Question

​​prove that sum of rational and an irrational number is an irrational number

Answer 1

consider if a number such as begin mathsize 12px style square root of 2 space plus space 5 end style is rational or irrational. Guide the student to reason that (given begin mathsize 12px style square root of 2 end style is irrational and 5 is rational) begin mathsize 12px style square root of 2 space plus space 5 end style cannot be rational. If it is rational, then begin mathsize 12px style square root of 2 space plus space 5 end style is equal to some rational number x which means that square root of size 12px 2 size 12px equals size 12px x size 12px minus size 12px 5. But the rational numbers are closed for subtraction, so if x is rational, x - 5 is rational, which contradicts the fact that square root of size 12px 2 is irrational. Ask the student to use similar reasoning to explain why size 12px 8 square root of size 12px 3 is irrational. Then, ask the student to develop a general explanation for why the sum of a rational number and an irrational number must be irrational.