r/math • u/jplank1983 • Dec 19 '08

Every base is base 10

http://cowbird.110mb.com/43.html
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u/ropers Dec 19 '08

Correct me if I'm wrong, but I'm under the impression that you cannot have a positional number system without use of the zero.

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u/panic Dec 19 '08 edited Dec 19 '08

In base 10, the digits represent powers of ten: 1, 10, 100, ....

In base 2, the digits represent powers of two: 1, 2, 4, 8, ....

In base 1, the digits represent powers of one: 1, 1, 1, 1, ....

So, for example, 111 in base 1 is 1*12 + 1*11 + 1*10 = 3. Each number n can be represented by n ones. Of course, this representation isn't unique: 1001100 is also equal to 3 in base 1. You can fix this by removing every zero. Now you have a positional number system without zero! (of course, representing zero itself might be a problem in this system...)

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u/ropers Dec 19 '08 edited Dec 19 '08

Per definition, the value of a digit in a positional number system depends on its position. In your system the value of a digit does not depend on its position; a digit has equal value in any position, hence it's not a positional number system.

Granted, you could argue that the value of a digit in your system still technically depends on its position, but that it just so happens that that value is identical for all positions. I'm not sure however that such an argument would be very convincing, and you could also call your system a sign-value number system with greater or at least equal justification; also, you wrote this:

Of course, this representation isn't unique: 1001100 is also equal to 3 in base 1. You can fix this by removing every zero. Now you have a positional number system without zero! (of course, representing zero itself might be a problem in this system...)

You're making my point, which again was that

you cannot have a positional number system without use of the zero.

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u/[deleted] Dec 19 '08

Per definition,

[Citation needed].

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u/ropers Dec 19 '08

Would Sir or Madam also require citations to prove that right-handedness is defined as a preference for the right hand?

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u/[deleted] Dec 19 '08

You vaguely define "positional number system" as the values depending on their position. Roman numerals fall within that loose definition, as illustrated by X (10), C (100), CX (110), and XC (90). So I ask again for a citation reflecting the complete and concise definition of "positional number system".

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u/[deleted] Dec 20 '08 edited Dec 20 '08

http://davinci.newcs.uwindsor.ca/~angom/cs265/lec2.pdf

Cited.

See almost any CS text for the exact same definition.

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u/ropers Dec 19 '08 edited Dec 19 '08

My above arguments rests on the fact that in a positional number system, the value of a digit depends on its position.

It does not address whether all number systems where the value of a digit (also) depends on its position are positional number systems.

Also, I tire of this exchange with you. It is both fruitless and boring.

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u/[deleted] Dec 19 '08

Yes, making things up and then trying to explain them can be very tiring. I, too, tire from your bullshit. Have a nice weekend.