Question

x - rational number, y - not rational number, How to prove (formal) that x + y - not rational number

Answer 1

are you saying x is irrational and y is irrational
therefore x + y is irrational?

you cannot prove this because it is not true

Answer 2

for example
\[\sqrt{2}\] is irrational, as is
\[3-\sqrt{2}\] but their sum is
\[\sqrt{2}+3-\sqrt{2}=3\]which is certainly rational

Answer 3

But x is rational

Answer 4

ooooooooh sorry i read incorrectly. sum of rational and irrational is irrational,

Answer 5

I think i should show that x+y cannot be presented as a / b (where a,b - integers and b is not 0)

Answer 6

ok so try this. if x is rational and y is irrational, let us assume for a moment that
\[x+y\] is rational and derive a contradiction. since rational numbers are closed under subtraction , that means
\[x+y-x=y \] is rational, but you are assuming that y is not rational, so you have a contradiction.

Answer 7

a is a rational number
\[a = \frac{m}{n}\]

b is a irrational number

Suppose that a + b is rational
\[\frac{m}{n} +b = \frac{o}{p}\]

\[b = \frac{o}{p} -\frac{m}{n}\]

Since a rational number minus a rational number must equal to a rational number.

Therefore a + b must be irrational

Answer 8

Thanks a lot, moneybird and satelite73 ! I think that moneybird answer is better for me since it's based on a/b definition of rational numbers.