Swap the positions of two addends, and the sum remains the same. This is called additive commutative law.

A+B=B+A

A+B+C=A+C+B=C+B+A

For example: 8+1=1+8 = 9100+2 = 2+100 =102.

2. Additive associative law

Add up the first two numbers, or add up the last two numbers, and the total remains the same. This is the so-called law of additive association.

(A+B)+C=A+(B+C)

Example: 7+4+1= 7+(4+1) = (7+4)+1=1210-5+2 = (10)

Other than that, there is no distribution law.

Extended data

1. Proof: additive associative law (a+b)+c = a+(b+c)

When a = 0, (a+b)+c = (0+b)+c = b+c = 0+(b+c) = a+(b+c).

If it is true for a = n and (n+b)+c = n+(b+c), then it is true for a = n+ 1 = n'

(a+b)+c =(n '+b)+c =(n+b)'+c =((n+b)+c)' =(n+(b+c))' = n '+(b+c)= a+(b+ c)

So the law of additive association holds.

2. Proof: additive commutative law A+B = B+A.

First, it is proved that 0+m = m+0 = m.

According to the algorithm of addition 1, there is 0+m = m.

So 0+0 = 0.

Then1+0 = 0'+0 = (0+0)' = 0' =1.

So for m = 0 and 1, there is m+0 = m.

Using mathematical induction, assuming that n+0 = n holds when m = n, then when m = n+ 1

m+0 = n'+0 = (n+0)' = n' = n+ 1 = m

So 0+m = m+0 = m holds.

Then, mathematical induction proves that m+n = n+m.

For m = 0, 0+n = n+0, we have proved that this is the first card of dominoes. I dropped this card.

For m = 1,1+n = 0'+n = (0+n)' = n' = n+1,the second card is also dropped.

Then we need to prove that if a domino falls, it can be guaranteed that his next one will fall.

Suppose m = k and k+n = n+k, then when m = k+ 1

m+n = k+ 1+n = k '+n =(k+n)' =(n+k)' = n '+k =(n+ 1)+k = n+( 1+k)= n。

To sum up, additive commutative law was established.