Prove or disprove: the sum of a rational number and an irrational number is an irrational number.

Answer 1

By contradiction, suppose the sum of a rational and an irrational is rational.

Then for rational x and irrational y, x+ y = z where x=a/b and z = c/d for integers a,b,c,d

a/b + y = c/d

y = c/d - a/b = (bc - ad)/bd

by closure property of multiplication and subtraction bc-ad and bd are integer

therefore y is the quotient of two integers which makes is rational, a DIRECT CONTRADICTION