Commutative property : It says that we can change the order of numbers (operand). Given a expression where we have multiple additions (say A+B+C+D, where A,B,C,D are numbers), commutative property says that we can place the numbers A,B,C,D wherever we want in that expression. Something like this B+C+A+D. In short, (A+B)=(B+A).
Associative property : It says that we can change the order of addition (operation). Consider the same expression A+B+C+D, associative property says that we can perform additions in any order. Say we can compute B+C first, and then add D to it and finally we could add A to it. In short A+(B+C) = (A+B)+C.
The above two properties though looking simple, are very effective and helps us in doing fast computation. Say we need to compute 23+65+12+35. Here we can compute 65+35 first with the above two properties and then compute 23 and 12. This makes it easy for us to add, ie 65+35=100 and 23+12=35 and then 100+35 = 135.
Additive Identity : Additive Identity, is any number which when added to a number N will result in the same number. Ie, N+0=N. Hence 0 is the additive identity.
Additive Inverse : Inverse identity, is any number which when added to a number N, will result in zero. Here N + (-N) = 0. Hence, in general, additive inverse of N is –N.
Properties of subtraction : The following are the properties for subtraction.
Commutative property does not exist for subtraction : Say we have to compute 2 - 3 , now if we do 2-3 = -1 and if we place the number as 3-2, it equals 1. Hence Commutative property is not applicable to subtraction.
Associative property also does not exist for subtraction. Say we have 2-(3-4) = 2-(-1) = 3. Now if we change the order of subtraction, (2-3)-4 = -1-4=-5. Hence associative property also does not exist for subtraction.
Identity : Same as that of Additive identity. 0+(-N) = -N , where N is positive.
Inverse : N is the inverse of a number –N where N is positive, since N-N=0.