Is there a mathematical significance to the linguistic names for orders of magnitude? This may be more a English History question as far as naming, but I'm more concerned if there are any mathematical reasons for the way we represent numbers.
When written, by convention, every 3 orders of magnitude is separated by a delimiter/comma (ie: 1,000).  But its placement seems arbitrary.  Why didn't we decide to write it every 4 orders of magnitude (ie 1,0000) or 5 magnitudes?  Along the same lines we have different names for each multiple of 3 in the orders.  Those names do have structures built into them at the higher levels (ie tredecillion -> quatrodecillion) so there is a system in place for this convention.
However, you would think that if Base10 is the conventional numerical system that the naming conventions would share some correlation.  Instead we put delimiters/markers for the orders in a Base4 fashion.
So, is there any mathematical significance for orders of magnitude using multiples of 3 (Base4)?  What brought about this convention?  Is the placement completely arbitrary?  And are there any real math consequences produced from this decision?
 A: The divisions are cultural or linguistic. We break the numbers at powers of 1000 because we only have new individual words for the powers $10^{3k}$ for $k>0$: thousand, million, billion, etc. (These are the “short scale” words; a “long scale” system used in England and parts of Europe is becoming less common, where the words for powers of 1000 could be one, thousand, million, millard, billion, billiard, etc. See https://en.wikipedia.org/wiki/Long_and_short_scale)
In Chinese, for example, groupings are in 4-digit groups, and the words for fundamental large numbers represent powers of ten thousand, as opposed to powers of 1000 as is the case in English. One million is written using arabic numerals as 100,0000, and is written 一百万 (Yī bǎi wàn; one hundred ten-thousands) See https://vividchinese.com/big-numbers-in-chinese/.
In India, the group size changes: There are three digits in the rightmost group and two digits from each group left of the first. The base ten number 123456789 is written 12,34,56,789, which means twelve crore thirty-four lakh fifty-six thousand seven hundred and eighty-nine. See https://en.wikipedia.org/wiki/Indian_numbering_system
The divisions are not only to make it easier to see and count the number of digits; they also break the number up so it aligns with the individual words that exist for larger powers of ten.
