The System Z of Integers
The systems of natural numbers and whole numbers are closed with respect to addition and multiplication and they have the basic properties mentioned in the previous chapter. However, these two systems are inadequate to solve every linear equation a + x = b, for x if given a and b in N. The difficulty is that solving a + x =b is essentially a problem of subtracting. In Exercise 18.6, one can see why this cannot always be done. The inability to subtract a from b for arbitrary a and b is overcome by forming a new system from N in which subtraction is always possible. The system so formed is the system of integers.