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If I give you one of each letter in the alphabet what's the largest you can spell (in word form)?

Bonus: What's the smallest?

Bonus 2: What if you can use the words "minus", "plus" and "times"?

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    $\begingroup$ I don't think this question is answerable. Various people have invented competing or mutually exclusive large number systems with various levels of acceptance. $\endgroup$
    – Muqo
    Nov 7 '14 at 14:04
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    $\begingroup$ @warspyking calm down about everyone using loopholes. That's the fun of it. $\endgroup$
    – Bobo
    Nov 7 '14 at 20:46
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    $\begingroup$ I understand that you would want an answer that interprets the question as you see it (since it is your puzzle). You're free to accept whichever answer you agree with (or were expecting). I'm just pointing out that some people like thinking outside the box and finding interesting answers (which we know are loopholes) so you didn't have to comment that on every answer. $\endgroup$
    – Bobo
    Nov 7 '14 at 20:54
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    $\begingroup$ You did not precisely define the question in the question. It is nonsense to then claim that answers are cheating/loopholes when they meet the little criteria you specified, but weren't quite what you had in mind. (you might redefine the question to exclude certain types of answers, but do so in the question for all to see, and know that means you might tick off the earlier answerers) $\endgroup$
    – Tim S.
    Nov 9 '14 at 22:25
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    $\begingroup$ Oh the irony. The one trying to score 'points' by giving attempted witty or 'out of the box' answers, is annoyed by people trying to do it on his own question. $\endgroup$ Nov 10 '14 at 8:37

12 Answers 12

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For a number immensely bigger than $\omega$, consider the uncountably infinite number hidden below. (Note, by the way, that $\omega$ is countably infinite, and rather than being the biggest something, it is in fact “the smallest infinite ordinal ... as it is the least upper bound of the natural numbers” [1]). So omega is a good candidate for the first bonus, the smallest number one can spell if given one of each letter in the alphabet.

Answer:

The transfinite number aleph sixtyfour appears to be the biggest aleph ($\aleph$) number one can spell if given one of each letter in the alphabet.
Note that $\aleph_{64} > \aleph_{63} > ... \aleph_1 = 2^{\aleph_0} > \aleph_0 = \omega$.
(For a big number that doesn't quite work because it has two a's and e's, see wikipedia's Aleph-ω article; aleph omega is the least upper bound of ${\aleph_n : n\in\{0,1,2,\dots}\}$. But if we use five Roman and one Greek letter, or one Hebrew and one Greek letter, aleph $\omega$ or $\aleph_{\omega}$ work ok.)

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    $\begingroup$ @jwpat7 You totally missed what i'm saying. It's not a number, it's a concept. You wouldn't accept 'NAN' (not a number) even if it didn't have two 'N'. $\endgroup$
    – v010dya
    Nov 8 '14 at 23:40
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    $\begingroup$ Not that it affects the answer, but the assertion $\aleph_1 = 2^{\aleph_0}$ is the Continuum Hypothesis. It's independent of Zermelo-Fraenkel plus the Axiom of Choice, which is to say that it's basically up to the individual user whether to call it true or false. $\endgroup$ Nov 9 '14 at 17:53
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    $\begingroup$ How does "aleph omega" have two o's? $\endgroup$
    – Jasper
    Nov 10 '14 at 9:02
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    $\begingroup$ @jwpat7: You're misunderstanding the usages of ordinal numbers. The "infinity" on a calculator is not equal to $\omega$, so far as they're not referring to the same concept in the same domain. $\endgroup$
    – user88
    Nov 10 '14 at 16:31
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    $\begingroup$ @jwpat7 Oh wow. I was trying so hard to see the second "o" that I didn't see either the second "e" or the "a"... $\endgroup$
    – Jasper
    Nov 10 '14 at 16:49
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US Gov. Debt

17,907,911,809,200 at last glance. :P

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  • $\begingroup$ Wow... This is a loop hole, but even so US translates to United States. 3 of the letter "t"and 2 of "e" and "s" $\endgroup$
    – warspyking
    Nov 7 '14 at 20:21
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    $\begingroup$ US is fine, why do you need to translate that to United States? $\endgroup$
    – yuritsuki
    Nov 7 '14 at 23:36
  • $\begingroup$ I've decided to go with the loop holes since there are so many of them, you win! $\endgroup$
    – warspyking
    Nov 7 '14 at 23:43
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    $\begingroup$ @warspyking, this answer is undeserving because he didn't fully spell out the word Government. It is like saying inf. for infinity. $\endgroup$
    – Kenshin
    Nov 8 '14 at 1:12
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    $\begingroup$ @Mew give it a few weeks, and US GOV DEBT will probably be higher than INF. anyway. $\endgroup$
    – corsiKa
    Nov 9 '14 at 3:24
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How about:

Megiston aka. Megistron

A decimal representation of this would require by far more digits than there are estimated atoms in the universe.

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  • $\begingroup$ Clever but that's kinda looking for a loop hole. You have to spell it in word form, you wouldn't be able to use like "pi" because that means "three" then you've used 2 of the letter e so it don't count. $\endgroup$
    – warspyking
    Nov 7 '14 at 20:19
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    $\begingroup$ Hmm? Pi doesn't mean three. $\endgroup$
    – Kevin
    Nov 7 '14 at 20:37
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    $\begingroup$ I think this is a completely legit answer, the question didn't say you have to spell using only 0-9. $\endgroup$
    – Etheryte
    Nov 7 '14 at 23:29
  • $\begingroup$ @Kevin That's as far as you can get though. $\endgroup$
    – warspyking
    Nov 8 '14 at 1:42
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    $\begingroup$ @warspyking Megistron is just as much a loophole as thousand. It's the number's actual name. $\endgroup$
    – NobodyNada
    Dec 9 '14 at 20:31
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You did not specify which alphabet we should be using, so when using greek, the smallest would be

ε

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  • $\begingroup$ This is definitely a loophole, I mean seriously, the question was written in English. $\endgroup$
    – warspyking
    Nov 7 '14 at 20:24
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    $\begingroup$ Very clever trick. @warspyking: I changed ε to "epsilon", in order to use the Latin alphabet. How about now? $\endgroup$ Nov 7 '14 at 21:34
  • $\begingroup$ @Giulio I'm waiting for a response to a meta post.. $\endgroup$
    – warspyking
    Nov 7 '14 at 21:38
  • $\begingroup$ Epsilon is used in English mathematics. It is a widely accepted. $\endgroup$
    – Aura
    Jun 15 '15 at 17:40
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    $\begingroup$ It is widely accepted, yes, but not to refer to any specific number. $\endgroup$ Nov 25 '15 at 17:30
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There's also:

Oktria, which is between 3↑↑↑↑↑↑↑4 and 3↑↑↑↑↑↑↑5 in Knuth's up arrow notation.

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  • $\begingroup$ Just like @GOTO 0 you're kinda looking for a loop hole. $\endgroup$
    – warspyking
    Nov 7 '14 at 20:20
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    $\begingroup$ How is this a loophole? $\endgroup$
    – Dancrumb
    Nov 7 '14 at 22:22
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For the first bonus question:

Although there are lower numbers, none are smaller than zero.

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0
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Just as a fun answer, for a loop hole:

G

This refers to Graham's number (Wikipedia), and is the largest number used in a serious mathematical proof. It is so large it is best expressed as a recurrence relation of Knuth's up-arrow notation as $G = g_{64}$ where $g_1 = 3\uparrow\uparrow\uparrow\uparrow3$ and $g_n = 3\uparrow^{g_n-1}3$ (equations lifted shamelessly from Wikipedia). The phrase "many orders of magnitude" is negligible in the face of this number.

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    $\begingroup$ I thought about this number, too. Unfortunately it has multiple duplicates if you use more than just the first letter. In any case, this is definitely the biggest number mentioned so far. $\endgroup$
    – Rob Watts
    Nov 8 '14 at 6:33
  • $\begingroup$ Definitely just for fun answer, and yes, unfortunate that its full name has so many duplicates :/ $\endgroup$
    – wwarriner
    Nov 8 '14 at 6:43
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Taking the question literately, the best I can do is

five thousand

Venturing outside the box, I can get to

ULONG_MAX which (in the standard C library) comes out to (2^64)-1

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    $\begingroup$ Technically ULONG_MAX is another language. $\endgroup$
    – warspyking
    Nov 7 '14 at 20:27
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    $\begingroup$ Technically C doesn't specify which value is used for that var. On some systems this could be something different. $\endgroup$
    – Nova
    Nov 8 '14 at 7:51
  • $\begingroup$ Nova is correct so perhaps instead of saying in C, say in the C standard library. $\endgroup$ Nov 10 '14 at 15:13
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    $\begingroup$ @warspyking I suppose, although it's only one step of translation from english, which is how most of these answers work. Without the step of translation this question would be kind of boring. $\endgroup$
    – Alex N.
    Nov 10 '14 at 18:06
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    $\begingroup$ @Nova Somewhat. I'll presume that ULONG_MAX really is in the spec (without verifying), so the size would really vary. For the most part 4294967295, or 18446744073709551615, which are still indeed quite big :P so it's valid on pretty much any common implementation. $\endgroup$ Nov 10 '14 at 19:48
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For bonus 1, if you use the convention that negative numbers are "minus x", then I think

minus forty

wins. Checking everything down to minus one hundred is straightforward, and then anything less than minus one hundred must include either "hundred", "thousand", or one of the "-llion" words for higher powers of $10^3$ (or $10^6$, depending on convention, but it doesn't matter). And all of these contain an "n".

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    $\begingroup$ Negative two, although lower than minus one, is also larger. You might have identified the lowest number that can be written each letter of the alphabet only once, but that doesn't mean it's the smallest. Perhaps the question should be reworded, since there is no possible number smaller than the additive identity (zero), and it easily satisfies the conditions of the puzzle. $\endgroup$
    – supercat
    Nov 7 '14 at 14:09
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    $\begingroup$ @supercat -2 is smaller than -1. I don't think your suggested difference between smaller and lower is actually recognised. $\endgroup$
    – JamesRyan
    Nov 7 '14 at 15:16
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    $\begingroup$ @supercat: I think it is entirely normal to consider -2 lower than -1, but -1 smaller than -2. The very notion of size (smallness and largeness) relates to magnitude not ordering. $\endgroup$ Nov 7 '14 at 16:20
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    $\begingroup$ @JamesRyan: Sorry you are so totally wrong and you keep trying to couch your wrongness in my misunderstanding. You're the one declaring a "fact". "Smaller" is an English word and so is "lower". There is no mathematical definition for the word smaller which give you the right to claim some imagined "fact". In math, the symbol '<' has a well defined meaning depending on the number set being used. The terms "smaller" and "lower" don't mean anything mathematically rigorous without a mathematically rigorous definition in the paper (e.g.) to which they are being applied. $\endgroup$ Nov 7 '14 at 18:33
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    $\begingroup$ @JamesRyan, displacement from the origin can indeed have an negative value because displacement is a vector. So if you define one direction as positive and the other as negative, the sign in front of displacement will depend on which direction the displacement is in. But you still didn't answer my question, what velocity is smaller, -300,000,000m/s or -1m/s? It is just a question I pose and a question can't be incorrect. $\endgroup$
    – Kenshin
    Nov 9 '14 at 1:49
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Omega.

From Wikipedia, omega is

the smallest ordinal [number] greater than every natural number.

Note that omega is an transfinite number.

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First question

It depends on the language :P. For example, "centomila" is Italian form of "a hundred thousand") :D

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In hexadecimal

FEDCBA0 (read zero as 'Oh')

Which would be 267242400

In dozenal ( http://www.dozenalsociety.org.uk/roses/devlieger.html )

forsen triliad

Which is 4.279728215×10¹⁴ Doh! Just noticed that i used 'r' twice. So the answer is

twosen triliad

And the value is 1/2 of the prev one.

And i have just noticed the Bonus 2 for using a plus, so here it is:

8+ (Eight plus) This is regex, no less than one eight.

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  • $\begingroup$ This demonstrates a double - i, and a missing double - l in trilliad. Since I don't know the language, I looked up the accepted spellings. $\endgroup$
    – wbogacz
    Nov 7 '14 at 13:39
  • $\begingroup$ Double doh! i guess. $\endgroup$
    – v010dya
    Nov 7 '14 at 14:01
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    $\begingroup$ In fairness, the question does stipulate "in word form", which reasonably eliminates hex representations. Of course, you could exploit the double-entendre that "word" also refers to a 16-bit quantity, or four hex digits, hence "FEDC" would constitute a number "in word form". $\endgroup$
    – COTO
    Nov 7 '14 at 15:58
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    $\begingroup$ @warspyking: "Five thousand" is the largest "non-loophole" number, which can be trivially deduced. If you ask a question with a trivial answer, people tend to get creative. ;) $\endgroup$
    – COTO
    Nov 7 '14 at 23:11
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    $\begingroup$ @COTO Actually i can make an argument that even "five thousand" is a "loophole" number since it uses space, which is not a letter. Basically some people read the question and think "these extra rules are implied" and some say to themselves "i wonder what rules were assumed by the author, that i don't have to assume". $\endgroup$
    – v010dya
    Nov 8 '14 at 7:42

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