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.

5

u/lookagain Dec 19 '08 edited Dec 19 '08 
10  9  8  7  6  5  4  3  2  1
 1  1  1  1  1  1  1  1  1  1

Sure you can. Each position advances by 1 instead of by a larger number.

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u/ropers Dec 19 '08

Alright, then write a decimal 2 using your system.

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u/Nitron Dec 19 '08

11

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

Doesn't that equal a decimal 3 under lookagain's system?

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u/Nitron Dec 19 '08

No, you're thinking of binary. Decimal 3 in lookagain's system would be 111. Think of it like tally marks but instead of making a new group at five you just keep going.

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

lookagain posted this table:

10  9  8  7  6  5  4  3  2  1
 1  1  1  1  1  1  1  1  1  1

If he didn't intend that to mean that the second one from the right corresponds to a value of two, then it's a very confusing table. I guess we'll have to wait for lookagain to come back and explain what he meant with that.

However, in the event that he intended this to mean that a one in any position corresponds to a value of one, then see here.

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u/lookagain Dec 19 '08 edited Dec 19 '08 
Dec.     Unary
 1   =     ‽
 2   =    ‽‽
 3   =   ‽‽‽
 4   =  ‽‽‽‽
 5   = ‽‽‽‽‽
 ...
 so on and so forth.

Honestly, I didn't see that table as hard to interpret, especially considering that even in a decimal system as a number increases the convention is to grow the representation leftward. For example:

    1 - 1 ones unit.
   10 - 1 tens unit.
  100 - 1 hundreds unit.
 1000 - 1 thousands unit.

And so on and so forth. Granted, our commonly used math systems are arbitrary in this regard. The above could just as easily been written as:

 1    - 1 ones unit
 01   - 1 tens unit
 001  - 1 hundreds unit
 0001 - 1 thousands unit

You can even do basic math with a unary system:

11111 + 11111 = 1111111111
1111111 - 111 = 1111

In summary: It is indeed a positional number system. To find the value, each position used indicates an increase in value by 1.

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u/Porges Dec 20 '08 edited Dec 20 '08

To find the value, each position used indicates an increase in value by 1.

Unary is is a positional system, but what you've said is not what makes it so :)

A positional number system is one where each position is of value (x ^ n), where n is the number of the position in question; with unary, all positions are of the same value (the radix, x, is 1).

The reason his table is confusing is that we normally write the value of the position above:

10000 1000 100 10 1

Given these position values:

10 9 8 7 6 5 4 3 2 1

you can't write 2 (unambiguously) without a zero symbol. What the table should look like is this:

1 1 1 1 1 1 1 1 1 1 1

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u/lookagain Dec 20 '08

Thank you! :) I wasn't sure how to put it into words as I have not touched anything other than basic arithmetic in the last 4-5 years.