Question

Suppose that each point (x, y) in the plane, both of whose coordinates are rational numbers, represents a tree. If you are standing at the point (0, 0), how far could you see in this forest?

Answer 1

wow <.< where are you getting these from? o.O

Answer 2

my textbook. I was looking through it and I found this whole problem solving section.

Answer 3

I wouldn't think you could see anywhere

Answer 4

because the rational numbers are dense right?

Answer 5

er..."dense", idk, i read it in an analysis book somewhere lol

Answer 6

something like "in between any two real numbers there is always a rational number"

Answer 7

say your line of sight is along the function f(x)=ax
then for e>0

the interval (0,e) will contain an irrational number..and thus a tree will be in your immediate line of sight

Answer 8

for all a

Answer 9

oops looking at it backwards

Answer 10

Joe? & Zarkon
http://openstudy.com/groups/mathematics#/groups/mathematics/updates/4e3231b90b8ba7b2da415e84

Answer 11

the argument still works though, just for rationals

Answer 12

replace irrational with rational

Answer 13

for some reason i thought a tree was at irrational points ;)

Answer 14

now I think it might not work...jas...let me think

Answer 15

what about the line of sight ...

\[f(x)=\pi x\]

Answer 16

x and y cant both be rational at the same time

Answer 17

so you can see as far as you want in that direction

Answer 18

The way i had started thing about the problem was that last statement i made, "Between any real numbers there is a rational number." If i have 2 trees some distance e apart, i would try to look in between them. But there would have to be another tree in between them. So i try to look in between those trees, but there would also be a tree in between them. So on and so forth~

Answer 19

but for any pair of points that satisfy the line of sight \[y=\pi x\]

there are no rational pairs...so nothing is blocking your sight in that direction

Answer 20

i take back my statement, i agree with yours Zarkon.

Answer 21

cute problem

Answer 22

i'm not sure that I fully grasp the problem...

Answer 23

any help?

Answer 24

can you be more specific?

Answer 25

well I thought that the coordinates of (x, y) are both rational numbers, but then you said that there are no rational pairs... anyways that confused me

Answer 26

Can you help me PLEASE!

Answer 27

well we put a tree for any pair (x,y) that are rational

but, using the line of sight from above \[y=\pi x\]
we get the points \[(x,\pi x)\]

is this case it is impossible for both the x and y coordinate to rational

if x is irrational the there is not tree at (x,y)
if x is rational then y is irrational and again there is no tree at (x,y)

does that help?

Answer 28

well I understand that, but where did you get the \[y = \pi x\]?

Answer 29

it just came to mind. it was a function that gave me what i needed.

I could have used others

Answer 30

let a be any irrational number

then the line of sight y=ax has no trees

by the same argument above

Answer 31

I use ax since it is a line that goes through the point (0,0)...which is the point we are standing at

Answer 32

oh ok thank you so much for the help! :)