# The largest 3-digit number you can write

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That only includes one digit.

I like that, though. If f(3) is the largest number that can be expressed using 3 digits, though (in combination with f() itself), then f(333) must be less than or equal to f(3), f(3...3) (containing 333 3's) must be less than or equal to f(333), and so on. For arbitrarily large x, then, f(x) must be less than or equal to f(3), which suggests f(3) being infinite.

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I'm going to define a function f(x) such that f(x) is the largest possible number that can be expressed using x digits.

Let me know if you can beat f(3).

The problem I see here is that you are not using a predefined operation such as the ones listed in my previous post. You have simply defined some function and named your own operation. Anyone can do that.

I see this particular game as an exercise for one to extend their knowledge of predefined operations that result in large numbers with some additional rules as outlined throughout the thread. Simply defining your own function as an operator seems to defeat the purpose of this exercise. Please note that while I didn't start the thread, I'm trying to uphold the spirit of the game as stated in the OP.

I thought we were only ruling out division by 0, not infinity itself. The set of hyperreals *R includes infinities and infinitesimals, and we can divide a finite number by an infinitesimal to yield infinity. But, no matter.

So now the rules are:

1. Use three digits and at most one operator (of any convenient arity--and is this one operator per digit, or one operator for the entirety of whatever is constructed by those digits?).

2. We cannot define new operations, though operations defined in the places mentioned in your post are fine.

3. The result must be a real number.

Anything I'm missing?

1.) The rules of the game should be to post an expression that uses three single digit base 10 whole numbers to achieve the highest resulting value possible (excluding infinities).

As stated in my first post in the spoiler:

Use an infinite base such that we have an infinite number of single digit numbers or symbols to represent a number. However a more practicle base would be something like base $2^{64}$. That way a single 64 bit number would represent one symbol / digit. We can always choose a larger base. The point being made is that we can always choose a set that has more symbols / elements and push the resulting value of the operation, whichever one you choose, higher.

2.) We cannot define new operations. Only use predefined operators that can be found in Wikipedia, Wolfram, or a website / paper published by an educational institution such as Cornell University's arXiv. Any such operation must be referenced in the post.

3.) We should rule out using an operation recursively such as nesting a factorial over and over again, and that only one unary operation can be applied to a number to prevent nesting such number in a sea of unary operations. Nesting a unary operation implies $n$ recursions, which hides yet another variable or single digit number. For instance, Knuth's up-arrow notation is extended by incorporating a variable that defines such recursion:

$a \uparrow^{n} b$

However, using a predefined operation such as the extended Knuth's up-arrow notation is fine. You just can't invent an operation that cannot be found as defined in number 2.

4.) The result must be a finite real number. Of course, integers are also acceptable.

Any other rules we should impose?

Edited by Daedalus

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So, speaking for the numerically illiterate, when I ended a sentence with a number and exclamation mark, I was 1) upsetting the mindset of a mathematician, 2) displaying my numerical illiteracy to all?

So, no change, then...

I'd just like to register a protest at the wholescale thievery of the already spoken for letters and symbols of the language of alphabets. It's not like you're thick. Keep to the original symbols you come up with, and leave us to bunker down with the last few letters and symbols we can be sure you haven't seduced and taken to your mathematical Las Vegas, where every figure probably has a 2nd, secret life, in an amoral, louche existance...

To those who left, remember we will always love you, and you can always come home, even if your reputation is dust.

Have you taken the innocence of alL, already?

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So, speaking for the numerically illiterate, when I ended a sentence with a number and exclamation mark, I was 1) upsetting the mindset of a mathematician, 2) displaying my numerical illiteracy to all?

http://xkcd.com/859/

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What is the largest number value in base-10 you can write with just 3 digits?

No symbols and characters allowed.

Hints: it's not 999

Ask someone to write the largest 3-digit number and they'll respond with 999.

Logical answer, but we can go bigger.

Some may get the "power" brainwave and think of 999 (99 to the power of 9), which calculates out as 99×99×99×99×99×99×99× 99×99.

Even better is 999 (9 to the power of 99) which calculates out as 9×9×9×9×9×9×9 ... and so on 99 times.

The correct answer, however, if you extend the idea even further ends up as... 99^9 (9 to the 9th power of 9).

Work out the second and third powers first (9×9×9×9×9×9×9×9×9 = 387420489.) We can therefore restate the sum as 9387420489

which works out as.... very very big indeed.

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Many have ignored the criterium "No symbols and characters allowed". Of course there is an implied "other than the three digits used to form the number".

A base-10 digit is implied to be any one of the ten Arabic numerals. A number is an expression used to represent an arithmetical value. 123 is an expression. The position of each digit defines the arithmetical value: 1×102 + 2×101 + 3×100. As can be witnessed, an operation is automatically performed on the digits, thus operations are implied as acceptable in expressing a number.

Exponentiation is an operation that does not require any addtional symbols, Positioning of the digits or expression is used, with a "base" part and a "raised" part (superscription). 456 is equivalent to the arithmetical value: (4×101 + 5×100)6.

Tetration is another operation that does not require any additional symbols. Similar to exponentiation, tetration also uses superscription, but rather the "raised" number appears to the left of the base number instead of the right. 35 is equivalent to the arithmetical value (55)5, which is equivalent to 298,023,223,876,953,125. [Parentheses are used here as it is difficult to superscript beyond one level of superscripting using ASCII text.]

The largest number using only 3-digits and no other symbols is 9 tetrated to the 9th tetrated to the 9th. [using parentheses to indicate superscripting beyond the first level, this can be written 9(99).]

It is possible to define a new operation by using only 3 digits and positioning that is unambigous to already defined operations, yet any new operation that has not yet found acceptance by the authorative individuals of the mathematical community remains unacceptable by the given criteria. As the operation can be considered not pre-defined, and hence invalid.

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Daudalus

The problem I see here is that you are not using a predefined operation such as the ones listed in my previous post. You have simply defined some function and named your own operation. Anyone can do that.

Actually you will find the Max function defined in many maths textbooks, particularly those that deal with numerical methods (as we are) or metrics.

This is a fun thread though and it shows just how ingenious people can be.

Edited by studiot

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My question is, though, if we're using operators, aren't we now dealing with expressions, not numbers?

I mean I can express 81 as 99, but 81 is the numeric representation of that expression.

Edit: Wrote this before I saw dejmar's post, above.

I'm not sure I agree with the gist of the idea, but I can't fault the logic. If you look at numbers in that fashion, then a certain amount of operations would be permissable to write a "number".

Edited by Greg H.