B Ideal Base for a Number System

[Isaac0427]

This is just a fun question I thought of:
If you take away all knowledge of base-10 being the most natural number system, something we were just taught to think, and you could decide what number system we use, what would you pick? What do you think would make the most sense? Personally, I think base-8, but I'm curious to see what others on PF would choose.

 

[ProfuselyQuarky]

Hm, base-8 is an interesting choice! Why do you choose that one? I think that base-12 would be the most reasonable number system to use. The transition from the decimal system to dozenal would be really easy. Base-12 is also convenient because 12 has a lot of factors--1, 2, 3, 4,and 6. Quick arithmetic while you're at the store or trying to divide something in your head during daily life would be simpler, I think.

I quite like binary, but the thought of living in a world that runs on the impractical base-1 is just ugly . . .

 

[ProfuselyQuarky]

I quite like binary, but the thought of living in a world that runs on the impractical base-1 is just ugly . . .

On that note, I know someone who is fluent in binary and can have an entire conversation consisting of 1s and 0s. I imagine that he would like a base-1 world

 

[micromass]

Base 6. Division by $2$ and $3$ would be easy since they divide $6$. Testing for division by $5$ is also easy by adding sum of digits. Testing for diviison by $7$ is also easy by the alternating sum of digits. So you can test for division for the smallest 4 primes.

 

[jbriggs444]

Balanced ternary. No ambiguity about unary minus signs on numeric literals because there are none. The highest radix efficiency of any integer base.

 

[fresh_42]

Base 60 is one of the oldest used systems. (3300 B.C.)
https://en.wikipedia.org/wiki/Sexagesimal
A few decades ago as Assembler was still often in use 8 and 16 were useful to know.

 

[Mark44]

Base 60 is one of the oldest used systems. (3300 B.C.)
https://en.wikipedia.org/wiki/Sexagesimal

And the reason that we have 1 hour = 60 minutes, and 1 minute = 60 seconds, as well as 1 degree = 60 minutes, and 1 minute = 60 seconds.

A few decades ago as Assembler was still often in use 8 and 16 were useful to know.

Base-16 (hexadecimal) is still very useful to know if you do any programming.

 

[Isaac0427]

Hm, base-8 is an interesting choice! Why do you choose that one?

Most things in life are split into twos, and it seems quite pleasant to have 2³=10, 2⁶=100, 2⁹=10000, etc, and all base numbers (10, 100, 1000, etc.) to have prime factors of all twos. This logic would also bring binary, base-4 (which I don't know the name of), hexidecimal, etc. into the mix, but binary and base-4 seem too long and ugly, and hexidecimal and up have too many digits for me. I find base-8 to be the sweet spot.

 

[Isaac0427]

On that note, I know someone who is fluent in binary and can have an entire conversation consisting of 1s and 0s. I imagine that he would like a base-1 world

Is base one even possible?

 

[ProfuselyQuarky]

Is base one even possible?

Technically, yes. Practically, no.

I find base-8 to be the sweet spot.

How is the dozenal system not a sweet spot also? :)

 

[Mark44]

Is base one even possible?

Technically, yes. Practically, no.

Not even technically, unless you're talking about tally marks.
In base-2, the digits are 0 and 1. In base-3, the digits are 0, 1, and 2. In base-n (n > 1), the digits are 0, 1, ..., n - 1. In "base-1" the only possible digit is 0.

In any base-n system, and arbitrary number is the sum of multiples of powers of the base. For example, the decimal number 123 means $1 \times 10^2 + 2 \times 10^1 + 3 \times 10^0$. In "base 1" the only multiplier available is 0, and every power of 1 is also 1.

I had a long discussion on Compuserve about this more than 20 years ago.

 

[Isaac0427]

How is the dozenal system not a sweet spot also? :)

In dozenal, only 3/10, 6/10 and 9/10 can be written in the form of a decimal with a finite number of didgets. Something about that bothers me, but that's just me.

 

[Mark44]

I think that base-12 would be the most reasonable number system to use. The transition from the decimal system to dozenal would be really easy.

There's a lot of history for base-12 counting (duodecimal). We buy eggs in dozens, and there are still some items that you can buy by the gross (12 dozen). From the Wikipedia article on Inch (https://en.wikipedia.org/wiki/Inch):

The English word inch comes from Latin uncia meaning "one-twelfth part" (in this case, one twelfth of a foot); the word ounce (one twelfth of a troy pound) has the same origin.

I quite like binary, but the thought of living in a world that runs on the impractical base-1 is just ugly . . .

Binary is not base-1 -- it's base-2. In fact, we do live in a world that runs on binary, at least those of us who use computers. Each transistor that makes up the billions and trillions of bytes of memory can take on one of two values, which we choose to label as either 0 or 1.

If you don't like dealing with $0101110010010001$, you can divide it up into groups of four bits (short for binary digit) as $0101~ 1100~ 1001~ 0001$. There's a straightforward conversion from binary to hexadecimal (base-16), or $5C91_{16}$, or as usually written in C,C++, etc., 0x5C91.

 

[Mark44]

finite number of didgets

There are midgets, gadgets, and even Gidget, who might fidget, but not didgets. The word you want is digit.

 

[ProfuselyQuarky]

Binary is not base-1 -- it's base-2.

Oh, gosh, yeah. Infinite apologies! I've always had the idea that binary = base-1 because . . . okay, fine, I don't know why but, yeah, I mix base-1 with base 2. Thanks for those corrections

@Isaac0427 So when I said "technically yes, practically no" I thoght you were talking about base -2. Sorry 'bout that. I think I'm going to log off and doodle on a sheet of paper right now. I'm tired.

 

[Mark44]

@ProfuselyQuarky, "bi" in binary, bicycle, biplane, etc means "two". It even shows up in the word "biscuit" from the French word bescuit, meaning twice (bis) cooked (cuit). The "cuit" part is related to "cotta" as in "terra cotta."

 

[ProfuselyQuarky]

@ProfuselyQuarky, "bi" in binary, bicycle, biplane, etc means "two". It even shows up in the word "biscuit" from the French word bescuit, meaning twice (bis) cooked (cuit). The "cuit" part is related to "cotta" as in "terra cotta."

I'm aware of that, thanks

It's just one of those odd things I forget, I guess. I'd edit my posts to make them accurate, but I no longer can edit them.

 

[Isaac0427]

There are midgets, gadgets, and even Gidget, who might fidget, but not didgets. The word you want is digit.

That is why I want to go into physics, not spelling. I don't even know how I'm so bad at spelling basic words.

 

[SammyS]

There is a numbering system which has no zero digit but is place based much like we are used to. It's perfectly adequate for expressing any counting number (positive integer). It works for any natural number base.

For some base, b, instead of using digits 0 through (b-1) it uses digits 1 through b .

With this scheme, represent numbers in base 1 is possible, although this does simply amount to a tally system, e.g., the number five is simply 11111 .

 

[fresh_42]

There is a numbering system which has no zero digit but is place based much like we are used to. It's perfectly adequate for expressing any counting number (positive integer). It works for any natural number base.

For some base, b, instead of using digits 0 through (b-1) it uses digits 1 through b .

With this scheme, represent numbers in base 1 is possible, although this does simply amount to a tally system, e.g., the number five is simply 11111 .

These things occur if we neglect to define what we are talking about. The crucial point here is whether a "numbering system" is required to represent the integers which I would have implied or only (some) natural numbers. The example above is just counting sticks which already has been too poor 5,000 years ago.

 

[ProfuselyQuarky]

That is why I want to go into physics, not spelling. I don't even know how I'm so bad at spelling basic words.

Hehe, so I can't remember bases (even with logical prefixes) and you can't spell very good (even with spellcheck). Looks like we're even, Isaac!

 

[ProfuselyQuarky]

And . . . this is probably a horrible question that can be answered with common sense, but what base is roman numerals in? Is it even logical to say that roman numerals have a “base”?

To add to that, position doesn’t matter with roman numerals which makes it more confusing. In base 10, the digit “2” means the actual value of two in the number “12”, but in the number “125” the same digit “2” means twenty. Order doesn't matter in roman numerals. For example, LXXI is 71 because L=50, X=10, and I=1. “X” always equals 10 and both Xs in “LXXI” have the same value, regardless of their position in the number. Does this affect the base of roman numerals, if at all?

 

[Mark44]

And . . . this is probably a horrible question that can be answered with common sense, but what base is roman numerals in? Is it even logical to say that roman numerals have a “base”?

Roman numerals aren't position-based. They are really only a slight improvement over tally marks, with IIIII being abbreviated as V, which was supposed to represent a hand (with 5 fingers), and X representing two hands. C and M came from the Latin words for 100 and 1000, respectively, but I don't know the origin of the symbols for 50 (L) and 500 (D). It was a great advance in arithmetic to change to a decimal number system, using zero.

To add to that, position doesn’t matter with roman numerals which makes it more confusing. In base 10, the digit “2” means the actual value of two in the number “12”, but in the number “125” the same digit “2” means twenty. Order doesn't matter in roman numerals. For example, LXXI is 71 because L=50, X=10, and I=1. “X” always equals 10 and both Xs in “LXXI” have the same value, regardless of their position in the number. Does this affect the base of roman numerals, if at all?

There's no "base" in Roman numerals. The placement of a Roman numeral can affect the value of the number it represents. For example, VI is 6, but IV is 4 (IV is sometimes written as IIII). LXXI is 71, but LXIX is 69. Ordinary addition and subtraction are very difficult using Roman numerals, and even worse for multiplication. I've never seen anyone attempt division using Roman numerals.

 

[axmls]

I've never seen anyone attempt division using Roman numerals.

So naturally, @micromass will have to make his next integral challenge entirely in Roman numerals. I'll get us started: $$\int _0 ^\text{LXII} \frac{\text{MCV}}{\text{IX}} x^{\text{IV}} \sin(\text{DLII} x) \ dx$$ No converting to decimal! Try using partial fractions on that.

I'd suggest "base-Graham's number," but I don't suspect there are enough combinations of symbols in the universe to be able to reach $10$, and no one's got that good of a memory.

So, I'd like to live in a world where we use base-16. At the very least, it wouldn't hurt to make computers easier to deal with, and it's not like it's any less efficient than base-10. The only drawback is that we maybe would have to come up with 6 more symbols, because if we were starting from scratch, I doubt we'd want to borrow letters from the alphabet for our numbers. Also, binary and octal are more inefficient than base-10, so I wouldn't want those.

 

[ProfuselyQuarky]

So naturally, @micromass will have to make his next integral challenge entirely in Roman numerals. I'll get us started: $$\int _0 ^\text{LXII} \frac{\text{MCV}}{\text{IX}} x^{\text{IV}} \sin(\text{DLII} x) \ dx$$ No converting to decimal! Try using partial fractions on that.

I'd suggest "base-Graham's number," but I don't suspect there are enough combinations of symbols in the universe to be able to reach $10$, and no one's got that good of a memory.

So, I'd like to live in a world where we use base-16. At the very least, it wouldn't hurt to make computers easier to deal with, and it's not like it's any less efficient than base-10. The only drawback is that we maybe would have to come up with 6 more symbols, because if we were starting from scratch, I doubt we'd want to borrow letters from the alphabet for our numbers. Also, binary and octal are more inefficient than base-10, so I wouldn't want those.

Jeez, I wouldn't know where to start . . .

 

[fresh_42]

Jeez, I wouldn't know where to start . . .

Easy. Since the Romans didn't have fractions, replace the entire thing with a Taylor expansion, integrate a few terms, look at the leading integer and pray it's neither zero nor negative.

 

[phinds]

On that note, I know someone who is fluent in binary and can have an entire conversation consisting of 1s and 0s. I imagine that he would like a base-1 world

Ah you youngsters. When I was a kid we didn't even HAVE 1s, just 0s. We could carry on an entire conversation in just 0s. It's all a matter of inflection.

 

[SammyS]

Ah you youngsters. When I was a kid we didn't even HAVE 1s, just 0s. We could carry on an entire conversation in just 0s. It's all a matter of inflection.

OOOOOOOOOO I see.!

Actually that was misspelled.

0000000000 I see !

 

[ProfuselyQuarky]

Ah you youngsters. When I was a kid we didn't even HAVE 1s, just 0s. We could carry on an entire conversation in just 0s. It's all a matter of inflection.

Ah you oldsters. I'm beginning to gain the impression that the ye olden days were quite shady

 

[phinds]

Ah you oldsters. I'm beginning to gain the impression that the ye olden days were quite shady

Nah, we didn't have no stinking shade either. Had to walk everywhere in the sun.

 

[ProfuselyQuarky]

Nah, we didn't have no stinking shade either.

And school wasn't that great either?? That didn't teach that the use of double negatives was poor grammar? Sheesh!

 

[fresh_42]

Ah you youngsters. When I was a kid we didn't even HAVE 1s, just 0s. We could carry on an entire conversation in just 0s. It's all a matter of inflection.

I've read yesterday: When we were young we had a number which we could dial and had been told the accurate time. This was our internet.
(I remember those days ...)

If nowadays everything is "digital" why don't we have binary phone numbers?

 

[Isaac0427]

Also, binary and octal are more inefficient than base-10, so I wouldn't want those.

I understand binary, but why octal?

 

[Isaac0427]

So naturally, @micromass will have to make his next integral challenge entirely in Roman numerals. I'll get us started:

∫LXII0MCVIXxIVsin(DLIIx) dx∫0LXIIMCVIXxIVsin⁡(DLIIx) dx​

\int _0 ^\text{LXII} \frac{\text{MCV}}{\text{IX}} x^{\text{IV}} \sin(\text{DLII} x) \ dx No converting to decimal! Try using partial fractions on that.

I'm doing it. All you need is integration by parts, n-substitution (as u was taken by integration by parts) and patience.

EDIT: Nevermind, I think it is impossible.

 

[axmls]

I understand binary, but why octal?

Much like you can break binary strings up into groups of 4 and associate them with a hexadecimal number, e.g. 0010110010111111 becomes 0010 1100 1011 1111, which becomes 2CBF in hexadecimal, you can also represent groups of 3 binary numbers with one octal number.
It's not as common as hexadecimal, mainly because hexadecimal is useful since a common unit used in computing is a byte, which is 8 bits, which is divisible by 4, which is good when using hexadecimal, but it is possible, since $8=2^3$.

Fun fact: Half of a byte is called a nibble!

I'm doing it. All you need is integration by parts, n-substitution (as u was taken by integration by parts) and patience.

EDIT: Nevermind, I think it is impossible.

In our system of numbers, integrate by parts 4 times starting with $u = x^4$, then eventually you'll just be left with one trig function. It's a bit trickier with Roman numerals.

 

[Isaac0427]

In our system of numbers, integrate by parts 4 times starting with u=x4u=x4u = x^4, then eventually you'll just be left with one trig function. It's a bit trickier with Roman numerals.

That's what I did. I got a huge expression that required me to multiply Roman numerals at least 5 times. The only way to do that is to distribute and simplify, which would take way too long.

 

[micromass]

That's what I did. I got a huge expression that required me to multiply Roman numerals at least 5 times. The only way to do that is to distribute and simplify, which would take way too long.

That's of course different from saying it's impossible!

 

[ProfuselyQuarky]

That's of course different from saying it's impossible!

So let's see you do it, micromass

 

[micromass]

So let's see you do it, micromass

I have proven the answer exists and is unique. That is enough for a mathematician.

 

[ProfuselyQuarky]

I have proven the answer exists and is unique. That is enough for a mathematician.

How is that enough? That's a very unsatisfactory way to do things.

I know that I can bake a cake, but who cares until I actually do it?

 

[micromass]

How is that enough? That's a very unsatisfactory way to do things.

I know that I can bake a cake, but who cares until I actually do it?

You must be very interested in engineering and experimental physics

 

[ProfuselyQuarky]

You must be very interested in engineering and experimental physics

Not so much engineering, but I like biochemistry and, yes, experimental physics. Of course, there's not so much "experimenting" I can do considering that I'm limited to a garage that doesn't meet the standards of "lab"

 

[micromass]

Not so much engineering, but I like biochemistry and, yes, experimental physics. Of course, there's not so much "experimenting" I can do considering that I'm limited to a garage that doesn't meet the standards of "lab"

Did you try the famous egg-drop experiment yet?

 

[jim mcnamara]

On a different note: Athabascan languages use base 4. Everything relates to the cyclic nature of things; 4 seasons. If Navajos had invented baseball it would be four strikes and you're out. The Navajo speakers I knew used decimal numbers. Why? Not because they view them as better, rather just to be able to function in business transactions since everyone else uses base 10. Langauge is meant for communication.
Numbers are an important part of communication in Western culture.

Answer to the OP;
So, if you want to see what numbering would be like if we 'started over', look at other language families. You will find lots of differences.
Humans are disgustingly inventive from that point of view.

As @Mark44 points out base 60 was also used and is still part of everyday life.

I would like to suggest a concept: there is not always a best, rather a set of choices of varying subjective usefulness. For a number base, consider the Lingua Franca, decimal, as the first choice, then go from there if changes are really needed - other than as an intellectual fun exercise.

 

[phinds]

How is that enough? That's a very unsatisfactory way to do things.

I know that I can bake a cake, but who cares until I actually do it?

I have to side with micromass on this. To a mathemetician, if you have proven conclusively that you can do something then actually doing it is just a detail.

In engineering on the other hand knowing that something CAN be done is a far cry from actually doing it and the difference matters.

 

[Dr_Zinj]

The title of the thread is actually a deception.
You use different based number systems for different purposes. Depending on the purpose, some systems work better than others.
For instance, I like base 2 for counting to 32 on a single hand. (I only have the normal 5 fingered hand.)

 

[fresh_42]

The title of the thread is actually a deception.
You use different based number systems for different purposes. Depending on the purpose, some systems work better than others.
For instance, I like base 2 for counting to 32 on a single hand. (I only have the normal 5 fingered hand.)

31, I guess. And there are some very tough numbers in between ...

 

[phinds]

For instance, I like base 2 for counting to 32 on a single hand. (I only have the normal 5 fingered hand.)

Uh ... you probably shouldn't do that in public or the 4 might get you in trouble

 

[fresh_42]

Uh ... you probably shouldn't do that in public or the 4 might get you in trouble

18 can be even more dangerous!

 

[phinds]

18 can be even more dangerous!

That one depends more on the culture