LEARN MATH STEP BY STEP THROUGH VERY EASY PROCESS

SOME THEOREM IN IRRATIONAL NUMBER

**Some Theorem In The Irrational Number –**

**Theorem 1.) Prove that √2 is an irrational number.**

__Proof:-__ If possible, let √2 be rational.

**Lets its the simplest form be √2 = x/y, where ‘x’ & ‘y’ are integers having no common factor, other than 1 and y ≠ 0.**

** x**

**Then, √2 = ------**

** Y**

**Then, √2 y = x**

**Or, x² = (√2 y)²**

**Or, x² = 2y² …………………(1)**

**=> x² is even**

**=> x is even [ so, only squares of even numbers are even]**

**Let x = 2m for some integer m**

**Then, x = 2m => x² = (2m)² = 4m²**

** => 2y² = 4m²**

** => y² = 2m²**

** => y² is even**

** => y is even**

**Thus, ‘x’ is even and ‘y’ is even.**

**This shows that 2 is a common factor of ‘x’ and ‘y’.**

**This contradicts the hypothesis that ‘x’ and ‘y’ have no common factor, other than 1.**

**So, √2 is not rational and hence it is irrational. **

**Theorem 2.) Prove that √3 is irrational.**

**Proof.) If possible, let √3 be rational.**

**Let its simplest form be √3 = x / y, where ‘x’ & ‘y’ are integers, having no common factors, other than 1 and y ≠ 0**

**Then, √3 = x / y**

**Or, √3 y = x**

**Or, x² = (√3 y)²**

**Or, x² = 3y²…………………….(1)**

**=> x² is a multiple of 3**

**=> x is multiple of 3**

**Let, x = 3m for some positive integer m. **

**Then, x = 3m => x² = (3m)² = 9m²**

** => 3y² = 9m² [from equation (1), x² = 3y²]**

** => y² = 3m²**

** => y² is a multiple of 3**

** => y is a multiple of 3**

**Thus, x as well as y is a multiple of 3,**

**This shows that 3 is a common factor of x & y. this contradicts the hypothesis that x & y have no common factor, other than 1.**

**So, √3 is not a rational number and hence it is irrational.**

**Similarly, we can prove that each of the numbers √5, √6, √7, √8, √10, √11, √12,………., etc. is irrational**