Optimal number system?

[Chaotic42]

Hey everyone.

I was just curious if there is a number system that nature prefers more than base-10. Something like always getting 169 which would be 100 in base-13.

 

[TuxDave]

Nature prefers the fibanocci sequence... heh, that would be a messed up number system if each digit in a number represented a different base.

 

[Chaotic42]

Originally posted by: TuxDave
Nature prefers the fibanocci sequence... heh, that would be a messed up number system if each digit in a number represented a different base.

You aren't kidding.

I might have to change my major to something other than math if that happens.

 

[uec0]

This is possibly a bit off-topic, but your post reminded me of something one ECE TA mentioned; while going over binary and hex, he claimed that the most efficient number system would actually be base e.

I actually found an article that seems to support his statement.

http://www.americanscientist.o...l/assetid/14405/page/2

 

[Shalmanese]

Well, heres counting to 20 in base e (to 8 dec places):

1 0.21211112
2 1.21211112
3 10.02001120
4 11.02001120
5 12.02001120
6 20.11101110
7 21.11101110
8 100.11201011
9 101.11201011
10 102.11201011
11 110.21010102
12 111.21010102
13 120.01020021
14 121.01020021
15 200.01120000
16 201.01120000
17 202.01120000
18 210.11000001
19 211.11000001
20 212.11000001

Doesn't look all that simple to me

 

[DrPizza]

Hey, thanks uec0... I found that article to be pretty interesting...

Plus, I was also looking for more material to supplement my pre-calculus course. Because our state curriculum and standards changed, a lot of my pre-calculus course has become redundant. I'll be able to go more in depth with my students as well as supplement the material. Currently, there is no place where our students are exposed to the types of mathematics that are taught (I presume, it's been many years) in computer science. I think that when I'm covering logs and exponents this year, I'll also cover other number systems.

 

[Chaotic42]

Cool link, man.

Number systems have interested me for a long time.

 

[Geniere]

Uec0 - Nice link. As I am the ?Odd Couple? embodied in one person I found this paragraph (base 3 numbers) very interesting.

?In my own files I use third-cut rather than half-cut folders; the tabs appear in three positions, left, middle and right. Nevertheless, I had long thought?or rather I had assumed without bothering to think?that a similar analysis would apply, and that I couldn't be sure of avoiding conflicts between adjacent folders unless I was willing to shift files to new folders after every insertion. Then came my Epiphany of the File Cabinet a few weeks ago: Suddenly I understood that going from half-cut to third-cut folders makes all the difference.?

 

[kcthomas]

i think base 10 is optimal. we got 10 fingers dont we? maybe for complex mathematics a different number system is better but for what normal non mathematicians do base 10 is the easiest.

 

[Witling]

The problem with the post expressing counting to 20 in base e, and what makes it look dumb, is that it is expressed in base 10 numbers. It might not be as tough if the appropriate symbols were adopted, as was done in hexadecimal.

Second, I don't think the Fibanocci series is nature's favorite. It's used, but I think it's not the most popular. For example, I'm pretty sure that shell spiraling is base e. No doubt someone has a more complete knowledge of the subject.

 

[Shalmanese]

Well, counting from 1 to 14 in Hex looks exactly the same. I was making the point that integers become irrationals in base e which makes it slightly less than useful.

 

[Vee]

Originally posted by: kcthomas
i think base 10 is optimal. we got 10 fingers dont we? maybe for complex mathematics a different number system is better but for what normal non mathematicians do base 10 is the easiest.

It doesn't matter. We're used to 10, that's why we feel it's *natural* or easy. And in a way it is, but just because we have 10 fingers. (I don't think any base for positionary number systems, other than 2, 8, 10, 16 and 60 have ever been in real use? Non position systems, are of course a different issue.)

Personally, I think there would be some small gains in convenience, by using a base that can be factorized by 2's. Like 8 or 16. But I also think that it really doesn't matter. In line with that conviction, I can't believe that any whole number base has any significant real disadvantages, compared to any fractional or irrational number, for any use. - So, from practical reasons follows that a whole/natural number is preferable.


P.S.
Just a sidetrack: While on the subject of number systems, do you really know the "number of the beast"?

- 666?
- Wrong! That's not at all what the original text in the book of revelation said.
(I'm not going to answer any questions, or post on this subject again. I'm a bad girl today. :laugh: )

 

[Kibbo]

IIRC, the ancient Sumerians used a number system of base 12. It's why we still have 360 deg in a circle. And 12 hrs on a clock.

I once read an essay by Asimov that mused that if only our ancestors had counted without using our thumbs (leading to a base 8 system), we would all have a much more natural connection to the binary number system.

 

[imported_pak9rabid]

Originally posted by: kcthomas
i think base 10 is optimal. we got 10 fingers dont we? maybe for complex mathematics a different number system is better but for what normal non mathematicians do base 10 is the easiest.

it all depends on what you're trying to do. true, we have 10 fingers, which makes the decimal system easier to understand, but using a binary system, those same 10 fingers can represent 1024 different values as opposed to only 10.

 

[rgwalt]

Instead of thinking about numerical bases for a number system, think about changing the number system as a whole, and moving to an interval number system. Numbers will no longer be represented by a single set of digits, but by a pair of digits representing the upper and lower limit of the interval. There is a set of mathematical rules available for performing computations with intervals, and one consequence is that a gobal equation solving and optimization method has been developed using interval mathematics.

Ryan

 

[TheBoy]

Originally posted by: pak9rabid

it all depends on what you're trying to do. true, we have 10 fingers, which makes the decimal system easier to understand, but using a binary system, those same 10 fingers can represent 1024 different values as opposed to only 10.

That means each finger could represent a 1 or a 0, so with base 10 you would be able to count up 1,111,111,111. No matter what the base you would only be able to count up to 10 (1010 in binary) on your fingers because we only have that many fingers

 

[iwantanewcomputer]

binary sure would be a lot easier if we had grown up using base 8. 10 isn't really good for anything other than we can count 2 more things on our fingers. base e does have some mathematical usefulness, but not for counting things as shown above. base 2 is just too small to be useful in counting things in everyday life

I think base 8 would be the best considering the way we write computer code. 8 bits is already a byte. so that would make a lot of computer things more natural. just think, our hard drives would actually be as big as they are advertised as I was so annoyed that my 200 GB only gives me 186. Drive manufacturers wouln't be able to cheap us out in base 8

 

[iwantanewcomputer]

ohhh! i just had an idea. i'm not a comp engineer, but is it possible to make transistors with more than 2 possible outputs? so then we could make base 10 transistors? the only way i think of to do this would be to put a branch of transistors spreading out from an original one which wouldn't make sense

 

[TheBoy]

more than 2 outputs? that would be an analogue component, binary is used in computers because it is faster and less prone to noise

 

[DonPMitchell]

In the original FFT paper, they state that base 3 is more efficient than base 2. I've also heard the same said about the complexity of logic ciruits in general. Soviet scientists (Ukranian to be specific) actually built some base-3 computers, but they weren't really used much.

I mentioned this to a friend at Microsoft once, and he suggested that maybe base e is optimal and 3 is just the closest integer to that. Interesting and bizarre idea!

 

[Mday]

hmm that base e thing looks interesting.

On the other hand, if we use our fingers to count using binary, we can count to 2^10-1 (1023 decimal). Just watch how you do 4, and 256, not to mention 260.

 

[everman]

I'd heard that base 3 is the most efficient, and also that base 4 is very efficient as well. Isn't DNA a base 4 system? Our brains seem to do a pretty good job sometimes, so would that be a good guess? I'm not a mathmetician and haven't really a clue

 

[TuxDave]

Originally posted by: iwantanewcomputer
ohhh! i just had an idea. i'm not a comp engineer, but is it possible to make transistors with more than 2 possible outputs? so then we could make base 10 transistors? the only way i think of to do this would be to put a branch of transistors spreading out from an original one which wouldn't make sense

Also known as a digital to analog converter which generates several voltages levels depending on the digital inputs. But processing data in the analog domain is not very easy, nor is it very compact and power efficient.

 

[Mday]

Originally posted by: iwantanewcomputer
ohhh! i just had an idea. i'm not a comp engineer, but is it possible to make transistors with more than 2 possible outputs? so then we could make base 10 transistors? the only way i think of to do this would be to put a branch of transistors spreading out from an original one which wouldn't make sense

well, the way transistor network work is this: you input a voltage, and that voltage is modified. Now, the output can be anything, depending on the input.

binary logic works like this: you have N inputs, and you output 0 or 1 (high or low, low or high), or a series of 0s and 1s.

When you apply transistors into binary logic, what you get is, you have N inputs, and you output a 0 or a 1, or a series of 0s and 1s in the form of 2 separate voltages. Typical designs today use 0 volts or 5 volts (or lower, 3.3, 2 etc).

So, yes, you can have a transistor output any number of voltages, as long it is below the VCC, or the power source of the device. However, there is an issue involved here. Since you have a VCC that's fixed, lets say at 5 V. If you want 2 possible outputs (0 or 1), you will have 0V representing 0, and 5V representing 1. If you want 10 possible outputs, you will have 0V representing 0, and 5V representing 9. Notice the difference here. To distinguish between signal, 2 outputs require a 5 V difference, which means all your system has to do is differentiate between 0 and 5. That is, 0.1 V is a 0, 4.5 V is a 1 and so forth. Of course there is the issue of 2.5, but unless your power source and circuit design is THAT BAD, it wont be an issue. Once you get into the problem of 10 possible outputs, is a 4.5 a 9 or an 8.

 

[eigen]

Originally posted by: rgwalt
Instead of thinking about numerical bases for a number system, think about changing the number system as a whole, and moving to an interval number system. Numbers will no longer be represented by a single set of digits, but by a pair of digits representing the upper and lower limit of the interval. There is a set of mathematical rules available for performing computations with intervals, and one consequence is that a gobal equation solving and optimization method has been developed using interval mathematics.

Ryan

Well then to describe a single number you would need a infinite amount of numbers.AS each the lb and the ub would bothe be number which would then be expressed as a lb and ub. You essentially creating a binary tree with 2^n vertices where n-> oo.Doesnt sound very efficient. This would would require a infinite of memory to represent. Were as with all other bases it take about log n bits to represent a number.A far as addding interval etc. that is just an example of fuzzy arithmetic.

But as far as the original post. It doesnt matter. As the systems are equaivalent and show beauty and possess utitilty. But I like base2 personally .But that is just the aesthetic of it 1/0 on/off good/evil ......
The dualism is nice.