# Why are digits written in groups of three?

This may be a simple question, but I'm intrigued and am not having much luck looking it up. At least in the US and other major countries, you have a units place, tens place, then hundreds place. After a comma, you go to thousands, ten thousands, hundred thousands.

So why was it set up like that? What's the logic behind it? I'm creating my own number system for fun, and I'm wondering "what's to stop me from having 4 or 7 main value places?".

• I can't seem to find the magic words to return a search result, but I remember reading that humans can instantly identify the size of a set of up to (I forget) 3 or 4 objects, without breaking into smaller groups and adding (e.g., instantly recognizing the 3 dots on a dice as "3", versus evaluating the 5 as $4 + 1$). I can't help but wonder if that's relevant. Either way, it's purely conventional. – pjs36 Jun 13 '17 at 2:54
• @pjs I don't believe that people generally recognize 5 on a die as 4+1. We're so good at pattern matching that we instantly recognize a huge number of words simply by their shape -- I very much doubt that you read any of the words in this comment by breaking them down into single letters. So it might be true that people recognize five points in general position as 4+1 (though I'd be surprised if the cut-off were quite that low) but the arrangement on a die is very special and recognizable. Even crows can tell the difference between seven and eight. – David Richerby Jun 13 '17 at 9:05
• It seems obvious from the words thousand, million, billion, trillion etc. – Jim Balter Jun 13 '17 at 9:22
• @JimBalter But those words used to have different meanings (e.g., a billion used to be a million million, rather than a thousand million, etc.) – David Richerby Jun 13 '17 at 9:45
• Japanese has four: 1=ichi, 10=jou, 100=hyaku, 1000=sen, 10000=ichi man, 100000=jou man, 1000000 = hyaku man etc. – Nathaniel Jun 13 '17 at 9:48

It's certainly true that this is an instance of "chunking", but I think that writing numerals that way follows the way we name the numbers in the first place. Consider $123,456,789$. Each $3$-digit block is read as a stand-alone three digit number, followed by an appropriate big-number word: "One hundred twenty three.... million," then "four hundred fifty six... thousand," and finally "seven hundred eighty nine."

Thus, the question is really, why did we stop making new words for each place value after "thousand"? Rather than sticking with "myriad", a somewhat disused word for $10^4$, we call it "ten thousand", and then $10^5$ is "one hundred thousand", with no new word being introduced until "a thousand thousand", which we call a "million".

I suspect - and this is entirely speculative - that this happened because, in the time when this aspect of language was being developed, there wasn't much use for numbers as big as $10,000$, so they were described in terms of smaller numbers, rather than being named independently. Looking at the etymology of the word "million", it originally would have meant "a great thousand", which sounds a little less silly than "a thousand thousand". Note that, after that, the words for additional multiples of $1000$ use prefixes for $2$ (bi-llion), $3$ (tri-llion), etc.

• That makes sense. So if one were to try to make a different system, it'd likely need to be rather easy/conventional to say/write, or people would make their own shortcuts (like they do with words, especially in text). – Iter Jun 14 '17 at 12:13

Nothing. It's just a matter of convenience and also convention. There are systems that use different spacings. As an example, in India, the first group is of length 3 and the subsequent are of length 2. For instance, we'll write 23,25,963 instead of 2,325,963. So, what's stopping you? Nothing really. But, I don't think it's going to affect your number system in any ways.

• It probably won't affect the number system drastically, you're right. Still, quite intriguing. It wouldn't change the number system so much as how it was seen. – Iter Jun 14 '17 at 12:15

I think a lot of it has to do with chunking.

For example: it's easier to remember (123) 456-7890 as somebody's phone number than 1234567890.

• Either way, that's a particularly easy-to-remember phone number ;) – G Tony Jacobs Jun 13 '17 at 4:45
• The easier the better ;D – Cody Mathews Jun 13 '17 at 4:47
• This seems speculative and it doesn't have a lot to do with the actual question. Why specifically do you remember phone numbers as (123) 456-7890? Partly, it's because of the area code which, in this form, is specific to north American phone numbers and doesn't apply to general integers. But why 456-7890 rather than, say, 4-56-78-90 as it would be in France? (Again, there are probably technical reasons to do with local analogue telephone exchanges but that's not the point I'm getting at.) – David Richerby Jun 13 '17 at 8:59