Numbers: Rational Numbers

 Rational numbers as the name suggests are ratios of two numbers. They are derived from Integers and are defined as follows.

Let $p,q$ be two integers, i.e. $p,q \in \mathbb{Z}$, then $\frac{p}{q}$ where $q \neq 0$ is a rational number. 

in more advanced mathematical notation the rational numbers can be defined as follows

$\mathbb{Q}=\{\frac{p}{q}: p,q \in \mathbb{Z}, q \neq  0 \}$                                                           $(1)$

It can be shown that the set of rational numbers is closed with respect to 

• addition
• subtraction 
• multiplication
• division

Let us discuss the closure of rational numbers with respect to the above mentioned operations in detail. The aim here is to show some very basic mathematical proofs. But first a question arises. How do we add two rational numbers?

Let $\frac{a}{b},\frac{c}{d}$  be two rational numbers, then we can add them as follows

$\frac{ad+bc}{bd} \Longleftrightarrow \frac{a}{b} +\frac{c}{d}$                                                  $(2)$

 From equation $1$ it can be seen that $a,b,c,d$ are all integers. Thus, from the closure of integers under multiplication and addition it can be shown that $ad, bc, bd$ are all integers. Following directly from this $(ad + bc)$ is also an integer. Now as $ad+bc =g$ is an integer and so is $bd=f$ , we can write equation $2$ as

$\frac{ad+bc}{bd} = \frac{g}{f}$               where    $g,f \in \mathbb{Z}$                                       $(3)$

From equation $1$, $\frac{g}{f}$ or equivalently $\frac{ad+bc}{bd}$ is a rational number. Hence, set $Q$ is closed under addition.

Using the same arguments we can show that the rational numbers are also closed under subtraction. 

Let $\frac{a}{b},\frac{c}{d}$ be two rational numbers then

$\frac{a}{b}-\frac{c}{d} \Longleftrightarrow \frac{ad-bc}{bd}$                                                     $(4)$

From the closure of integers under multiplication $ad,bc$ and $bd$ are all integers. Thus, similar to equation $3$ we can now write

$\frac{ad-bc}{bd}=\frac{i}{j}$                            where   $i,j \in  \mathbb{Z}$                                $(5)$

Hence, from equation $1$ it can be shown that $\frac{i}{j}$ and equivalently $\frac{ad-bc}{bd}$ is a rational number. Thus, set $\mathbb{Q}$ is closed under subtraction.

 Let us now move on to an other question. How can we multiply and divide rational numbers?

Let $\frac{a}{b},\frac{c}{d}$  be two rational numbers, then 

$\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$                                                                             $(6)$

By the closure of integers under multiplication $ac,bd$ are integers hence,

$\frac{ac}{bd}=\frac{k}{l}$                                  where  $k,l \in  \mathbb{Z}$                              $(7)$

Hence, from equaiton $1$, $\frac{k}{l}$ and equivalently $\frac{ac}{bd}$ is a rational number. Thus, rational numbers are closed under multiplication.

We know from basic arithmetic that $a \div b =\frac{a}{b}$ and that is equivalent to $a  \times \frac{1}{b}$. Using this we can infer that if $\frac{a}{b},\frac{c}{d}$ are two rational numbers then 

$\frac{a}{b} \div \frac{c}{d} \Longleftrightarrow \frac{a}{b} \times \frac{c}{d} =  \frac{ad}{bc}$                                                                $(8)$ 

From the closure of integers under multiplication both $ad$ and $bc$ are integers hence,

$\frac{ad}{bc} = \frac{m}{n}$                            where  $m,n \in \mathbb{Z}$                               $(9)$

Thus, from equation $1$ it can be concluded that $\frac{m}{n}$ and equivalently $\frac{ad}{bc}$ are rational numbers. Hence, we can state that the rational numbers are closed under division.

Physical Applications of Rational Numbers

There may not be a direct relevance between the rational numbers and the physical universe under classical physics, but in these play a crucial role in quantum physics. The quantum probabilities of states are rational numbers. These are advanced topics which we may discuss later on. 

As we have discussed earlier the arithmetic operations namely addition, subtraction, multiplication and division have analogues in physical geometry.

Addition and subtraction have applications in arithmetic manipulation of lengths and distances

Multiplication is analogous to calculation of areas

Division is analogous to calculating shares of a whole.

The utility of rational numbers is that they allow the representation of quantities as a large as required and as small as desired. It should be noted that large quantities could have been represented using natural numbers. Rational numbers allow us the freedom of representing quantities between two adjacent natural number $a$ and $a+1$.

Moreover, the closureo f rational numbers under the operations of addition, subtraction, multiplication and division build a very strong case that rational numbers are sufficient for the basic geometrical requirements.