Rational and Irrational Numbers

Rational and Irrational Number

Imagine you are standing at the origin (0,0) of a perfectly square, infinite forest.Trees are planted at every single whole-number coordinate intersection—for example, at (1,1), (2,1), (5,3), and so on.Every tree trunk is infinitely thin (like a literal mathematical point).

1. Looking at Rational Numbers (Visible Trees)If you turn your head and look out into the forest at a specific angle, your line of sight represents a slope.If you look along a slope that is a rational number (like 0.5, which is 1/2, your line of sight will eventually hit a tree (the tree at coordinate x=2, y=1).In fact, because the forest is infinite, you will hit an infinite number of trees perfectly lined up behind it (4,2), (6,3), etc.

2. Looking at Irrational Numbers (The Hidden Pathways)What if you look along a slope that is an irrational number,

 like square root 2(approximately 1.4142...) 

or π = approx 3.14...? 

Fractions used in everyday math—such as 22/7 are only rational approximations. 

Because an irrational number cannot be written as a fraction your line of sight will never hit a single tree, no matter how far you look into the infinite forest.You are looking down a "secret pathway" where you can see infinitely far without any tree blocking your view

In this analogy, rational numbers are the trees, and irrational numbers are the empty spaces between them. Even though the forest is packed with an infinite number of trees, there are actually infinitely more empty pathways (irrationals) than there are trees (rationals).

Compiled by 

Ms Naresh kuwar