 1/w as . This is because you need 1/w (and therefore w) to even get to 1/3. Without w, you can only get fractions with 2^n as the denominator.
+arcuesfanatic lol, what's up with your definition of fraction? You can totes have them in any numerator and denominator before dabbling in the infinitely small.
***** Before you get to omega (written as "w" here), you can only have fractions with 2^n as the denominator and anything as the numerator, due to the way surreal numbers are built. And while you can get a whole load of fractions that way, you never quite get to stuff like 1/3 or 1/5.
If you have a surreal number where neither the left set nor the right set are empty, the resultant number is right in between the largest element in the left set and the smallest number in the right set. For example: is 1/2 as that is right in between 0 and 1. Due to this exact middle behavior, fractions before the infiniteth generation is dyadic (denominator is 2^n).
Once you get to the infiniteth generation, every real number can be created. For 1/3, you can use for any "y" value on either side that fits that side's inequality. These inequalities simplify down to y1/3. The number right in the middle of that is y=1/3.
As for adding infintely, we have shortcuts to simplify the equations.
Interesting but I still feel that you could have alternate interpretations of the pair of sets that make for a more deep structure
Do surreal numbers contain irrational numbers?
Is Surreal Numbers algebraic?
Interesting point!
I'd say no, because they include all the real numbers, which include all the (non-algebraic) transcendental numbers. And that's before you even get to the transfinite or infinitesimal ones.
We discovered the surreal numbers before numbers - we called them many things - rocks, stars, crystals, swarms, hopes : )
aw! how cute that he think we can reach infinity because he doesnt understand how the internet work.
The principals that man understands would give you nightmares.
To the OP: How cute that you completely missed his point.
Show some respect man, this guy's a legend
Stunning display of the Dunning-Kruger effect, wow...
Ironic
So whatever are surreal numbers useful for? Absolutely nothing? Why have all these alternate number systems, when we already have a number system that works wonderfully?
What does infinity + 1 even mean? The ordinary number system simply describes reality in that ∞ + 1 = ∞. Numbers are already abstract, so what is the point of surreal numbers making an abstraction of an abstraction? I would rather just have surreal art.
infinity isn't a number
Romário Rios
Infinity is not a constant.
Thus, ∞ - ∞ = undef, according to my TI89 graphing calculator.
+Yosef MacGruber of course it is undefined, you're subtracting two non-numbers.
similarly, infinity + 1 is also undefined, because infinity is not a number
Romário Rios
Wrong. Infinity is a concept, and such a calculator would simply give the answer to ∞ + 1 as infinity.
Infinity is an answer on my calculator, and it is also a type-able character on my TI89 calculator. Calculators that do symbolic math, rather than just decimal approximations, seem to be able to deal with infinity, and also complex numbers. Well at least that is the case with my TI89.
undef simply means there is not enough information to determine an answer. For example, my TI89 says that 1/0 = undef. Simply looking at a graph of 1/x as x approaches 0, shows what the problem is. The graph diverges at 0 going towards both -∞ and +∞. Since it does not converge towards a single answer, the calculator says undef. But when I put it in as limit(1/x) as x approaches 0 from the positive direction, my TI89 gave the expected answer of ∞.
∞ - ∞ = undef, because it is one of the listed indeterminate forms. We can not simplify, because ∞ is not a constant, and not all infinities are equal.
A line can not be defined by only one point, right? So consider what happens with the line slope formula, based upon 2 points, when both points are actually the same point. We have
Slope = (y2-y1)/(x2-x1)
We know that y1=y2 and x1=x2, so let's substitute.
Slope = (y1-y1)/(x1-x1) or 0/0, which is a recognized indeterminate form. Slope = undef. We just can not find the slope of the line, because one point is not enough information. Based on only that point, the slope could be of any value. If you can find the coordinates of another different point, then we can find the slope.
Why do we even care what the slope is? One reason is, with that information, plus the y value when x=0, we can give the equation of the line. All lines that have a y=0 point and a x=0 on them, can be written in the form of y=ax+b. (Well except for the line x=0.) Or Slope • x + b, in which b is the value of y when x=0. The exception would be, horizontal lines, which are written in form of y=b. But that is not an exception is it? The slope of a horizontal line is 0, and 0x is of course 0, so y=ax+b simplifies to y=b. Maybe the exception is x=c, a vertical line, in which slope = ∞, or is that undef, as do we say it is +∞ or -∞?
Come to think of it, that would be an interesting program assignment. Given any 2 points, the program should correctly calculate the equation of the line connecting those points, handling all normal special cases of horizontal and vertical lines correctly. If both points are the same point, it should correctly give the result as Undefined. It should be decided whether allowed input values are to be integers, rational numbers, irrational numbers, decimal approximations, mathematical expression of the combinations of these (example: [2*√3,2/π] [2*√3,2/π] Answer => Undefined), or what? Maybe we would just go with decimal approximations, since that is what programming languages generally use.
Can any graphing calculators solve for this already? I know some calculators can solve equations, but points are not equations?
Surreal numbers have been useful in determining the outcome of games. If we have two perfect players, L and R, and a surreal number x = , we can get four different outcomes:
x > 0, Player L is guaranteed to win regardless of who goes first
x < 0, Player R is guaranteed to win regardless of who goes first
x = 0, the player who moves second is guaranteed to win
x || 0(incomparable), the player who moves first is guaranteed to win.