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Creation of the rational number system

What is the process, in terms of sequences of definitions, theorems, etc., in extending the system of positive rational numbers to the system of rational numbers i.e. in defining negative numbers and creating the system of positive-negative numbers known as the rational number system?

Step 1. Define the concept of a negative number.

Definition 1. Negative number. Let a be a positive rational number. Then negative a, that is -a, is the additive inverse of a i.e. -a is that number a* for which a* + a = 0.

NOTE. We note that since the inverse a* of a is not in the set of positive rational numbers, it is a “created” number, a number that only exists by definition and only in our imagination. It may not correspond to anything we can visualize physically at all.

We have thus, in this definition, defined a number a* with the property that a* + a = 0 and called it the negative of a.

Step 2. Define the sum a + b* = a + (-b) where a and b are positive rational numbers.

Definition 2. The sum a + b* = a + (-b) is defined as being obtained by the following procedure: Subtract the smaller of the two numbers a, b from the larger and attach the sign of the larger.

Step 3. Define the sum a* + b* = (-a) + (-b) where a and b are positive rational numbers.

Definition 3. The sum a* + b* = (-a) + (-b) is defined to be (a + b)* = - (a + b).

Step 4. Define the products ab* = a(-b) and b*a = (-b)a where a and b are positive rational numbers.

Definition 4. The products ab* = a(-b) and b*a = (-b)a are both defined to be (ab)* = -ab.

Step 5. Define the product a*b* = (-a)(-b) where a and b are positive rational numbers.

Definition 5. The product a*b* = (-a)(-b) is defined to be ab.

Derivation of the quotients a/(-b), (-a)/b and (-a)/(-b). For the case of the positive rational numbers the quotient a/b of two numbers a and b is defined as the solution of the equation bx = a. We extend this definition to our new positive -negative number system that we are creating. Thus the quotient a/(-b) is the solution to the equation (-b)x = a. The solution is x = ole.gif Thus the quotient a/(-b) is given by ole1.gif By the same reasoning the quotient (-a)/b is given by ole2.gif and the quotient (-a)/(-b) is given by ole3.gif

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