Real mechanism behind addition, subtraction, multiplication, division

[pratikaman]

we all know the basic rules for operations of addition, subtraction, multiplication and division.

but what i don't know is why these rules (of addition, subtraction, multiplication and division) works.

as if we have been given algorithm to do these operations but not explained how they were derived.

if anyone can explain please expain .

 

[arildno]

How they are said to work is given in the field axioms.

How do you know that the rules of a poker game, or the game of football "Works"?

They are stated as, essentially, arbitrary rules for how the game should be played.

Furthermore, you can create new games (and new maths) by laying down different basic rules defining the New game.
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Thus, the only real theoretical problem With the set of rules governing (i.e, the rules we have chose to govern) arithmetic is whether that rule set does not contain contradictions in their Application.
That would be similar that in a game of Soccer that you could come up in a situation where different rules of the game says that a goal is valid and an other Application of the same rules say the goal is invalid.

The issue whether arithmetics represent a consistent set of rules is a very subtle one, and not really understood prior to Gödel's work in the 1930s.
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The actual performance of, for example, multiplication, in the algorithm we learn in School is a condensation of multiple usages of the said axioms.