Density

Density Of Rational in Rational Numbers

Given two rational numbers, you can always find a rational number between the two. This is easy to prove, just say $x,y$ are two different rational numbers then the rational number $\frac{x+y}{2}$ is between them

Density Of Rational In the Real Numbers

$Q$ (A Rational Number) is dense in $R$ for every 2 real numbers $a,b$ with $a<b$. There exists a rational number $r$ satisfying $a<r<b$

Implications

This means that you can have a Rational Number squeezed between two Irrational Number

Proof

• Density of Rationals in Reals Proof

Density Of Irrational in Real Numbers

$\forall a,b\in R,a<b⟹\exists t\notin Q$ s.t $a<t<b$