Comment by maxxxxx | Hacker News Reader

Comment by maxxxxx | Hacker News Reader

Since we have a lot of math experts here I thought I'd ask a question I was always wondering about: Is there an inherent advantage or disadvantage to using the decimal system as we do? Somehow I think octal or hexadecimal would be easier but I am not sure.

I apologize for not being one of the experts you speak of, but I do know of others that say there are superior number bases to use for everyday counting.

http://www.dozenal.org/drupal/content/brief-introduction-doz...

These people believe that base 12 would be superior to base 10. And they do make good points, expressing fractions like those in 3rds become easier. Is it superior in the grand scheme? I'll leave that debate to actual experts. Will people see the utility and switch? I'm going to keep my base 10 skills up, you know, just in case duodecimal doesn't catch on. ;-)

Maybe this is just me being very naive, but how does it help? It doesn't actually change the properties of the underlying numbers.

10 (decimal) is not divisible by 3, and rewriting it in base 12 as A won't change that. You simply get 29.4 instead of 33.3...

If a kilo, for example, was made of 12 x 12 x 12 (1728) grams instead of 1000 grams, then you could sell stuff by the half-kilo, third-kilo, and 1/12 kilo without going into decimals. The number 1728 would still be written as 1000 in base 12, but 1/3 kilo would be 400 grams instead of 333.3333... as it is now. Useful if you're a grocer.

Other applications, like documenting how much oil to put in a car or whatever, there would be more options for picking memorable numbers. Like, my motorcycle takes 1.6L of oil. It could be 1 represented as 1 2/3 L, which is more visually intuitive. Not sure if I explained that well enough, maybe someone else has got a better metaphor handy...

Is it some American thing that measuring things as fractions is easier than decimals? To me being used with metric system 1.6L is equally intuitive as 1 2/3, maybe even more intuitive.

It's just hardwired in the brain how much 0.6, or 60%, represents, I think internally my brain finds the closest easy references like 10%, 25%, 33%, 50% or 75% and then interpolates or adds or removes chunks of 10%, the rest is rounding errors. I mean, i dont have to think consciously if 0.69 is more or less than 75% and i know exactly where both 66.66% and 75% is and can place it somewhere in between.

I agree that fractions are a very American thing. A lot of stuff is expressed as fractions here. But I bet a lot of people couldn't tell you whether 5/16 is more then 1/3 or not.

Another thing is that multiplying and dividing by 2 and 5 is easier in base-10. For example, to divide by 5 you multiply the number by 2 (just add it to itself) and then divide by 10 (a right shift).

On the other hand, to divide by 3 you basically need to do long division.

I was about to post that in my opinion base-12 is superior to base-10. But someone beat me to it. In a six-fi novel I'm writing, an advanced alien civilisation uses base-12.

As to your question specifically regarding base-16 instead of base-12, it depends.

Decimal itself is just a bizarre choice, most likely due to humans having literally ten digits. In decimal we can represent exact fractions of 1/2, 1/5, and 1/10 (without repeated decimals like 0.33333 for 1/3). Counting by fives (and twos) is very easy.

But choosing prime factors of 2 and 5 is a strange choice in itself. Why skip 3? Why is it more useful to easily represent fraction 1/5th as 0.2 instead of 1/3rd? How often do we use fifths?

Hexadecimal in one sense is easier, all prime factors are two. So we can represent 1/2, 1/4, 1/8, and 1/16 exactly.

Duodecimal (Base 12) is very convenient for having a high proportion of exact fractions. Eg - 1/12, 1/6, 1/4, 1/3, and 1/2 can all be represented exactly. I'd argue in everyday use we're more likely to consider 1/3rd of something than 1/5th. Counting by twos, threes, fours, and sixes is easy. Watch, let's count to 20 (24 in decimal) by 3's : 3, 6, 9, 10, 13, 16, 19, 20. In 4's : 4, 8, 10, 14, 18, 20. By 6 : 6, 10, 16, 20.

Base-12 offers four handy subdivisions (excluding 1) instead of two for decimal or three for hexadecimal. That beats hexadecimal using fewer unique digits. It beats decimal by two using only two extra unique digits.

And I think it's these reasons it was chosen for various historical subdivisional units (inches per foot, pence per shilling).

The other item to consider is the relative number of unique values per digit. I'm not sure of the utility of having 10, 12, or 16 here.

At one extreme, while binary is useful for discretising signals in digital logic, using only zeroes and ones becomes cumbersome for daily use at higher numbers.

Once we're at base 10 and higher, I'm not sure how much here extra digits help or hurt.

>I'd argue in everyday use we're more likely to consider 1/3rd of something than 1/5th.

Because, I'd think, many things follow power laws which makes values closer to 1 occur more frequently.

It seems the consensus is that base<prime number> like base 7 or base 11 definitely would a bad idea.

Unless you believe in the consistency of inconsistency prime bases would be annoying to work with.

> In a six-fi novel I'm writing, an advanced alien civilisation uses base-12

Six-fi? Typo or genre I'm not aware of?

Haha, yeah, sci-fi.

Luckily I'm not yet at the copy-editing stage ;-)

Make them use base-23 and somehow this will allow them find the secrets of the universe because the question to life, the universe and everything can be solved easily with base-23 math.

The higher the base the more combinations of numbers you have to memorize to do basic arithmetic, and fewer steps you have to take when doing things like addition. I'm sure there's a sweet spot. I don't know that decimal is necessarily it.

The most efficient basis when you take into account both storage and calculation work is binary. That's not exactly the reason our computers use it, but it's one reason why they never moved away from it.

But humans are kinda complex, and don't trade well long term learning for boring tasks. We seem to be great exactly around base 10 (probably 12 would be perfect, like people already said).

According to Wikipedia [0], the most efficient radix is Euler's Number.

But what would it even mean to construct a numeral-system with a non-integral radix?

[0] https://en.wikipedia.org/wiki/Radix_economy

I am going to guess base 10 is prevalent because we have 10 fingers ? :)

I speak a language (Mende) that has no formal mathematics but counting is done in base 10; there are 10 individual words for 1 to 10 , the word for one is "yaella", the word for 10 is "poo", the word (really expression) for 11 is "poo mahun yaella", which literally translates to 10 on top of 1 ie 10+1...I can only think they stopped at 10 words because there are ten fingers to do elementary arithmetic?? granted there are other words for high magnitudes...there are apparently other languages with other bases besides 10 so who knows...possibly due to using other cultural artifacts to serve as digits.

> I am going to guess base 10 is prevalent because we have 10 fingers ? :)

Yeah, basically. There are other ways to count on your hands though. I seem to recall there are places that count on their hands using the 12 finger joints (or segments), and it works just as well.

Edit: To clarify, they use their thumb to point to the joints or segments on each of their fingers, in turn.

interesting :)...if you use all the bone segments in the fingers you can also get base 14 or 28, there is probably a culture out there doing just that.

An odd base is better in the sense that it doesn't need the weird "round to even" and "round to odd" rounding modes (https://en.wikipedia.org/wiki/Rounding#Round_half_to_even) to do unbiased computations when computing with fixed-width numbers.

That would mean 1/2 doesn't have a finite expansion, but why would it? Most numbers don't.

Not really. Different decimal systems will make divisibity by the primes of the base number more obvious (and thus polynomials whose solution set contain those primes will be more obvious), and fractions whose prime factorizations are contained in the primes of the base will be finite as opposed to "repeating", but all the structures beneath remain the same.

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