How did the rule of addition come to be and why does it give the correct answer when compared empirically? I'm still a high school student and very interested in maths but none of my school books describe these kind of things, and only say how to do it not the whys. My question is very simple, for example:
        19
       +25
      = 44

because the one from adding 9 and 5 goes on to add with 1 and two. How did this rule of addition come to be?
Here's a bit of explanation that can be useful(sorry if it is frustrating):
Suppose we are a 3 year old child and no one teaches us how to add and we recognize 1 is our index and 5 all palm fingers held up. Someone gives us the problem add: 1+5 so we hold 'em up, right? and again someone gives us to add 8564+2345 so we can't lift 'em up. So we try to device a rule but we don't recognize 6+4= 10 in which 0 stays and one jumps neither can we say that only the digits from rightmost positions are to be added. This is what i meant.

A visual representation:
  19
+ 25
= 44

Let's remove the answer and split the numbers into tens and units:
    10 + 9 {= 19}
+   20 + 5 {= 25}

This give us:
    30 + 14

We can make a new addition out of these numbers:
    14
+   30

Let's split these numbers into tens and units too:
    10 + 4 {= 14}
+   30 + 0 {= 30}

The 10 represents the 1 that was carried from the units column to the tens column. The second number, 30, must end in a zero because we only used digits from the tens column to calculate it. This means that at this stage we will never get a carry from the units column to the tens column, allowing us to progress past this stage, although for larger numbers we could get a carry from the tens column to the hundreds column (not shown). We can now complete the sum:
    10 + 4
+   30 + 0
=   40 + 4 {= 44}

This is why this method works. It is because breaking a number into parts (not digits, $10\ne1$) and adding all the parts up again will always give you the original number. $14=10+4=4+10=7+7=11+3=1+2+3+8$, etc. This is called associativity. When you carry the 1 from 14, you are actually carrying 10, but in the tens column you write 10 as 1.
