# Expressing a number in the least possible space

I need to express a list of integers in a book, all falling between 1 and about 50,000, but I have limited space to print them. For e.g. "300009" takes up 6 characters. Put simply, I need to save paper. I'd like to either find a way to reduce the number of characters needed to express the numbers or replace the numbers with something that takes less horizontal or vertical space, but still represents the information reasonably well.

• If the numbers are above 100, their exact value is unimportant. I only need to display a ballpark figure, i.e., to say that the value is somewhere between 100 and 199, 200 and 299, ... 1000 and 1999, 2000 and 2999, ... 10000 and 19999, ... 100000 and 199999, etc. Unfortunately, I find this does not seem to help me, as rounding the number up or down doesn't reduce the number of characters used and displaying the range for "300009" takes up 13 characters (300000-399999), which is considerably longer.
• I have tried looking at using other bases, but these seem to make the numbers unreadable. The numbers should not require too much effort to understand.
• I explored various alternative numeral systems from other cultures (past and present), as these are not so difficult to learn, but could not find anything more condensed way to express "300009" or a ballpark range for this.

Does math notation provide any solution? Is there a short-hand notation which can express these kinds of ranges with fewer than 6 characters to express "300009"? Are there some other numeral systems I should consider?

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You could count the number of irrelevant digits. For instance, 300000-399999 could be 3+5. Depending on how important easy human readability is and on how often the number of irrelevant digits is zero, you could consider omitting the + and always letting the last digit stand for the number of irrelevant digits; in that case 300000-399999 would be just 35, but then you'd have to add a 0 when there are no irrelevant digits to avoid ambiguity.

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Something custom like this could work. If I can't find a more standard way (some system used by other people), I'll consider this. I might need to change the 5 in some way though, as numbers below 100 are more important, so the actual number 35 is not confused with something between 300000 and 399999. –  Village Nov 24 '11 at 13:48
@Village: You could use letters -- either a regular sequence like A, B, C, ... or perhaps roman numerals: X, C, M, continued with some letters of your choice. –  joriki Nov 24 '11 at 13:57
I did see a document some time back displaying numbers like this: 1^IV, 2^III. That could probably work well. I'm not sure what meaning was intended by that document. –  Village Nov 24 '11 at 14:04
That seems like a huge waste of space. You don't need the caret, and there's no reason to have multiple characters for the exponent if it doesn't exceed 5 or 6 in your case. I didn't suggest to express the exponent in Roman numerals, but to use Roman numerals as single letters that are readily associated with the right exponent. This could be interpreted as multiplication of the two numbers, i.e. 3M would stand for 3000, and thus for 3000-3999. –  joriki Nov 24 '11 at 14:08

Scientific notation such as $3.53 \times 10^4$ for 35300. Round to the number of significant digits you want and/or use compact notation (i.e 3.53E4 is used frequently in programming).

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Oh, I spent much time these past few days trying to remember and find the name for that. It seems difficult to do a Web search for numbers. Is the use of "E" ever used in print or only in the context of calculators? –  Village Nov 24 '11 at 13:44
I've only seen the E in calculators and programming languages (it's often the default for floats et al.). In any case, you can devise a pretty compact notation based on scientific, describe it in a footnote and be done with it. –  alex Nov 24 '11 at 13:54
That seems okay. Is it fairly common that people will create a new or alternative when they want to describe something and introduce it to readers (such as when publishing)? –  Village Nov 24 '11 at 14:06

If the numbers are close together, you can use xxyyyy, where xx is set in the leader columns, and the tabular entries give yyyy. This is fairly common on old books of tables.

Railway enthusiest use a scheme where only the last changed digit are shown, eg the number list 81003/8/14/04 means that engines 810003, 810008, 81014 and 81004 are hooked up in that order.

If you want to display digits, where the known digits are important, use X for the missing digits, eg 31XXXX. You can use other letters if the XXXX has several meanings, eg 31XYYY works.

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