Question

prove that the sum of a ratinal number and an irrational number is always irrational

Answer 1

Example 1: let our rational number be 3 and irrational be√3
3+√3= 3+2.2360...=5.2360.... which is irrational.

Example 2: here our rational number is a mixed fraction I.e, 4 1/3 and irrational is π.
4 1/3+ π = 13/3+22/7 = 4.3333...+3.1415..=7.4748... which is irrational.
therefore, the sum of any rational and irrational number is always irrational.
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Answer 2

let a rational number be 3 and a irrational number be 1.73233233332....
3-1.732332333332....=2.732332333332....