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

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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 99

^{9}(99 to the power of 9), which calculates out as 99×99×99×99×99×99×99× 99×99.

Even better is 9

^{99}(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... 9**.

^{9^9}(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

**9**which works out as.... very very big indeed.^{387420489}
What about 9^9^9?

ReplyDeleteBut they are not digits. They are exponents, or indices.

ReplyDeleteWrong: 999 is correct.

ReplyDeleteThe question states ..."just 3 digits?"

As I understand the English language, the question is therefore exclusive of all characters, representations and symbols that are not [digits]. This would then exclude operators, including implied such as in exponentiation, and special symbols such as (!) factorial.

Had the question been worded "...with three digits" there might be more room to argue that operators and symbols are not excluded however this then becomes a trick question. To be direct why not go with "Whats the largest numerical value that can be represented with 3 numbers and 3 symbols?"

David, Benjamin, Paul, et al.,

ReplyDeleteYou are mistaken. We can not exclude implied operators, and indices and exponents can be represented, though not limited to, with digits, just as a number is not always expressed in numerals. A number can be expressed as an expression or variable, as well. The criteria given for the answer to the question does exclude variables, but it does not exclude expressions that are implied through positioning that require no symbols other than any of the included three digits (numerals).

In the positional numeral system, the number 999 has the implied operators of multiplication, exponentiation, and addition -- i.e, 999 = 9*(10^2)+9*(10^1)+9*(10^0). Not shown in the given expression is the radix, also often called the base. The notation of the radix, often subscripted to the right of the base number when given, is given when their are mixed bases or the number is represented in a base different than that of the default -- which is not always 10. For the given question, we are given that the number represented is base-10. It is not required to be shown to be represented.

The answer given is, indeed, not the correct answer. The number 9^(9^9) = 9^387420489 is very large, yet there is an operation that produces higher numbers than exponentiation. The operation, like exponentiation, can be expressed with superscription. The difference to exponentiation is the superscripted number, variable, or expression is superscripted to the left as opposed to the right of the base number. The ASCII notation of exponentiation is using a single caret following the base number, variable, or expression -- e.g., 3^2. The ASCII notation for tetration is using a double caret following the base number, variable, or expression -- e.g., 3^^2. As stated, these two ASCII notations can be represented without the caret symbols by use of superscription, a type of positional notation. Tetration produces the larger results. The number 9^^9 is equivalent to 9^(9^(9^(9^(9^(9^(9^(9^9))))))). 9^^(9^^9) is immensely larger. It may be the largest number represented with only three base-10 digits and no other symbols.

What about (((((999!)!)!)!)!)!

ReplyDeleteNo symbols and characters (other than the three digits) are allowed. The factorial symbol (the exclamation point) is disallowed, so are the grouping notation (bracketing) symbols.

Delete9!! is the largest

ReplyDeleteYea the technically correct but it ends up as a bigger number after that...

ReplyDelete