Why rational number plus an irrational number is equal to an irrational number?

Answer 1

You can easily prove (doing some basic manipulations with fractions) that the addition of two rational numbers gives you a rational number. As a corollary, a subtraction of two rational numbers also results in a rational number.Now, assume an addition:

a + b = c

Also, assume that "a" and "c" are rational. Solving for "b":

b = c - a

Since "c" and "a" are rational, so is "c - a" - this is a contradiction to the assumption.

Answer 2

Let X + Y = Z where X is rational, Y is irrational, and suppose, if possible, that Z is rational.Then X - Z = -Y.

Now X and Z are rational.

Therefore X can be written, in simplest from as a/b where a and b are integers and b>0.

And Z can be written, in simplest from as c/d where c and d are integers and d>0.

Then X - Z = a/b - c/d = (ad - bc)/bd.

By the closure of integers over multiplication and addition, (ad - bc) and bd are integers and bd>0.

Therefore, X - Z is rational.

Therefore, LHS = X - Z is rational and RHS = -Y is irrational. This is a contradiction and consequently, X + Y cannot be rational.