An Irrational Number is a real number that cannot be written as a simple fraction or we can say Irrational means not Rational For Example: π (pi) is an Irrational number which has the value 3.14 Now we will come to the Square ROOTS:- A square root of a number is a value that can be multiplied by itself to give the original number. For Example: A square root of 4 is 16, because when 4 is multiplied by itself we get 16. = 16 here 2 is the power of 4. Now, Prove That Square Root Of 2 Is Irrational:- = Irrational number. Yes, The square root of 2 is Irrational. Let us see how!! First we Square a Rational Number, If the rational number is a/b, then it will become a2/b2 when squared. For Example: ( 3/4)2 = 32/42 Now we will see that in this the exponent is 2 which is an even number. But to do this we should need to break the Numbers down into their prime factors .
Example: (3/4)2 = (3/(2×2) )2 = 32/24 We notice that still the exponents are Even Numbers. The 3 has an exponent of 2 in (32) and the 2 has an exponent of 4 in (24). Now one thing becomes obvious that Every exponent is an even number!! So we can see that when we square a rational number, the result will be made up of Prime Numbers whose exponents are all even numbers. Now, When we square a rational number, each prime factor has an even exponent. Now, let just look at the number 2 As a fraction, 2 = 2/1 Which is 21/11, and that has odd exponents!! We can write 1 as 12 (so it has an even exponent), and then we have: 2 = 21/12. Now, it can be simpler to solve it to 21, but in other way: here it is an odd exponent, We could even try things like 2 = 4/2 = 22/21, but we still cannot get rid of an odd exponent. So it could not have been made by squaring a rational number! This means that the value that was squared to make 2 (i.e. the square root of 2) cannot be a rational number. Hence we can say that the square root of 2 is Irrational.