# Ideal Base for a Number System

• B
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

ProfuselyQuarky
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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 . . .

Ralph Dratman
ProfuselyQuarky
Gold Member
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
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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
jbriggs444
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Balanced ternary. No ambiguity about unary minus signs on numeric literals because there are none. The highest radix efficiency of any integer base.

Mark44
Mentor
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.
fresh_42 said:
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.

ProfuselyQuarky, Isaac0427 and fresh_42
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.

ProfuselyQuarky
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
Gold Member
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
Mentor
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.

theBin and ProfuselyQuarky
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
Mentor
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.
ProfuselyQuarky said:
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.

ProfuselyQuarky
Mark44
Mentor
finite number of didgets
There are midgets, gadgets, and even Gidget, who might fidget, but not didgets. The word you want is digit.

Isaac0427
ProfuselyQuarky
Gold Member
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
Mentor
@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
Gold Member
@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.

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
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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
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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.

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ProfuselyQuarky
Gold Member
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
Gold Member
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
Mentor
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.
ProfuselyQuarky said:
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.

ProfuselyQuarky
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.

Samy_A, micromass and ProfuselyQuarky
ProfuselyQuarky
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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
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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.

ProfuselyQuarky
phinds
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2021 Award
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.

ProfuselyQuarky
SammyS
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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
ProfuselyQuarky
Gold Member
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
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2021 Award
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
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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
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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?

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

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.

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ProfuselyQuarky
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.

ProfuselyQuarky and Isaac0427