Hello Everyone. Before going to the proof, let us go into the basics of Rational and Irrational numbers. Rational Number : A number that can be expressed as the fraction p/q of two integers p and q, where q not equal to zero. Irrational Number : A number that cannot be expressed as the fraction p/q of two integers p and q, where q not equal to zero. What is required to prove : Sum of a rational number and an irrational number will be an irrational number. Proof : Let "m" be an irrational number. --------- (1) Let "n" be a rational number. ------------(2) Let us think that we don't know whether m + n is rational or irrational number. So, let us assume m + n as a rational number initially. Then, m + n = p/q where p and q are some integers and q is not equal to zero. (According to the definition) So, m + n = p/q Subtract "n" on both sides. Then, m = (p/q) - n = difference of two rational numbers Difference of two rational numbers will be a...
Comments
Post a Comment