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Table of contents

I. Definitions

Algebraic number
Irrational number
Rational number
Transcendental number

II. Examples
Proof that the square root of 2 is irrational

I. Definitions

Algebraic number

An algebraic number is the root of a non-zero polynomial with integer (or equivalently rational) coefficients. Numbers that are not algebraic are called transcendental numbers.

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Irrational number

An irrational number is a number that is not rational.

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Rational number

A rational number is a number that can be expressed as a fraction p/q of two integers p and q such that q is not zero.

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Transcendental number

A number is transcendental if it is not the root of a polynomial with integer or rational coefficients, which is to say that it is not algebraic. Transcendental numbers of course are irrational.

Keywords: transcendental, number, algebraic, root, polynomial, irrational

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II. Examples

Proof that the square root of 2 is irrational

Let's assume that is rational

There exists two strictly positive relatively prime integers m and n such that

= m/n (1)

By taking the square of each member, it follows that

2 = m^2/n^2 (2)

Therefore

2 * n^2 = m^2 (3)

2 divides m^2, since 2 is a prime number

2 divides m (4)

2 divides m implies that

4 divides m^2 (5)

Using (3) and (5), we deduce that

2 divides n^2 (6)

2 divides n^2, since 2 is a prime number

2 divides n (7)

2 divides both n and m. Our hypothesis was that n and m are relatively prime. This contradiction proves that our initial hypothesis that is rational is incorrect.

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