## I have decided!

I have decided to restart my Calculus learning. Reference book is ‘Introduction to Calculus and Analysis’ by Richard Courant and Fritz John.

## Introduction

### How to represent rational numbers

For better understand numbers, we need to represent the rational numbers at first.

Imagine a number axis L, with a unit length 1, we can rule the positive direction and negative direction.

And then we need to represent the rational numbers. Arbitrarily set a point ‘P’. Segment arbitrary unit, like +- 1, +-2 into smaller pieces, and one of the pieces is 1/p. As you can see in the Figure 2 below.

In this way, we can find the every point rational point p/q, the P is on the p/q or between two rational point p/q and (p+1)/q. In the other word, we find any rational point p/q on this number axis L.

We call this two quantities whose ratio is a rational number **Commensurable Number**. Because it can be expressed as integral multiples of a common unit. And there is **Incommensurable Number** too. But why?

### What is irrational number?

Imagine there is triangle with (1^2 + 1^2 = 2)