Fractions

The fraction is a special operation used in the construction of the rationals. A fraction is defined as defined as an ordered pair of integers. There is no real difference between ordered pairs and fractions except fractions are displayed differently and used in the theorems constructing the rationals for this reason. The top and bottom operators grab the numerator and denominator of the fraction.

At first glance, fractions and division appear to be identical. The number one half could be represented as the fraction (</> (1) (2)) or as a division (/ (1) (2)). But there is an important distinction. The fraction (</> (1) (2)) represents a pair of the integer (1) and the integer (2), but division takes the rational number (1) and divides it by the rational number (2). The fraction definition is useful for the construction of the rationals, but once the concept of rational division has been properly defined it is prefered to use the concept of division since it keeps everything in terms of rational numbers.