# Could we define the decimal notation of a natural number as a series of operations applied to 0? [closed]

People have made mistakes. For that reason, there probably is a demand by some people for a computer program that's easier to verify makes the calculations properly. I think that according to the conventional definition, for the 10's column, you add 10 times the digit, for the 100's column, you add 100 times the digit and so on. We could alternatively define the decimal notation as follows. Start from 0. Multiply by 10 then add the first digit then multiply by 10 and add the second digit then multiply by 10 and add the third digit and so on.

I think in Polish notation, operations such as + proceed the operands but you still write numbers in decimal notation normally so people use spaces to make notations unambiguous. That's because if they didn't use spaces, +222 could mean 2 + 22 or 22 + 2 so they instead write + 2 22 to mean 2 + 22 and + 22 2 to mean 22 + 2. I think that in addition to that, we could also describe natural numbers as a series of operations applied to 0 each of which is of the form of left multiplying by 10 then left adding a number between 0 and 9. For a reason I will described later, spaces might not even be needed. Also, you'll see later why I'm using the character ∅ instead of the character 0 to represent 0. We could let ∅ denote the number zero, S, the successor operation, + the addition operation and $$\times$$ the multiplication operation. So 2 + S(2 + 2) would be denoted +SS∅S+SS∅SS∅. Now how do we denote the method of getting the number 122 by starting from 0 then multiplying by 10 and adding 1 then multiplying by 10 and adding 2 then multiplying by 10 then adding 2? It's by writing +SS∅×SSSSSSSSSS∅+SS∅×SSSSSSSSSS∅+S∅×SSSSSSSSSS∅∅. To make it more compact, we could invent 10 single digit characters each of which is short hand for a string of characters as follows:

• 0 means +∅×SSSSSSSSSS∅
• 1 means +S∅×SSSSSSSSSS∅
• 2 means +SS∅×SSSSSSSSSS∅
• 3 means +SSS∅×SSSSSSSSSS∅
• 4 means +SSSS∅×SSSSSSSSSS∅
• 5 means +SSSSS∅×SSSSSSSSSS∅
• 6 means +SSSSSS∅×SSSSSSSSSS∅
• 7 means +SSSSSSS∅×SSSSSSSSSS∅
• 8 means +SSSSSSSS∅×SSSSSSSSSS∅
• 9 means +SSSSSSSSS∅×SSSSSSSSSS∅

Note that each of those strings of characters can be considered a unary operation so the 10 digits can be considered short hand for those operations. Then the expression 2 × 1008 can be denoted ×2∅8001∅. It's so much more straight forward than interpreting the standard Polish notation where you would probably write × 2 1008. I know that according to my notation, 8×2∅2∅ is also a valid notation for the number 48 even though there's no notation in standard Polish notation that's analogous to it. My question is

Could this notation be a nice simple way to tell a computer to make calculations on numbers given in decimal notation? A human using a Python like program could type in ×2∅8001∅ to be given the answer 6102∅ and know how to interpret it as meaning 2 × 1008 = 2016. They can put simple problems to the test to verify whether it actually gave the answer they would expect it to give.

• I don't see any real answerable question here, except the title, to which you've already proven that the answer is "yes". Also, skimming your post, I see that you somehow get from decimal representation to chess computers: one topic per post, please, and that topic should be a question. – David Richerby Jun 23 '19 at 8:17
• Your question is very hard to read; it may discourage possible answers – lox Jun 23 '19 at 23:01
• @DavidRicherby My question was whether people thought it was a good idea to express natural numbers in that way because they can be expressed as a series of operations on the number zero given in Polish notation, making it a very straight forward set of instructions to type into a computer program. I don't think Polish notation currently does that which is why I thought this question might be useful. I know there's a guideline not to write questions for discussion but it's not written in stone. The question – Timothy Jun 24 '19 at 0:54
• @Timothy "appears to be a question for discussion and it got a score of 19." Some other question on some other site getting a high score isn't really relevant. "For those who are great at paying attention, I think this question is explained pretty clearly." Two people have told you that it's unclear. Take the hint. – David Richerby Jun 24 '19 at 9:53
• What is your question? What should be in the answer? This looks like discussion starter, not objectively answerable question. – Evil Jun 24 '19 at 17:11