We recognize that every natural number can be expressed as 10 times a natural number plus a number from 0 to 9. Take any natural number and express it that way in the form $ (10 \times x) + y$ where $ y$ is between 0 and 9. Again for $ x$ , express that number that way and keep going until the second term of the expression is 0. This shows that every natural number can be gotten by starting from 0 and repeatedly applying operations each of which is of the form of multiplying by 10 then adding a number from 0 to 9. It seems so intuitive to define the decimal notation of a number by the method of getting it in that way where you start from 0 then multiply by 10 and add the first digit then multiply by 10 and add the second digit and so on. For example, we can define the notation 122 to literally mean $ (10 \times ((10 \times ((10 \times 0) + 1)) + 2)) + 2$ .

I know that’s probably not actually the way it was defined but it turns out to be correct. Since it is correct anyway, maybe it is better to define it that way as an instruction to submit into computers. It’s not that hard to show that that definition agrees with the conventional definition. Some people might have a demand to understand why proven math results such as the statement that long division to get a quotient and remainder works but have so many things to learn and don’t want to bother understanding why that definition agrees with the conventional definition. Despite that, could we still define it that way without creating a problem for those people? They might figure out a proof that long division to get a quotient and remainder works using that definition of decimal notation works and then be like “That’s fine, I accept it only as a proof that long division works using that definition of the decimal notation of a natural number and not as proof that long division works using the conventional definition of the decimal notation of a number.”

Also, could we treat English like Python and and say something means something because we defined it to mean that, and then use Polish notation to describe expressions with symbols we just invented a meaning of? Let’s say we already have a meaning for 0 which is ∅ and can also express the successor operation $ S$ and the addition operation + in Polish notation so $ 2 + S(2 + 2)$ would be written $ +SS∅S+SS∅SS∅$ . Next we define $ 0 = ∅, 1 = S∅, 2 = SS∅, 3 = SSS∅, 4 = SSSS∅, 5 = SSSSS∅, 6 = SSSSSS∅, 7 = SSSSSSS∅, 8 = SSSSSSSS∅, 9 = SSSSSSSSS∅, X = SSSSSSSSSS∅$ . Now, the number 122 can be described in Polish notation as $ +×X+×X+×X∅122$ . If you decide to replace the operations of right addition with left addition of the same number, then the Polish notation is $ +2×X+2×X+1×X∅$ . If you have a Python like program you can then again redefine $ 0 = +0×X, 1 = +1×X, 2 = +2×X, 3 = +3×X, 4 = +4×X, 5 = +5×X, 6 = +6×X, 7 = +7×X, 8 = +8×X, 9 = +9×X$ so now the digits are defined as operations so you can now type in $ 221∅$ to mean 122. Also, $ ×2∅8001∅$ could be the way to write 2 × 1008. Although some people use the symbol ∅ in place of the symbol 0, in this case 0 and ∅ don’t mean the same thing it at all. I originally defined 0 to mean the same thing as ∅, the number zero. Later, I redefined the meaning of all the digits to be operations where as ∅ still kept its original meaning, and the new notation for a natural number does not end until you get to the character ∅ making the multiplication expression unambiguous. Then those picky mathematicians will be satisfied with knowing that computer programs programmed that way can use the following law to compute the quotient and remainder of a division problem of natural numbers using the fact that the quotient and remainder of division of a one number by another number can be determined from the quotient and remainder of the floor function of a tenth of the former by the latter. There might be a lot of them who are picky because other mistakes in computer programs have happened such as the Chess Titans glitch in the YouTube video Windows Vista Chess Titans Castling Bug. They will consider long division of natural numbers to get a quotient and remainder to be entirely a function from ordered pairs of natural numbers to ordered pairs of natural number, and will not accept that as proof of how to compute the quotient as a rational number given in mixed fraction notation.