Base 10 is not a good base
teamten.com
teamten.com
- It's not consistently base 2 (or base 12), it's more like base random.
- Imperial calculations still use base 10 numbers for representation! This is the worst of both worlds. You can argue that base 10 is inefficient but at least the Metric system aligns perfectly with it, as compared to the Imperial system which uses base random and base 10 numbers which cannot perfectly model many Imperial system values.
> Metric’s entire foundation is a bad base.
You're stuck with base 10 numbers. Don't blame the base, pick a better system.
It was a very short comment.
What's the connection between those two mentions?
Surely you're not saying mention 1 is unrelated to mention 2. And it really looks like an attempt to do the same calculation with both system.
It would be very weird if Foo and Bar were an analogy for feet and inches for example, and even weirder if they weren't an analogy for anything.
12 / 4 = 3 Vs 100 / 4 = 25.
See, math is easy when you do it correctly!
Metric may be not perfect, but is not in the same category of mess that is the imperial one, specially if he is complaining about proportions between different units of measurements.
We can celebrate the fact that our notion of numbers is flexible enough that we can easily represent other bases without significant reach. Eg: Imagine trying to express another base with Roman numerals.
Is what I thought you were going to say. But I do sometimes wish we had 12
If we can convince everyone to have 12 fingers, maybe we can convince them to switch to dozenal.
This scheme allows for binary counting and even generalizes for people with different numbers of fingers. Lose a finger? You can still count to 511. Gain two fingers through genetic manipulation? Now you can count to 4095!
Maybe the binary coded decimal fans can reduce their counting space and count to 99 with two extra bits left over for holding their pencil!
Base 12 or maybe “base 10” where 10 - 1 = B could more efficiently be encoded in the same way so that one can still count to BB and have two bits left over!
The world is chock full of possibility. In the meantime, we have really cheap calculators available so we can divide 7 by 3 and not have to cut one of our fingers partially off to express the result.
Sumerians used Base 60.
https://en.wikipedia.org/wiki/Sexagesimal
It’s the most divisional number under 100. Hence why 60 is still used today for so many things (clocks, degrees, etc).
My understanding is that it's "finger bones between knuckles reachable by thumb" and "count of times counted through all knuckles". Meaning you count to 12 with one hand (thumb incrementing through finger bones) and then count 13-60 by incrementing the other.
It just so happens that 12 * 5 ends up having a fantastic number of divisors.
So in some ways it's not entirely coincidental.
In fact, 12 is derived the same way, 4 fingers times 3 knuckles, so the only coincidence is that 4 is divisible by 2.
If people were Disney cartoon characters (3 fingers + thumb) they'd likely have used base 36 (9 segments, 4 total digits on other hand).
And yes -- by having a mixed system like this you'll end up with more divisors than if you only count digits.
OP didn't say or imply anything about the origins of the Sumerians' use of base 60, they just observed that it has nice properties.
60 and 12 are both divisible by 2, 3, 4, and 6, which are some of the most common divisions that people need in daily life.
Base 60 has the additional advantage of being divisible by 5, 10, 15, and 20 as well. Note that we still use bases 60 and 24 for our timekeeping, and it's extremely convenient to be able to divide up the hour or day evenly into these chunks.
The metric system is great for scientific purposes, but the properties that make it good for scientific use are not terribly relevant to household or daily use.
零一二三四五六七八九十百千萬億 is all you need to get into the billions. It's not as practical for computation, though.
But yes, having a standard everyone agrees on is way more important than how well thought out the standard is.
30 is not divisible by 4.
Given how this very short rant of a blog post is technically right that 10 is not a very integer-division-friendly base and how it uses imperial measures to illustrate the point—the pain point with imperial measures is not that they don't take base ten, the point is that it takes a plethora of multiplicators between commensurate units and that none of those multiplicators lines up with the way we say and write numbers. The obstacle from switching from colloquial "four in the afternoon" to "16:00" (not e.g. "14:00" which for a learner would be the obvious but wrong answer) is not to blame on the multiplicator, it's the fact that the multiplicator is not the base of our counting. Another obstacle in adding up seconds, minutes and hours and to say the results in terms of days, hours and so on is that several multiplicators—12, 60 and 24—are involved, none of which is a power or at least a multiple of the base that we use for expressing the results.
Of course none of this even touches about the great utility of a system where one liter equals a thousand cubic centimeters, with water weighing (OK, "massing") one kilogram (at standard conditions, imagine all the hedges), and where all mass measurements are based on the kilogram, all lengths on the meter and so on. In imperial, the horizontal distance from you to that tower is given in miles, yrds or feet depending on your habits and how far away you are; the height of the tower will most likely be given in feet, the vertical distance to a plane high in the sky curiously in feet, too, but the distance to the ISS in miles. In metric you do something similar in choosing between centimeters, meters, kilometers as you see fit, but crucially the digits remain the same, only the decimal point shifts somewhere else. This is what they wanted when they initially proposed the metric system near the end of the 18th century; SI is building upon this and greatly expands the system.
You are an apprentice house builder. Your boss hands you meter-stick, with 100 lines on it to mark out the centemeters, and asks you to cut him a piece of wood which is 1/3 meter long. This puzzles you, because you can't find a line on that stick which indicates 1/3rd of a meter.
So you go back to your boss, and he grumbles, but gives you another meter stick, with 1,000 lines on it. You still can't find a line on the stick which indicates 1/3rd of a meter....
Yeah, you can't represent 1/10, but why does that matter? When 1/10 comes up in practice it's almost always as a side effect of us using base 10, not a natural requirement to partition something 10 ways.
You can't represent 1/5 either while base 10 can. You gain 3 but lose 5, it has the same number of primes.
Here is a thumbnail sketch of a reason for why they do:
1. Large structures tend to be build from smaller substructures. (E.g., a train is typically built of a number of cars, a 6-pack is built from 6 cans).
2. And large structures tend to be built from integral multiples of smaller substructures, because a fraction of a substructure is typically not as useful as a whole substructure. (e.g. a train car missing its wheels isn't as useful as an intact train car, and a partially drunk-up can of beer isn't nearly as desirable as a whole can of beer.)
3. The fundamental theorem of arithmetic says that every integer can be factored into prime numbers.
4. Small primes are more common factors of integers than large primes are. (e.g. you more often want twice of something than 37 times of something)
Ergo:
5. Most of the numbers involved in designing and building something are going to be divisible by 2 and 3--because the number of subsystems it contains will likely be multiples of 2 and 3. Prime factors of 5 and over are relatively rare.
6. Therefore, when you are measuring the larger system, it is handy to use units which are easily divisible by 2 and 3. (like a foot is 12 inches).
Its why donuts and eggs are sold by the dozen, and beer comes in 6-packs, and cases of 24. You are far more likely to divide up cans of beer or some donuts to a number of people which is divisible by 2 or 3, then by 5 or any higher prime factor.
TLDR reason: Every second number is divisible by 2. Every third number is divisible by 3.
So if you want to divide something (like donuts or cans of beer, or the length of a wood plank) by N, it's far more likely that N will have a factor of 2 or 3 than it will not have a factor of 3, but be divisible by 10. Which is to say, you are far more likely to need tick marks spaced by 1/2 and 1/3 than tick marks spaced by 1/10. Which is to say that a 12-inch ruler is preferable to a 10 cm ruler, Q.E.D.
That's why they sell donuts by the dozen, and beer in 6-packs. And why the US officially (and Canada and the UK unofficially) still use English units for everyday measurements. Its just more practical.
https://www.amazon.com/Universal-Stainless-Protractor-Precis...
This is why 12-inch rulers commonly have the inches divided into 10 inches.
There is a line on that ruler for 1/2, 1/3, 1/4, 1/5, 1/6, 1/8, and 1/10 of a foot. Uh, and 1/12th of a foot too, of course :-)
cf. with a 10-cm ruler, each cm divided into 10 mm. You've got a line for 1/2, 1/4, 1/5, 1/10, and 1/25, and 1/100th. That's it.
In reality, I would cut the board either 33.4 mm, or 13 1/8 inches (a bit long) then pound it in place (wood has a bit of give to it).
Arguing in favour of base 12 systems would make sense if it seemed doable to introduce more numerals and replace base 10 altogether, in my opinion, but that's not realistic.
Besides, if you live somewhere where the metric system is used, and you're unable to manually measure and cut one third of a meter (0.333m/33.3cm/333mm/333333µm) with the same speed and precision as you would be able to using a yard-stick - then you probably shouldn't be allowed near the tools needed in the first place.
PS. In practice, the boss would've asked you for a piece that is either 3dm, 33cm or 333mm long, indicating the expected precision.
There is no way to machine, grind and then lap the wood to a perfect 1/3.
Boss tells you to install 3 separate cabinets with shelves in 1 meter space. You measure that too to be sure it is. Then you see cabinet walls are 1 cm thick, you need 4 walls. 96 cm total shelve space. Actually this is nice 32 cm per shelve... And you even have some room to make extra cuts...
Instead, what you find is that they measure things in millimeters--and curiously, the lengths in millimeters tend to be numbers like 48, or 120, i.e. numbers which have lots of factors of 2 and 3 in them.
Don't just take my word for it!! Go browse Amazon.com and see what typical lengths of objects designed in those countries are, and what units they are stated in.
Which is to say, if you are really going to use the metric system, you are going to be using lots of lengths divisible by 12 anyways. Which means they are not powers of 10, and probably not divisible by 10, which means you've largely lost the biggest advantage of the metric system--easy arithmetic for numbers which are powers of 10 and divisible by 10.
Cf with a ruler, which is divided into 12 inches, and each inch into tenths. There is a line on that ruler for 1/2 a foot, 1/4 of a foot, 1/5th of a foot---and 1/3rd and 1/6th of a foot. All of the smallest and most commonly used prime numbers are present and accounted for.
cf with a 10-centimeter ruler, each cm divided into 10 milimeters. You've got lines for powers of 1/2 and 1/5--that's it. If you want to divide it by any other factor, you are squinting your eyes and guesstimating between lines.
"Why don't I just give my boss a 33mm piece?" -- and leave a 1mm gap in your house, where ants can come in, and heat goes out....
"Why would I ever need a board 1/3rd of a meter long?" -- sigh it could just as easily be a board 10/3 meters long, or 200/3 meters long. Because 3 is a small prime, and by the fundamental theorem of arithmetic, 3 is going to be a factor in a lot of integral lengths.
Say you are designing a switch with two buttons on it, "on" and "off". Well, the centers of those two buttons are going to be located at 1/3rd and 2/3rds of the length of the plate they are mounted on.
Say you design a crock pot with 3 buttons for the heat, "low", "medium", and "high". You don't want two of those buttons to be 1cm apart and the other 2cms apart!
If you look at countries like, say, Japan, or Germany, which really do use the metric system, you'll find that they measure things not in centimeters, but in millimeters. And the lengths they use are numbers like 48, or 120, which have lots of factors of 2 and 3 in them. Because you divide lengths into 2 or 3 parts all the time.
Which is to say, if you want to actually use the metric system in industrial design, you don't use a lot of lengths which are powers of 10. You use lengths which are divisible by 12 anyways.
There's a reason the U.S. never went off the English system, and why countries like Canada and the U.K., which actually make real efforts to try to go off of the English system still use a lot of English units in everyday measurements. And it's not just because of conservatism, or NIH syndrome.
Rather than presenting a rhetorical device, I’m genuinely wondering what would happen to math literacy rates when switching to something like base 12.
It isn’t a question about specialist use cases, because we already apply plenty of specialist systems as needed, such as binary and hexadecimal in computing.
Two just isn't a big enough divisor, ten works much better as a "step change" interval.
A measurement system that was uniformly 8 or 12 or 16 could avoid that problem by having equivalent steps to mm, cm, meter, etc. but still runs into the problem of having to change the entire number system too which is rather not the fault of the metric system in the first place.
And time isn't decimal-based at the larger-than-one-second level, after all.
But in the world of concrete objects "eyeball this into two pieces" is much less useful and less common than "ok so we need to account for the extra 3mm of this extra piece so add that to the 44mm base measurement" - which is much more straightforward than something like "add 3/32nds to 1 and 3/4 inches." especially if you don't know that you're gonna end up with a measurement in the 32nds when you make the first measurement.
Or larger-scale like "we went six hundred feet already, we needed to go a mile and a half, how much is left?"
Base 10 comprises ten digits, viz. count(0, 1, 2, 3, … 9) = 10z
Base 2 is count(0, 1) = 2
Base 16 (hexadecimal) is count(0, 1, 2, 3, … 9, a, b, … f) = 16
The other view is fine too: Base-9 would mean "max number is 9".
Base 10 is count(0, 1, 2, 3… 9) = ||||||||||
Base 16 is count(0, 1, 2,… a, b, …f) = ||||||||||||||||
The opposing view is that we'd be better off using only powers of one single factor—10—for all commensurable units, and prefer decimal fractions for calculations.
I've frequently seen people argue in favor of the imperial system arguing that one-third, one-fourth, three-eighths of an inch is so much more natural than, say, 0.5m, 0.33m and so on. I have hardly to point out that this perceived advantage (that I believe in to a degree) breaks down as soon as you want to add and multiply: what's 2 times 3/8ths plus 1/3rd? When calculations increase in complexity, using only fractions of a single base and its power—and using a base that is already the base of the number system that you think and count in—takes a lot of incidental complexity out of the system.
We could make the switch to 12 if we wanted. Ten fingers and two hands, so we can make a case that it is 'natural'. Then we have 2 and 3. Time is already on base 12/24. Then someone says hey, we got 5 fingers! And we're now doing base 60. Old school Sumerian. Time still works as well. All we need are glyphs for the digits, somehow reflecting the natural partitions into 12, 5, 3, and 2.
I do, because it's 2^16, a very important number in computer science (though less important now). But I won't spoil it by giving the answer.
<12> * <12> * <12> * <12> in base 12 is 10000
Assuming that you mean the decimal number "12" and not the number 12 in base 12 which is the decimal number 14.
In base 12 the number 12 is written "10"
If you write it out in base 12, it looks like 10 * 10 * 10 * 10 = 10,000. I can't tell you what number that is in English without working it out of course.
Edit: lol responses explode within 60 seconds saying the same thing.
Edit: You got nine of us with one comment. You win.
Do you know how many feet are in a mile? The answer is: Who cares.
The fact that you could pick a better base is irrelevant. They had their chance and came up with some silly numbers to scale from an inch to feet to yards to miles.
Metric: Just remember the base unit. Want a new unit? You change the prefix and move the dot appropriately. You can now scale down to atoms or up to galaxies with a single base unit.
Edit: yeah, wanting to add a symbol for twelve instead of two more symbols before 10 in base twelve is my hasty mistake.
It's only hard bc your "12" is in base 10. In the proper base it's easy.
It can make sense as a mistake without needing the base to actually be higher. The base isn't higher just because they carried a digit incorrectly.
I must say though that there’s a certain awe and glory in bouncing a dec number up and down by 10x by moving the decimal.
Bases aren’t “good” or “bad” you the user are either good or bad at dealing with them. If you think a base is bad you’re probably not working hard enough to use it well.
Or trolling. Maybe you’re just trolling.
When you count on your hands, you can use one hand for 1-5, and the other for your 10s (6s) so you can count much higher on your fingers.
The internet seems to increasingly just be a place to rewrite old thoughts but with less actual content (or more content with less actual thought behind it).
Could be confirmation bias, but I feel like there was a clear transition when comment threads on forums became more prone to top level repeats of the same thought rather than replies to the first person who had the same thought. ("Came here to post this" ain't great, but I'd take it over dozens of blindly typed repetitions)
Seems to have bled out into blog posts too now.
Can't imagine this will reduce with LLM (mis)usage.
Is it just that we read less? That there's so much new information and such pressure to produce that there's no time for discourse? Has information reached such critical mass that we're all emboldened with the assumption that all which came before was cruft?
Genuine question, not rhetoric (and I'm certainly guilty of the same thing - perhaps someone already expressed this in a comment below and I didn't even look) but I'd love to see it balanced by something if not solved.
Can we still somehow return to building on the shoulders of giants, or are we doomed to an eternity of posturing and feigned originality?
Seems like it'd be an interesting topic to explore if it hasn't been already - any sociologists here?
Base8 would have been workable.
Base10 is just crap. It is quite unfortunate how it is embedded in society now.
It's been this way for a couple thousand years
100 in base 12 (144 in base 10) is divisible by
1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144 (base 10)
The mental model of these proportions would be accessible to us relative to quantities of 100 base 12, in full precision, if we used base 12.
We could readily divide common quantities into halves, thirds, quarters, sixths, eighths etc. and get results in whole integers.
No more errors due to estimations of 1/3 * 100 as 33.333333.. and 1/9 as 11.1111111..,
100 / 3 = 40 in base 12, done.
100 / 4 = 30
100 / 8 = 16
100 / 9 = 14
100 / 12 = 10
Look how nice that is.
100 base 10 is only divisible by
1, 2, 4, 5, 10, 20, 25, 50, 100 (base 10)
It's really a tragedy that we use base 10.
just kiddin, interesting observation though.
At least some, perhaps all, of the giants in the Old Testament of the Bible had 6 fingers on each hand:
"In another battle at Gath, there was a tall man who had a total of 24 fingers and toes: six fingers on each hand and six toes on each foot. He also was a descendant of Haraphah." - 2 Samuel 21:20
The Babylonians had base 60 (more or less) and they did serious astronomical calculations way back in the day. Maybe we could do that again.
source: https://space.stackexchange.com/questions/37607/why-did-nasa...
If you keep reading the very source that you yourself linked you’ll find that NASA generally used customary units. Hence my qualifier “mostly” in my entirely correct comment you replied to.
An advantage of a small base is that you can use different symbols for each numeral and make reading much quicker/easier/less error-prone. 60 is probably too many to be optimal, but most alphabets work well with a few dozen.