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| | Hmmm... What do you mean by '''base doesn't need to be an integer'''? Especially suspicious is the idea to use '''complex''' bases. What numbers are you going to represent in such bases and what are the digits? In my opinion, this part (at least, as written) is rather confusing than revealing... --[[User:Fedja|Fedja]] 19:34, 22 June 2006 (EDT) | | Hmmm... What do you mean by '''base doesn't need to be an integer'''? Especially suspicious is the idea to use '''complex''' bases. What numbers are you going to represent in such bases and what are the digits? In my opinion, this part (at least, as written) is rather confusing than revealing... --[[User:Fedja|Fedja]] 19:34, 22 June 2006 (EDT) |
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| − | We can have base <math>i</math>, base <math>\phi</math>, and improper fractional bases like 3/2. In a rational base, any integer may be represented without a decimal point in that base. For complex and irrational bases, we use (I think) 0 or 1 as digits. For a fractional base p/q with max(p,q)=a, we use 0 up to a-1 for digits, i.e. for 3/2, 0,1,2 are digits; for 1/4, we use 0,1,2,3. --[[User:Solafidefarms|solafidefarms]] 21:11, 22 June 2006 (EDT)</math> | + | We can have base <math>i</math>, base <math>\phi</math>, and improper fractional bases like 3/2. In a rational base, any integer may be represented without a decimal point in that base. For complex and irrational bases, we use (I think) 0 or 1 as digits. For a fractional base p/q with max(p,q)=a, we use 0 up to a-1 for digits, i.e. for 3/2, 0,1,2 are digits; for 1/4, we use 0,1,2,3. --[[User:Solafidefarms|solafidefarms]] 21:11, 22 June 2006 (EDT) |
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| | Hmmm..., so how would you represent, say 4 using base 3/2 and digits 0,1, and 2? Also, what would, say, 4 be when written in base <math>i</math> (the latter question is especially interesting, because all possible powers of <math>i</math> are <math>1</math>, <math>i</math>, <math>-1</math>, and <math>-i</math>). Maybe you mean something here but, since even I cannot understand you, there is no hope that 12 year old kids will...--[[User:Fedja|Fedja]] 21:42, 22 June 2006 (EDT) | | Hmmm..., so how would you represent, say 4 using base 3/2 and digits 0,1, and 2? Also, what would, say, 4 be when written in base <math>i</math> (the latter question is especially interesting, because all possible powers of <math>i</math> are <math>1</math>, <math>i</math>, <math>-1</math>, and <math>-i</math>). Maybe you mean something here but, since even I cannot understand you, there is no hope that 12 year old kids will...--[[User:Fedja|Fedja]] 21:42, 22 June 2006 (EDT) |
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| | After closer examination of this article, I decided that the link to [[Improper fractional base]] belongs in the Base Number Topics section. | | After closer examination of this article, I decided that the link to [[Improper fractional base]] belongs in the Base Number Topics section. |
| | --[[User:Xantos C. Guin|Xantos C. Guin]] 23:28, 7 August 2006 (EDT) | | --[[User:Xantos C. Guin|Xantos C. Guin]] 23:28, 7 August 2006 (EDT) |
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This is a great start to an article. The early part could use a little white space. Students who really need to learn about base numbers from this article are going to need a mental pause, and multiple examples. A section on conversions would be nice.--MCrawford 00:42, 20 June 2006 (EDT)
Hmmm... What do you mean by base doesn't need to be an integer? Especially suspicious is the idea to use complex bases. What numbers are you going to represent in such bases and what are the digits? In my opinion, this part (at least, as written) is rather confusing than revealing... --Fedja 19:34, 22 June 2006 (EDT)
We can have base 
, base 
, and improper fractional bases like 3/2. In a rational base, any integer may be represented without a decimal point in that base. For complex and irrational bases, we use (I think) 0 or 1 as digits. For a fractional base p/q with max(p,q)=a, we use 0 up to a-1 for digits, i.e. for 3/2, 0,1,2 are digits; for 1/4, we use 0,1,2,3. --solafidefarms 21:11, 22 June 2006 (EDT)
Hmmm..., so how would you represent, say 4 using base 3/2 and digits 0,1, and 2? Also, what would, say, 4 be when written in base (the latter question is especially interesting, because all possible powers of are 
, , 
, and 
). Maybe you mean something here but, since even I cannot understand you, there is no hope that 12 year old kids will...--Fedja 21:42, 22 June 2006 (EDT)
First, non-integral bases are not an introductory topic; more of Intermediate or Olympiad. (I have heard it said by several that my work in improper fractional bases is above even that level.)
There is a special method of conversion which works on any rational base greater than 1, not just integral bases, which we use instead of power methods for improper fractional bases. Basically, we let the number we wish to convert be the rightmost digit. Then, while any of the "digits" of our number are greater than/equal to the numerator of our base, we subtract the numerator from that digit and add the denominator to the next digit to the left. I.e., to convert the number 23 to base 7 using this method, we start with the digit 23. 23>7, so we subtract 7 and add 1 to the next digit: 1|16. (We use | as a matter of convenience, to separate the digits.) Again we do it, till we get 3|2. Now none of the digits are more than 7, so we are through, and 23_7=32_10.
For, say, 7, in base 3/2, we start with 7. Subtract 2 3s and add 2 2s: 4|1. Again, and we have 211_3/2=7_10. If you want to read more, I have a paper on improper fractional bases at [1]. (I went to the International Science and Engineering Fair with improper fractional bases :) )
I'm not sure about complex bases. Complex bases are not terribly well-explored. A google doesn't turn up much.
Now then, I am strongly in favor of putting links to individual pages on complex/irrational/fractional bases, because I could go on forever on improper fractional bases.--solafidefarms 22:08, 22 June 2006 (EDT)
OK, I looked at your paper and, at least, understand now what you mean by "fractional bases" (though complex and irrational ones remain a mystery to me). I think you should ask some of the admins whether it is an appropriate stuff for AoPSWiki, and, if they decide it is, write a separate article on it first. If you succeed, there will always be time to link it to this one.--Fedja 23:01, 22 June 2006 (EDT)
Indeed, yes, complex bases I know little about. Irrational bases I've been exposed to in Art and Craft of Computer Programming a little, but they are really strange. I'll work sometime on an improper fractional base article sometime separately. --solafidefarms 23:09, 22 June 2006 (EDT)
All math topics are fine to write about. Just create new article appropriately when the amount of content justifies its own article.--MCrawford 23:28, 22 June 2006 (EDT)
Complex number bases do exist as proven by this AIME problem: http://www.artofproblemsolving.com/Forum/viewtopic.php?p=436603#p436603 But that's about all I know about complex numbers for number bases :( Joe 10:07, 23 June 2006 (EDT)
Neither of the really important properties of bases are mentioned in this article -- nowhere does it say that base representations exist nor that they are unique. Nor, in fact, does it address at all the notion of representing any numbers other than smallish integers. --JBL 09:00, 17 July 2006 (EDT)
Ack, don't ya'll think we should have just a /little/ more text on here? On Wikipedia, generally even if there is a sub-article an overview remains on the main page (see [2])
for an example). Besides which, improper fractional bases needs a link :D. --solafidefarms 22:13, 7 August 2006 (EDT)
Do you mean you want a link to Improper fractional base in the See Also section? If so, I'll add it. Remove it if it doesn't belong here.
--Xantos C. Guin 23:25, 7 August 2006 (EDT)
After closer examination of this article, I decided that the link to Improper fractional base belongs in the Base Number Topics section.
--Xantos C. Guin 23:28, 7 August 2006 (EDT)
, base 
, and improper fractional bases like 3/2. In a rational base, any integer may be represented without a decimal point in that base. For complex and irrational bases, we use (I think) 0 or 1 as digits. For a fractional base p/q with max(p,q)=a, we use 0 up to a-1 for digits, i.e. for 3/2, 0,1,2 are digits; for 1/4, we use 0,1,2,3. --solafidefarms 21:11, 22 June 2006 (EDT)
, 
, and 
). Maybe you mean something here but, since even I cannot understand you, there is no hope that 12 year old kids will...--Fedja 21:42, 22 June 2006 (EDT)
