## Large Numbers, Few Characters

I wonder what the largest number is that can be represented with one character. Probably 9. How about two characters? Is it 9^{9}? What about three or four?

I guess I should define a character. Let’s have two separate cases. In

the first one you can only use keyboard characters. In the second one

you can use any Unicode characters.

I’m awaiting your answers to this.

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## udalrich:

Wouldn’t c (the continuum) be the largest single character number? I suppose that you should also define number. ðŸ˜› A transfinite number has “number” in it’s name.

When we allow Unicode characters, we can argue if ∞ is a valid answer.

If we disallow transfinite numbers and ∞, NA (Avagadro’s number ≈ 6 × 1023) beats 99.

8 June 2011, 10:48 am## udalrich:

It appears that you can’t used sup and sub tags in comments. 1023 should be 10^23 and 99 should be 9^9. The A in NA is supposed to be a subscript.

8 June 2011, 10:50 am## Sachin:

Write an 8, and tip it over.

8 June 2011, 11:09 am## Brenda:

or 1/0 for three characters… If we can use symbols that stand for large numbers, then my answer is “X”, where “X” is defined as one more than anyone else’s non-infinite answer. 1 character, I win.

8 June 2011, 11:16 am## Nathan:

H(9) for hyperfactorial 9.

H(9) = 1^1 * 2^2 * 3^3 * 4^4 * 5^5 * 6^6 * 7^7 * 8^8 * 9^9

8 June 2011, 11:51 am= 21,577,941,222,941,856,209,168,026,828,800,000

## Vortico:

The largest number is about 45,000,000,000, but mathematicians suspect there may be even larger numbers.

http://www.youtube.com/watch?v=Pj2NOTanzWI&feature=player_detailpage#t=78s

Joking aside, the tetration is a fast-growing function, so with two characters we can write “raise 9 of 9” (^9 9 in LaTeX) to denote 9^9^9^9^9^9^9^9^9. You could do better with Roman numerals: ^M M = 1000^ . . . ^1000. Since a “barred” M represents 1,000,000, here is my final solution for two characters.

http://files.trivialqed.com/MM.png

Other functions such as Ackermann and Busy Beaver grow faster than tetration, but these are of the form f(n) and cannot be represented by two characters.

8 June 2011, 2:55 pm## Andrew MW:

For 4 digits…

A tower of exponent 9s gets to a fairly substantial number, around 696 digits I believe.

In unicode, the obvious first choice would be Knuth up arrows… assuming they are in Unicode?

9 [uparrow] [uparrow] 9 is a 9-high tower of exponent 9s – a massive number.

Or how about Î£(9) – where Î£ represents the busy beaver function? I have no idea how many digits that has, buts it’s a lot.

8 June 2011, 4:28 pm## Richard:

We could represent the largest number imaginable as a single character (suppose: I) provided we all had a common definition for said character. Even to the characters 0,1,2,…,9 we still need our common definitions. Allowing for tetration we could see that a tetrated a-times (which can be written, and thus expressed, with only two characters is exceedingly large; to add value we can simply replace a with 9 (or even F if we allow for changes in base). This though more meaningful than my first answer is still only as meaningful as our common definitions.

A third idea: consider changing the definition of a character, a character is configuration of pixels within a rectangular container, since the resolution and area of this box is finite we can assign a natural number to each character. With this we could then say that dependent on the resolution there is a definite answer.

I prefer the first :); which brings me to the second question. If we could represent the largest number imaginable as a single character, then we could indeed represent an even larger number using two characters. this contradicts leaving us with this: the set of numbers (subset of the Natural numbers) representable by a single character is unbounded.

8 June 2011, 4:34 pm## Andrew MW:

And for two digits, what about the ninth term in the definition of Graham’s number, g[subscript]9… just a little bit bigger than Avogadro’s number…

That also gives new candidates for three and four digits… i.e. g[subscript]g[subscript]9 and g[subscript]g[subscript]g[subscript]9 … now we’re really talking. And no Unicode in sight.

8 June 2011, 4:37 pm## Andrew MW:

last post … this is a nice little article premised on a similar challenge… http://www.scottaaronson.com/writings/bignumbers.html

8 June 2011, 4:43 pm## Jonathan:

With keyboard only, why not go back to the Romans? M

9 June 2011, 12:18 am## prithwiraj:

with english letters, and using base 37, one can go upto 36 with “z”

9 June 2011, 1:57 am## Aaron F.:

I remember well from my Hebrew school days that Hebrew letters were once used as numerals. Alas, the greatest letter (character 05EA in the Unicode 6.0 standard) has a value of only 400.

Fortunately, I also went to public school, where I learned that the ancient Egyptians had hieroglyphs that stood for numbers. According to Wikipedia, Unicode character 1304F was sometimes used to represent the number one million. (Legend has it that the Israelites left Egypt carrying gold, silver, and clothing given to them their former masters. Maybe they should’ve taken some numbers, too…)

Fumbling around with a Chinese-English dictionary, I was disappointed when I only found two-character expressions for one million and one billion. It seems, though, that one trillion can be written in Chinese using just one character—Unicode character 5146!

9 June 2011, 3:39 am## prithwiraj:

oh, i forgot. if you are using latex, the escape sequence $infty$ will produce a single character that cannot be surpassed

9 June 2011, 6:56 pm## Gregory Marton:

I’m surprised that even Aaron F. didn’t think about Aleph as infinity, or for two characters, aleph_1, which, I would claim, is the largest infinity yet mentioned.

9 June 2011, 10:54 pm## Christ Schlacta:

with unicode, we can use UTF-32, which the highest character is U+4294967295, using 2 digits, you can use U+4294967295^U+4294967295, and for 3 digits, you can do (U+4294967295^4294967295)^4294967295 and so on ad infinitum, the formula is highest digit [raised to the highest digit[, the quantity raised to the highest digit] […] ]

20 June 2011, 5:19 pm## David Wilson:

“Love Symbol #2”, the unpronounceable squiggle which is the true name of the artist formerly known as Prince, has the value “one more than the largest number you can describe.”

29 June 2011, 1:26 pm## jim farrington:

Hoping that i might be allowed to count four as the fifth character after zero,

4 October 2011, 1:53 pmi would like to see if i could sneak in the old

9!^9!

for consideration.

## Moo:

Instead of 9!^9!, how about 9!!! ?

Or: Gamma sub(Gamma sub(Gamma sub(9)))

(http://www.greatplay.net/essays/large-numbers-part-4-epsilon-and-phi – Feferman-Schutte ordinal)

Bachmann-Howard ordinal?

Or just G, for Graham’s number.

10 October 2011, 7:13 pm