Question

"The sum of a rational and an irrational number is irrational."

How could I go about proving or disproving such a statement?

Answer 1

Use an indirect argument. Let x be a rational number and y an irrational number. Suppose now that x + y is rational. Show that this leads to a contradiction.

Answer 2

will do

Answer 3

This is one of the nicest examples of an indirect argument.

Answer 4

Proof:

Let \(x\) be rational, that is, \(x=a/b\) where \(a,b\in\mathbb{Z}\) and \(b\neq0\). Suppose that \(x+y\) is rational, that is, \(x+y=c/d\) where \(c,d\in\mathbb{Z}\) and \(d\neq0\). It follows that\[x+y=\frac{a}{b}+y=\frac{c}{d}\]\[\implies y=\frac{c}{d}-\frac{a}{b}\]\[\implies y=\frac{cb-ad}{db}.\]Since \(cb-ad,db\in\mathbb{Z}\), this leads to a contradiction since it implies that \(y\) is rational. \(\blacksquare\)

Is this okay?

Answer 5

Right. I'd add one more sentence at the end to make the logic very explicit.

"Hence it cannot be the case that x+y is rational and therefore x+y is irrational"

Answer 6

perfect, :)))) i just noticed that i also forgot to include a line at the beginning saying that y is irrational .-.

Answer 7

yes, I should have seen that.

Answer 8

i'm going to go off and use this same logic to answer the next three problems or so. hopefully i won't encounter anything tricky)))))

Answer 9

ok ... good luck.