# What are Rational Numbers?

Oct 24, 2014

Let's assume that we know what integer numbers are and what is an operation of their multiplication, so we know how to multiply any integer number by any other.

Considerations of symmetry and harmony lead us to a desire to reverse the operation of multiplication, that is to be able to divide any integer number by any other. Obviously, it's not always possible within the realm of integer numbers. Operation of multiplication is not really complete in the space of only integer numbers since its reverse, division, is not possible for some integer numbers. For instance, we can multiply $3$ by $7$ getting $21$ and we can divide $21$ by $7$ getting $3$, but we cannot divide $21$ by $6$ within a set of integer numbers.

Rational numbers are completely new entities that allow us to divide any integer number by any other (not equal to $0$).
So, by definition, a rational number is a set of two integer numbers, the first, usually called numerator, and the second (not equal to $0$), usually called denominator, that has one important property: if multiplied by a denominator, result is a numerator.

Traditionally, if a numerator is $M$ and a denominator is $N$, the rational number formed by them is written as $\frac{M}{N}$ with property defined for it: $\frac{M}{N} \cdot N = M$.

Introduction of rational numbers completes the operation of multiplication by enabling its reverse in a broader set of numbers. Now we can divide $21$ by $6$ using rational numbers and the result, by definition of rational numbers , is $\frac{21}{6}$.

The harmony has been restored by expanding the concept of numbers from integer to rational.

Obviously, we have to prove the correctness of our definition, that certain properties of operations of addition and multiplications of integer numbers are preserved within a set of rational numbers, but this is a different topic.