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Author Topic: EP Flash: Off Base  (Read 13391 times)

Russell Nash

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Reply #25 on: June 10, 2009, 12:09:36 PM
One must work with the material one has at hand...
Hey.
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Moderator - Do we permit puns in this forum?

It has at times been almost encouraged.




Unblinking

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Reply #27 on: January 28, 2010, 02:53:05 PM
This was a fun story, though a few seconds after the punchline I also wondered why they didn't translate the number into decimal base when translating everything else into English.  But no matter.

And an interesting sub-topic:  Did you know that ALL numbering systems are really base 10?  It sounds wrong on first read, but it's true!!  Think about it!



eytanz

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Reply #28 on: January 28, 2010, 03:54:10 PM
And an interesting sub-topic:  Did you know that ALL numbering systems are really base 10?  It sounds wrong on first read, but it's true!!  Think about it!

Not true; the string "10" is meaningless in base 1.



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Reply #29 on: January 28, 2010, 03:59:42 PM
And an interesting sub-topic:  Did you know that ALL numbering systems are really base 10?  It sounds wrong on first read, but it's true!!  Think about it!

Not true; the string "10" is meaningless in base 1.

But base 1 itself is meaningless.  You can't have a number system with only a single digit.  Well, you can, but it's not very useful.  "I have 0 hamburgers" could mean that you have no hamburgers or millions of hamburgers, there's no information contained in the number "0".  Sort of like Smurf language (or Marflar).



eytanz

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Reply #30 on: January 28, 2010, 04:03:34 PM
And an interesting sub-topic:  Did you know that ALL numbering systems are really base 10?  It sounds wrong on first read, but it's true!!  Think about it!

Not true; the string "10" is meaningless in base 1.

But base 1 itself is meaningless.  You can't have a number system with only a single digit.  Well, you can, but it's not very useful.  "I have 0 hamburgers" could mean that you have no hamburgers or millions of hamburgers, there's no information contained in the number "0".  Sort of like Smurf language (or Marflar).

Nonesense. Let me count from 1 to 10(decimal) in base 1 :

1
11
111
1111
11111
111111
1111111
11111111
111111111
1111111111

(or, alternatively, 0 00 000...)

Ok, so large numbers get quite long, but that doesn't make it useless. I'd argue that it's far more useful than base 126335745621, for example, as you'd run out of symbols before you can reach 2 digits in that base.



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Reply #31 on: January 28, 2010, 04:16:40 PM
There are still two possible values of each digit: " " and "1", which still makes it binary, but represented in a slightly different way.  :)  To carry information it has to be at least base 2.

Just like counting on your fingers could be seen as the same binary counting system, but with a limit of ten digits.  Every finger has a value of "up" or "down".  The human way of finger-counting is not an efficient way of using 10 bits--if you counted in the more typical binary way then you could have 1024 possible values--but it is still binary.

In any case, I suppose I could have stated in the original statement that my assumption that all numbering systems are base 10 was the digits are counted by having the rightmost digit count from 0 to the maximum value, then reset to 0 and simultaneously incrementing the digit to the left of it, with 0's implied whenever there is blank space.  But that wouldn't be any fun to state everything so explicitly.  :)



eytanz

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Reply #32 on: January 28, 2010, 04:33:02 PM
There are still two possible values of each digit: " " and "1", which still makes it binary, but represented in a slightly different way.  :)  To carry information it has to be at least base 2.


That's not true. Note that the unary system I proposed is limited - it cannot express 0. But it is a unary base system, and it does carry value. Note that if " " was really another symbol for "0", then "111 111 111" could be a single number, which it isn't.



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Reply #33 on: January 28, 2010, 04:51:33 PM
There are still two possible values of each digit: " " and "1", which still makes it binary, but represented in a slightly different way.  :)  To carry information it has to be at least base 2.


That's not true. Note that the unary system I proposed is limited - it cannot express 0. But it is a unary base system, and it does carry value. Note that if " " was really another symbol for "0", then "111 111 111" could be a single number, which it isn't.


It isn't a single number because the counting system you've laid out limits the information storage, but it's not inherently unary.  Let's say I'm counting on the fingers of one hand, and I have "11 11"  (all fingers extended except my middle finger).  You could say that means twenty-seven (as it means in computer binary), or four (by counting the digits), or a two and then another two (by counting separate groups of digits as different numbers).  But all of those are simply based on different counting systems--in all cases, there is binary information stored. 

Likewise if I make a fist, extending no fingers, that means something even if there is no digit "0".  Hypothetically:
eytanz:  How many hamburgers do you have?
Unblinking:  I have no hamburgers.

Even though there is no way to represent zero with numerals in this case, the value "zero" is implied by the word "no".

*I'm having fun with this, a sure sign that I picked the right field by going into engineering, but this is only somewhat related to the story topic.  Perhaps we should find a different place to debate numbering systems?  :)  *



eytanz

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Reply #34 on: January 28, 2010, 05:38:11 PM
It isn't a single number because the counting system you've laid out limits the information storage, but it's not inherently unary.  Let's say I'm counting on the fingers of one hand, and I have "11 11"  (all fingers extended except my middle finger).  You could say that means twenty-seven (as it means in computer binary), or four (by counting the digits), or a two and then another two (by counting separate groups of digits as different numbers).  But all of those are simply based on different counting systems--in all cases, there is binary information stored. 

Yes, in your example. But the point of my unary system is that you don't need to store any binary information. You can use it if you're only storing units. Your space in the middle finger plays absolutely no role in the maths. What you're doing is equivalent to me saying that the following two (base 10) numbers are different: 100 and 100, because one is black and one is red. But that's extraneous information, it plays no role in that math. All the 0/" " information you are describing is similarly incidental.

I'm not the person who came up with this concept. Here's the wikipedia page on unary numeral systems: http://en.wikipedia.org/wiki/Unary_numeral_system

(Note: I agree that this discussion could be happily split off, if the mods so wish)




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Reply #35 on: January 28, 2010, 05:53:53 PM
Yes, in your example. But the point of my unary system is that you don't need to store any binary information. You can use it if you're only storing units. Your space in the middle finger plays absolutely no role in the maths. What you're doing is equivalent to me saying that the following two (base 10) numbers are different: 100 and 100, because one is black and one is red. But that's extraneous information, it plays no role in that math. All the 0/" " information you are describing is similarly incidental.

I'm not the person who came up with this concept. Here's the wikipedia page on unary numeral systems: http://en.wikipedia.org/wiki/Unary_numeral_system

(Note: I agree that this discussion could be happily split off, if the mods so wish)

I have heard of unary numeral systems, I just disagree in concept.  The lack of a "1" digit is still information of a sort even if it's not explicitly represented. 

But in any case, the original statement was only meant to refer to numeral systems in which each successive digit represents an order of magnitude increase above the digit next to it.  Such as the usual use of binary, octal, decimal, and hexadecimal.  All of these, and other numbering systems that only differ by number of possible digits, are base 10. 



Yargling

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Reply #36 on: January 30, 2010, 03:49:22 PM
Very amusing once it clicked :) Remember the fingers! ;D

Yeah, and for me the amusement lasted until that Fridge Logic moment about 5 seconds later, when I decided that if they're using English words for numbers, they'd still have the same numeric values.

No matter. It was still a hoot, even after it collapsed under the weight of it's own logic.  ;D

well, at first they probably did; but give it a few generations of child counting on there fingers, and they'd probably switch over ;)



CryptoMe

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Reply #37 on: February 01, 2010, 10:19:47 PM
In any case, I suppose I could have stated in the original statement that my assumption that all numbering systems are base 10 was the digits are counted by having the rightmost digit count from 0 to the maximum value, then reset to 0 and simultaneously incrementing the digit to the left of it, with 0's implied whenever there is blank space.  But that wouldn't be any fun to state everything so explicitly.  :)

I think what you mean to say is that all the numbering systems with a base (like binary, hexidecimal, etc.) are decimal systems (meaning that moving one column to the left increases the value by one base, and a decimal point indicates where the value is base*0).
 
First of all, a binary system (for example) is by definition base 2. It can't possibly be base 10 and base 2 at the same time.
Second, not all numbering systems are decimal. The Roman system (I, II, III, IV, etc), for example, is not. And it doesn't have a base, really - though one could argue that it has several: I, V, X, L, C, M, etc. 



eytanz

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Reply #38 on: February 02, 2010, 08:36:14 PM
In any case, I suppose I could have stated in the original statement that my assumption that all numbering systems are base 10 was the digits are counted by having the rightmost digit count from 0 to the maximum value, then reset to 0 and simultaneously incrementing the digit to the left of it, with 0's implied whenever there is blank space.  But that wouldn't be any fun to state everything so explicitly.  :)

I think what you mean to say is that all the numbering systems with a base (like binary, hexidecimal, etc.) are decimal systems (meaning that moving one column to the left increases the value by one base, and a decimal point indicates where the value is base*0).
 

The name of the property you describe is "positional". "Decimal" literally means "base ten".

Unblinking is right that there can be no base-1 positional system; but he is wrong in assuming that positional numeral systems are the only ones that matter, and in assuming that you need a minimum of two symbols to represent things.
« Last Edit: February 02, 2010, 08:39:43 PM by eytanz »



CryptoMe

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Reply #39 on: February 02, 2010, 09:02:06 PM
I think what you mean to say is that all the numbering systems with a base (like binary, hexidecimal, etc.) are decimal systems (meaning that moving one column to the left increases the value by one base, and a decimal point indicates where the value is base*0).
 

The name of the property you describe is "positional". "Decimal" literally means "base ten".

I kinda see your point, eytanz. But don't we still call it a "decimal point" regardless of which base we are using? I have never heard of a "positional point". So, my take was that, in this case, "decimal" has been separated from it's "base 10" origins and taken on a life of it's own (much like "computer" no longer means a person who computes  ;)).



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Reply #40 on: February 03, 2010, 05:57:29 PM
In any case, I suppose I could have stated in the original statement that my assumption that all numbering systems are base 10 was the digits are counted by having the rightmost digit count from 0 to the maximum value, then reset to 0 and simultaneously incrementing the digit to the left of it, with 0's implied whenever there is blank space.  But that wouldn't be any fun to state everything so explicitly.  :)

I think what you mean to say is that all the numbering systems with a base (like binary, hexidecimal, etc.) are decimal systems (meaning that moving one column to the left increases the value by one base, and a decimal point indicates where the value is base*0).
 
First of all, a binary system (for example) is by definition base 2. It can't possibly be base 10 and base 2 at the same time.
Second, not all numbering systems are decimal. The Roman system (I, II, III, IV, etc), for example, is not. And it doesn't have a base, really - though one could argue that it has several: I, V, X, L, C, M, etc. 

Now you're getting to the heart of what I was trying to say!  :)

Binary is, of course, base two.  Two is represented as 10 in binary.
Octal is base eight.  Eight is represented as 10 in octal.
Decimal is base ten.  Ten is represented as 10 in decimal.
Hexadecimal is base 16.  Sixteen is represented as 10 in hexadecimal.

So, each of these numbering systems is base 10 (where the number 10 is self-referentially represented in its own numbering system).  When we refer to binary as being base 2, we know that the "2" is a decimal representation because the "2" digit does not exist in binary. 

So, if I say "I prefer to use base 10 for my mathematics", this statement is only useful if we know what base the number "10" is represented in.  This amuses me (though it wouldn't surprise me if I'm the only one it amuses). 



CryptoMe

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Reply #41 on: February 03, 2010, 08:35:55 PM
Now you're getting to the heart of what I was trying to say!  :)

Binary is, of course, base two.  Two is represented as 10 in binary.
Octal is base eight.  Eight is represented as 10 in octal.
Decimal is base ten.  Ten is represented as 10 in decimal.
Hexadecimal is base 16.  Sixteen is represented as 10 in hexadecimal.

So, each of these numbering systems is base 10 (where the number 10 is self-referentially represented in its own numbering system).  When we refer to binary as being base 2, we know that the "2" is a decimal representation because the "2" digit does not exist in binary. 

So, if I say "I prefer to use base 10 for my mathematics", this statement is only useful if we know what base the number "10" is represented in.  This amuses me (though it wouldn't surprise me if I'm the only one it amuses). 

Ahhh, another case of written vs spoken...
In my mind I heard "base ten", when you were thinking "base one-zero".

But, the only reason we use "base one-zero" is because we have defined things that way. Conceivably, we could have defined a base four system (say) that uses #, q, @, and y to represent zero, one, two , and three just so that we wouldn't confuse it with the base ten system.  In that case, this would be a "base q#" system. But that would require people to do even more mental gymnastics, which is why it probably wasn't done that way. So, there is nothing intrinsically special about "base one-zero", it's just an indicator of our laziness as a species - which I guess is kinda funny ;D



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Reply #42 on: February 04, 2010, 03:07:11 PM
So, there is nothing intrinsically special about "base one-zero", it's just an indicator of our laziness as a species - which I guess is kinda funny ;D

It wasn't meant to be earth-shatteringly deep, just something which amuses me.  :)