magine if we didn't have the zero.
Before the invention of the zero, people used positive counting marks (like Roman Numerals) to count. There were number systems based on 20 used in Old French. (They survive as "score" as "four score and seven").
If you have no zero, you only have assertions:
XX -> I have two groups of 10.
MMIX = I have two groups of 1000, and 1 less than one group of 10 - in arabic numerals, 2009.
In order to add, you simply wrote down all the numerals, and then grouped them according to the rules. To subtract, you canceled numerals - like:
a hundred years before 2009 it was 2009 - 100, or 1909. The same problem in Roman Numerals is:
MMIX - C = So you break an M and the number is: MDCCCCCIX -
Then you strike C to do the subtraction, and then you combine using subtraction:
MCMIX.
Now suppose you want to multiply. You have thirty things that cost 25 dinar:
XXX * XXV You take
75 X's - answers makes me shorten.
25 of the thirties - then you reduce, a C is 10 X's, so
CCCCCCCXXXXX And an L is 5 X's
CCCCCCCL There is a D that is 5 C's:
DCCL And that is your answer.
In this system, I always means one. It does not matter where it is in the number, it means 1. If it is to the left of a larger symbol it means subtract. X always means 10. Roman numerals are so obtuse that sites that try to explain them and where there are great calculators, do not use roman numerals to explain them.
So, the BIG invention was Arabic Numerals, and the zero.
How we count is a matter of definition. In Roman times, the Roman Numeral was how people counted. It was difficult - and people who were counting sheep would just have a string and make a knot for each sheep.
In the Arabic system, BY DEFINITION, there are a certain number of symbols.
The symbols have two values. One value is absolute: 0 means none, 1 means a single unit, two means two units, etc.
The other is positional: a symbol in the one's place is that value. One place to the left it is times 10, two places it is times 100, etc. To the right, the first place is times 1/10, the second place is 1/100, etc.
And now that you have positional values, you need a symbol that says, "Even though we have hundreds, we have no 10's: the zero. One hundred 5, roman numeral CV is 105 in Arabic, because we need a placeholder. This also means that we don't need a special symbol for every power of 10 (or a way to mark the symbol as some increase over the original value, like the overbar makes it increase a thousandfold. Because we have placeholders.
That zero, and the fact that the tens place used the same numbers as the one's place, with zero as the placeholder when the number was evenly divisible by 10 (I thought of it as every 10, back then) was what I had re-discovered at age five.
This is by definition. We are so used to Arabic Numerals and the zero, it is not clear we will ever be capable of counting any other way, nor is it clear why we would want to. Things like machines (computers) count in binary, but they still have zeros, placeholders, and to the left the positions mean 2^0, 2^1, 2^2, etc. The system of having digits that have an absolute value and a positional value just works. It allows us to write numbers easily, and to think about very large numbers.
The stuff you mentioned about borrowing, and carrying - this works because of the definition of the systems. It is because - each place is 10 times the preceding place - the places always have that relative definition, so borrowing works.
There is nothing natural about our way of counting. I have tried to explain that in the times when our calendar started counting from the year 1, there was no zero. This is why, when you shift from AD to BCE dates, you have to add and subtract one - because when those dates were numbered, it was not considered common to have a zero - it had not been invented in the year zero. Dates were numbered as "the first year of the rule of King so-and-so" until the dating system was commonized.
The Romans dated from the founding of the city of Rome, and the current system was invented by a monk in what is now thought of as 525. This was before the invention of zero - so there is no year zero. But in fact, the zero was invented over and over again, and the Indian subcontinent had several notable mathematics geniuses who described, mostly correctly, the use of and rules for zero. This was imported into Europe, most likely. But the Greeks had used zero for different things.
So here we have all these systems, and, well, some of them worked, some of them were miserable.
And borrowing/carrying works, not because there is anything natural about it, but because it is designed to work.