Help with understanding a floor proof

  • Thread starter Temp0
  • Start date
  • #1
79
0

Homework Statement


Prove that for any real number x, if x - floor(x) < 1/2, then floor(2x) = 2 floor(x)

Homework Equations




The Attempt at a Solution


Assuming that x is a real number. Suppose that x - floor(x) < 1/2
Multiplying both sides by 2, 2x < 2 floor(x) + 1
from the definition, 2 floor(x) <= 2x
2 floor(x) is an integer, and 2 floor(x) <= 2x < 2 floor(x) + 1
Implying floor(2x) = 2 floor(x)

My question is, how does the proof imply the result? I don't see how the logic works. Thank you for any help you can provide in advance.
 

Answers and Replies

  • #2
RUber
Homework Helper
1,687
344
2 floor(x) is an integer, and 2 floor(x) <= 2x < 2 floor(x) + 1
Implying floor(2x) = 2 floor(x)

My question is, how does the proof imply the result? I don't see how the logic works. Thank you for any help you can provide in advance.

The key is that floor(x) and floor(2x) are both integers and the relation you have above is 2floor(x) < 2 floor(x) + 1. You seem to be missing the part where 2floor(x) <= floor(2x).
 
  • #3
79
0
I see... I guess I missed that part, haha, just wondering, does it come from the inequality:

floor (2x) <= 2x < floor(2x) + 1 ?

I can see how it relates... kind of, since

2 floor (x) <= 2x as well, but how does

2floor(x) <= floor (2x) ?
 
  • #4
RUber
Homework Helper
1,687
344
2 floor(x) is always less than or equal to floor(2x).

If you let ## x = x_{int} + x_{frac} ## where ##x_{int}\in \mathbb{Z}, 0 \leq x_{frac}<1##
Then ##\text{floor}(x) = x_{int}=x-x_{frac}##
## 2\text{floor}(x) = 2x_{int}=2x-2x_{frac}##
But floor(2x) has 2 cases:
case 1:
##x_{frac}<.5 \implies 2x_{frac} < 1 \implies floor(2x) = 2 x_{int} = 2floor(x) ##
case 2:
##.5 \leq x_{frac}<1 \implies 2x_{frac} \geq 1 \implies floor(2x) = 2 x_{int} + 1=2floor(x) + 1 ##
In both cases, floor(2x) ##\geq ## 2floor(x) .
 
  • Like
Likes Greg Bernhardt

Related Threads on Help with understanding a floor proof

  • Last Post
Replies
11
Views
2K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
5
Views
7K
Replies
2
Views
5K
  • Last Post
Replies
5
Views
887
  • Last Post
Replies
8
Views
1K
  • Last Post
Replies
10
Views
2K
  • Last Post
Replies
15
Views
3K
  • Last Post
Replies
18
Views
968
  • Last Post
Replies
2
Views
2K
Top