Question

Rigorously prove that a finite sum of rationals cannot evaluate to an irrational.

Answer 1

Oh set theory... I used to know this proof off the back of my hand, gimme a min to remember it--or forget it, you never know.

Answer 2

actually, this is not true.

Answer 3

Alright, then, the question becomes rigorously prove that this isn't true. XD I guess by contrapositive assume.

Answer 4

i need to remember how you do that whole, series adding to different sums thing, lol

Answer 5

there is some sum I learned a while back that naturally sums to e, but if you add anything to it it becomes any other number you want, bleh

Answer 6

Well, because we have Euler's relationship, and we have the infinite sum of 1/n^2 converging at pi something, it shouldn't be too hard to evaluate the famous irrationals.

No, the issue is we need to prove, for all cases, that either a finite sum of rationals can evaluate to an irrational, or not.

Answer 7

oh, wait you said a finite sum, that was an infinite sum... i revert back to saying it's true

Answer 8

Yeah, I can't think of anything but a limit creating an irrational.

Answer 9

well, since all rational numbers can be expressed as a fraction a/b... and irrationals cannot be expressed as a/b thats probably where the proof lies.

Answer 10

a,b are integers.

Answer 11

Let x,y ∈ ℚ. Then x = a/b and y = c/d for a,b,c,d ∈ ℤ.

Then x + y = (a/b) + (c/d) = (ad + bc) / (bd) = p/q

for some p,q ∈ ℤ. Therefore (x + y) ∈ ℚ, i.e. the rationals are closed under addition.

Answer 12

Makes sense. I suppose I was hoping for a roundabout proof so that it could help me with a previous question. XD

The question was, "Rigorously demonstrate the conditions that must be satisfied such that the limit of an infinite sum of rationals converge to an irrational."

Which I had no idea how to start, so I decided to start by proving everything around it.

Answer 13

these are some really annoying questions. is this for like an algebraic structures or a proofs class?

Answer 14

Something like that. Informal challenges given out by the department once in a while.

Answer 15

Don't worry; I'm not cheating. XD

Answer 16

even worse.

Answer 17

this is bringing back bad memories of weeks spent in the library doing my undergrad research :P

Answer 18

so, as i mentioned, rationals must be able to satisfy a/b in order to become irrational this must no longer be able to happen. I'm sure the proof lies in an annoying limit dealing with this.

Answer 19

Bleaargghh. My school department is so helpful, giving out problems like this, but not the solutions.