# What is the biggest classified number [closed]

For example millions is a class. So what is the biggest number class identified ? The classified biggest number in the universe.

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## closed as not a real question by mixedmath♦, t.b., Fabian, Jonas Meyer, Andres CaicedoJul 4 '11 at 20:28

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You're going to have to define "class" and "classified" more precisely, I fear. Giving one example isn't enough. –  mac Jul 2 '11 at 23:25
i cant explain it very well. I mean you can write infinitely long number but you can't call it with a name. So i am looking for biggest number defined by humanity. –  MonsterMMORPG Jul 2 '11 at 23:29
Do you mean: the biggest number named by humanity? (This is different to "defined by humanity"). And what do you mean by a "number": should it be an integer? Either way, I don't think there is a satisfactory answer to this one, though. –  mac Jul 2 '11 at 23:30
This questions should be classified. –  The Chaz 2.0 Jul 3 '11 at 0:16
Incidentally, I really recommend Scott Aaronson's essay: scottaaronson.com/writings/bignumbers.html –  Akhil Mathew Jul 3 '11 at 1:12

Your question is a very delicate one, because the number of numbers that have been given individual names -- in the sense that someone stands up and declares "by this string of letters I mean the following number" -- is clearly finite at any given time, but certainly can change over time.

In fact, it's easy to do this. If someone else claims that some number $n$ is the largest named number, then I can come back and say, "Sorry, I decree that a Clarkillion is the number $n+1$." So now the largest number has changed. And of course it will change again shortly after that...

Along these lines, it amusing to read about the luring lottery in Douglas's Hofstadter's book Metamagical Themas (which collects articles he wrote over a three year period in Scientific American, along with some later commentary). As a parody of escalation he devised a lottery by which someone would increase their chances to win (at the cost of lowering the amount of money they would win) by naming the largest number they possibly could specify on an ordinary 3 x 5 card. The result was that people turned in so many ridiculously large numbers via iteration schemes, Ackerman-type functions and so forth that in fact it was not possible for him to figure out in the end who had actually named the largest number. So in practical terms this sort of question is impossible to answer.

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Maybe you want to read this:

http://en.wikipedia.org/wiki/Large_numbers#Examples_of_numbers_in_numerical_order

I guess the largest number named using English words on that page is Graham's number.

But that page only concerns integers; if you consider infinite cardinals to be numbers then there are plenty of bigger numbers with English names.

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yes biggest english named number –  MonsterMMORPG Jul 2 '11 at 23:37

The biggest number I've ever come across is Lowry's Big Number, which is similar to Graham's number except moreso. Using Knuth's up-arrow notation, Lowry's Big Number is defined to be $f \; \uparrow ^G (4)$, where $f(n) := 3 \; \uparrow ^n \; 3$ and $G := f \; \uparrow ^{64} (4)$. By the way, here, G is Graham's number. This number has no practical use whatsoever, and was named by the little-known David Lowry (very innocently) one afternoon.

Is that what you're looking for?

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As per the Guinness Book of World Records (1980), the highest lexicographically accepted named number in the system of successive powers of ten is the centillion ($10^{303}$). The highest named number outside the decimal system is Asaṃkhyeya ($10^{140}$). It is named in a Buddhist text. The highest named number in a mathematical proof is Graham's number, which is the bound of a specific value. It is so big that it cant be described using ordinary exponential notation! If all the material in the universe were turned into pen and ink it would not be enough to write the number down. We use Knuth's up-arrow notation to describe it.

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I dare guess the OP was looking for centillion. Fortunately for us Europeans centillion is $10^{600}$ by the same principle that a billion is $10^{12}$ over here :-) –  Jyrki Lahtonen Jul 3 '11 at 4:01

Numbers can be organized into size classes using bounds from the output of fast-growing functions. For example, the numbers we see in everyday life are often classified using bounds from the exponential function $n \mapsto 10^n$.

Once one has a way to organize fast-growing functions, one can make classes of very large integers relatively manageable. These classes of functions often take the form of an ordinal, like $\omega^\omega$, where you feed the ordinal into a recursive definition, such as the fast-growing hierarchy. Most methods of constructing large numbers that you routinely see have a fast-growth complexity that is quite low in the hierarchy of ordinals, and the problem of judging contests like the luring lottery is easier than you might at first expect. For example, Ackerman's function has complexity $\omega^2$ (as does the recursion defining Graham's number), and the Goodstein function has complexity $\epsilon_0$. This is still reasonably tame among computable ordinals.

You can define such functions as long as you can constructibly transfinitely induct, so the fastest-growing function in such a hierarchy depends on what axioms of set theory you assume. See Borcherds's blog post about this. If you want even larger numbers, you can use nonconstructive definitions like the Busy Beaver function, but you lose the ability to compare sizes in a provable way.

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The largest named integer is one more than the largest integer that can be named with no more than one hundred sixty letters, spaces, and punctuation marks.

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This is a bit of a non-question. There seems to have been something of a competition to create notation which expresses inconceivably large numbers in apparently simple ways. Look at Conway Chained Arrow Notation for example. You might also want to look at the Busy Beaver function - which grown enormously large very quickly and is known to be non-computable.

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