Question about point on a line?

[cragar]

I was reading the book one, two , three ... infinity . And he says that their are the same amount of points on a line as their are in a square. A line from 0 to 1 and then a 1x1 square.
He says when can represent any point in that square with 2 numbers like coordinates. And if we add these numbers together then we can draw a one to one correspondence between the numbers in the square and the line. But then couldn't I just add all the reals from 0 to 1 and then 1 to 2 , and then put these into a one to one correspondence with the natural numbers.
Or does this not work because they are uncountably infinite.

 

[HallsofIvy]

That is precisely the definition of "uncountably infinite"- that it cannot be put in one-to-one correspondence with the natural numbers. It can be shown that the rational numbers, between any two given numbers, are "countably infinite" (can be put in one-to-one correspondence with the natural numbers) but that the real numbers, between any two given number, and so the set of all points in an interval of the number line, is not.

 

[cragar]

so as long as what I am combining is countable then I can draw a one to one correspondence. But it kinda seems to me that in the book he is drawing a one to one correspondence between the set that has two numbers in it and a point. Not necessarily a one to one correspondence between the points themselves.

 

[HallsofIvy]

Two numbers will, of course, determine a point in the plane and so there is a one-to-one correspondence between pairs of numbers and points in the plane.

 

[cragar]

ok I can see that , Thanks for your response.