Subtraction is commonly known as the inverse of addition. Let's examine these two opposite operations at an elementary level. Addition involves taking two or more unsigned numbers and combining (adding) them to create a third number. For example, 3 + 4 = 7 means that given the two numbers 3 and 4, we perform addition to create 7. If we create a new number by adding two numbers, then the logical inverse process is to re-create the original two numbers from the result. In the example, the inverse process is to re-create the numbers 3 and 4 from 7. To do this, we must know one of the numbers, (as well as the fact that there are two numbers). Given this information, we perform the operation of subtraction: either 7 - 3 = 4 or 7 - 4 = 3 and re-create the original two numbers (3,4).
However, what if we have only the fact that there were two numbers? Then we have to "split" the number into two parts. This process of splitting numbers is known as partitioning, and is the focus of an entire field of mathematics called the partition theory of integers. In the case of the number 7, there are three partitions : (1,6), (2,5) and (3,4). Since we were given no additional information about the pair, we have a 1 in 3 chance of selecting the original pair (3,4). If we were not told how many integers created the result, then we would have to use the total number of possible ways to partition the number into from 1 to 7 parts (called the partition function), and there are 15 possibile ways to do this for the number 7.
Using this logic, subtraction is a special case of partitioning, where one of the parts is known. As a result, it is logical that the inverse of addition is partitioning rather than subtraction. In other words, the logical opposite of combining several numbers is splitting them apart again. The theory of partitioning integers occupied much time for many prominent mathematicians of the past, and continues to provide research opportunities for present mathematicians. There is still much to be learned regarding partition theory, and perhaps it's full understanding will reveal some of the mysteries of the positive integers.
The operation of multiplication is a condensed form of addition, and can therefore be viewed in a similar way. Using the example: 5 x 6 = 30, we have two factors, (5 and 6) and we again combine them (using multiplication this time) to create 30. The inverse of this is traditionally division, but the logical opposite idea is to re-create the pair of factors (5,6), given 30, by splitting it somehow. This operation is the focus of number theory, and such a function to easily split an integer into its prime factors has never been found. In other words, all the factors of a number can be found only if we are given the prime factors. In the absence of such a function to split numbers into their prime factors, a function that would provide the number of prime factors alone would be exciting. However this function as well is unknown. Of course if we had such a function to predict the number of prime factors, then we would be able to directly determine whether a number is prime or composite.
Although subtraction and division are not the logical opposites as described above, they are still valid operations. However, I prefer that all operations between integers be restricted such that all results are positive integers. So when I use subtraction, I choose that a - b = c be restricted such that a > b, and b >= 1. Similarly with division, where (a / b) = c must be such that b is a factor of a, and the division therefore produces a positive integer, c. In other words, b and c must be factors of a.