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Proving Irrationality of NumbersCovid19 has led the world to go through a phenomenal transition . Elearning is the future today. Stay Home , Stay Safe and keep learning!!! In this section, we will discuss proving irrationality of numbers.We will prove √2, √3 and √2/√5 etc. are irrational numbers using Fundamental Theorem of Arithmetic.In proving irrationality of these numbers, we will use the result that if a prime p divides a ^{2} then it divides ‘a’ also. We will prove the irrationality of numbers by using the method of contradiction. Examples : 1) Prove that √2 is an irrational number. Solution : Let us assume that √2 is a rational number. So, √2 = a/b (where a and b are prime numbers with HCF = 1) Squaring both sides 2 = a ^{2} /b ^{2} ⇒ 2b ^{2} = a ^{2} ⇒ 2  a ^{2} ⇒ 2  a > (1) [by Theorem > Let p be a prime number. If p divides a ^{2} , then p divides a, where a is a positive integer] a = 2c for some integer c. Squaring both sides a ^{2} = 4c ^{2} 2b ^{2} = 4c ^{2} [ Since a ^{2} =2b ^{2} ] ⇒ b ^{2} = 2c ^{2} ⇒ 2  b > (2) ∴ From (1) and (2) we obtain that 2 is a common factor of ‘a’ and ‘b’. But this contradicts the fact. So our assumption of √2 is a rational number is wrong. Hence, √2 is an irrational number.  2) √5 + √3 is an irrational number. Solution : Let √5 + √3 is a rational number equal to a/b. √5 + √3 = a/b √5 = a/b  √3 Squaring both sides 5 = (a/b  √3) ^{2} ⇒ 5 = a ^{2} /b ^{2}  2a√3/b + 3 5 – 3 = a ^{2} /b ^{2}  2a√3/b 2 = a ^{2} /b ^{2}  2a√3/b 2a√3/b = a ^{2} /b ^{2}  2 2a√3/b = (a ^{2}  2 b ^{2} )/ b ^{2} 2a√3 = (a ^{2}  2 b ^{2} )b / b ^{2} √3 = (a ^{2}  2 b ^{2} )/2ab ⇒ √3 is a rational number which contradicts our assumption. So, √5 + √3 is an irrational number. Euclid's Geometry • Euclid Geometry • Euclids division lemma • Euclids division Algorithm • Fundamental Theorem of Arithmetic • Finding HCF LCM of positive integers • Proving Irrationality of Numbers • Decimal expansion of Rational numbers Home Page Covid19 has affected physical interactions between people. Don't let it affect your learning.
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