# Elementary Number Theory and Its Application, 6th Edition by Kenneth H. Rosen Exercises 1.1.14

14. Show that $[2x] + [2y] >= [x] + [y] + [x + y]$ whenever x and y are real numbers.
Solution: We have to show that
$[2x]+[2y]>=[x]+[y]+[x+y]$
Where $x,y$ are real numbers
We write $x,y$ as
$x=n+\epsilon$
$y=m+\delta$
Where $n,m$ are integers and $\epsilon,\delta$ are non-negative real numbers less than 1.
There are two possibilities for the right hand side.
$n+m+(n+m)=2n+2m$
Or
$n+m+(n+m+1)=2n+2m+1$
If
$\epsilon+\delta>=1$
There are two possibilities for the left hand side.
$\lceil2x\rceil=\begin{cases} 2n & \epsilon<1/2\\ 2n+1 & otherwise \end{cases}$
And
$\lceil2y\rceil=\begin{cases} 2m &\delta<1/2\\ 2m+1 & otherwise \end{cases}$
We are trying to prove that the right hand side is always less than or equal to the left hand side. We construct a proof by contradiction and assume that right hand side is greater than left hand side. So the right hand side must be $2n+2m+1$ and the left hand side must be $2n+2m$
But then,
$\epsilon+\delta>=1$
So at least one of them must be equal to or greater than $1/2$
So the left hand side cannot equal $2n+2m$
Hence we arrive at a contradiction and end the proof.

## 17 Replies to “Elementary Number Theory and Its Application, 6th Edition by Kenneth H. Rosen Exercises 1.1.14”

1. Hi my friend! I want to say that this post is amazing, nice written and include approximately all important infos. I would like to see more posts like this.

2. Having read this I thought it was rather informative.

I appreciate you taking the time and energy to put this article together.

I once again find myself spending a significant amount of time both reading and leaving comments.
But so what, it was still worth it!

3. An impressive share! I have just forwarded this onto a co-worker who has been doing a little research on this.
And he actually ordered me dinner simply because I found it for him…
lol. So let me reword this…. Thanks for the meal!!
But yeah, thanks for spending some time to discuss this topic here on your
web site.

4. I’m not that much of a online reader to be honest but your blogs
really nice, keep it up! I’ll go ahead and bookmark your site
to come back down the road. Cheers

5. Thanks for finally writing about >Elementary Number Theory and Its Application, 6th Edition by Kenneth H.

Rosen Exercises 1.1.14 – Student Clan <Liked it!

6. Hello Dear, are you actually visiting this website on a regular basis, if so
after that you will absolutely take pleasant knowledge.

7. Wow that was unusual. I just wrote an extremely long comment
but after I clicked submit my comment didn’t appear.
Grrrr… well I’m not writing all that over again. Anyways, just wanted to say superb blog!

8. I regard something truly special in this web site .

9. Wow, fantastic blog layout! How long have you been blogging for? you made blogging look easy. The overall look of your website is excellent, let alone the content!

10. Having read this I believed it was rather enlightening. I appreciate you taking the time and energy to put this information together. I once again find myself spending a significant amount of time both reading and commenting. But so what, it was still worth it!

11. Hello i am kavin, its my first time to commenting anyplace, when i read this paragraph i thought i could
also create comment due to this sensible post.

12. But wanna tell that this is very useful , Thanks for taking your time to write this.

13. Thank you very much for the content. I wish you continued success.

14. Everything is very open with a clear description of the challenges.
It was definitely informative. Your site is very helpful.
Thanks for sharing!

15. Appreciate this post. Let me try it out.