The rational numbers

Rational numbers

Rational numbers

Defining rational numbers

We previously defined integers in terms of natural numbers. Similarly we can define rational numbers in terms of integers.

\(\forall ab \in \mathbb{I} (\neg (b=0)\rightarrow \exists c (b.c=a))\)

A rational is an ordered pair of integers.


So that:


Converting integers to rational numbers

Integers can be shown as rational numbers using:


Integers can then be turned into rational numbers:





Equivalence classes of rationals

There are an infinite number of ways to write any rational number, as with integers. \(\dfrac{1}{2}\) can be written as \(\dfrac{1}{2}\), \(\dfrac{-2}{-4}\) etc.

The class of these terms form an equivalence class.

We can show these are equal:





Ordering of rationals

Functions of rational numbers

Rational addition

Then we can define addition as:





Rational subtraction




Rational multiplication

Similarly, multiplication can be defined as:

\((a,b).(c,d)=(a.c, b.d)\)




Rational division




Cardinality of the rationals

Cardinality of rational numbers

We can see rational numbers as cartesian products of integers. That is:


We can order the rational numbers like so:


These can be mapped from natural numbers, so there is a bijunctive function.


\(|\mathbb{Q} |=|\mathbb{Z}.\mathbb{Z} |=|\mathbb{N} |=\aleph_0\)

As: \(|\mathbb{Z}.\mathbb{Z} |=|\mathbb{Z}|^2\)


Fraction rules





b Scaler addition


Scaler multiplication





Partial fraction decomposition

We have: \(\dfrac{1}{A.B}\)

We want this in the form of:


First, lets define \(M\) as the mean of these two numbers, and define \(\delta=M-B\). Then:


We can rearrange the latter two to find:


Now we need to find values of \(a\) and \(b\) to choose.

Let’s examine \(a\).


\(a=-\dfrac{bM+b\delta -1}{M-\delta}\)

\(a=-\dfrac{bM+b\delta -1}{M-\delta}\)

For this to divide neatly we need both the numerator to be a constant multiplier of the denominator. This means the ratio the multiplier for the left hand side of the denominator is equal to the right:

\(\dfrac{bM}{M}=\dfrac{b\delta -1}{-\delta}\)

\(b=\dfrac{b\delta -1}{-\delta}\)


We can do the same for \(a\).


We can plug these back into our original formula:



Density of the rationals

Rationals are dense in rationals

For any pair of rationals, there is another rational between them:



Where \(b>a\).

We define a new rational:



This is a rational number.

We can write:



As \(b>a\) we know \(2qm>2pn\)

So: \(a < c < b\)