# B Ideal Base for a Number System

1. Apr 22, 2016

### 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.

2. Apr 22, 2016

### 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 . . .

3. Apr 22, 2016

### 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

4. Apr 22, 2016

### 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.

5. Apr 22, 2016

### jbriggs444

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

6. Apr 22, 2016

### Staff: Mentor

7. Apr 22, 2016

### Staff: Mentor

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

8. Apr 22, 2016

### Isaac0427

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.

9. Apr 22, 2016

### Isaac0427

Is base one even possible?

10. Apr 22, 2016

### ProfuselyQuarky

Technically, yes. Practically, no.
How is the dozenal system not a sweet spot also? :)

11. Apr 22, 2016

### Staff: Mentor

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.

12. Apr 22, 2016

### Isaac0427

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.

13. Apr 22, 2016

### Staff: Mentor

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):
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.

14. Apr 22, 2016

### Staff: Mentor

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

15. Apr 22, 2016

### ProfuselyQuarky

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.

16. Apr 22, 2016

### Staff: 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."

17. Apr 22, 2016

### ProfuselyQuarky

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.

18. Apr 22, 2016

### Isaac0427

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.

19. Apr 22, 2016

### SammyS

Staff Emeritus
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 .

20. Apr 24, 2016

### Staff: Mentor

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.

Last edited: Apr 24, 2016
21. Apr 24, 2016

### ProfuselyQuarky

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!

22. Apr 24, 2016

### 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?

23. Apr 24, 2016

### Staff: Mentor

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.
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.

24. Apr 25, 2016

### axmls

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.

25. Apr 25, 2016

### ProfuselyQuarky

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