Straight lines with rational gradient
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Wondering if it makes sense to ask:
In $(x,y)$ plane what probability has an arbitrary straight line $y=mx+c$ to have a rational slope $m$? I.e., referencing to an infinitely large sheet of standard grid lines (with given origin, coordinate axes horizontal and vertical) made up of unit squares, what percentage of lines pass through rational points?
If a random straight line is drawn through a fixed point of rational coordinates then many near misses from other grid points can be observed.
probability
asked Nov 14 at 20:33
Narasimham
20.4k52158
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0
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favorite
Wondering if it makes sense to ask:
In $(x,y)$ plane what probability has an arbitrary straight line $y=mx+c$ to have a rational slope $m$? I.e., referencing to an infinitely large sheet of standard grid lines (with given origin, coordinate axes horizontal and vertical) made up of unit squares, what percentage of lines pass through rational points?
If a random straight line is drawn through a fixed point of rational coordinates then many near misses from other grid points can be observed.
probability
asked Nov 14 at 20:33
Narasimham
20.4k52158
It's actually not sufficient for $m$ to be rational to have the line pass through rational points. Regardless of whether $m$ is rational, you'll need either $c$ to also be rational or you will require $frac{c}{m}$ to be rational. Interestingly, if $m$ is irrational you can still pass through rational points provided that $c$ is a rational multiple of $m$.
– Michael Stachowsky
Nov 14 at 21:50
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Wondering if it makes sense to ask:
In $(x,y)$ plane what probability has an arbitrary straight line $y=mx+c$ to have a rational slope $m$? I.e., referencing to an infinitely large sheet of standard grid lines (with given origin, coordinate axes horizontal and vertical) made up of unit squares, what percentage of lines pass through rational points?
If a random straight line is drawn through a fixed point of rational coordinates then many near misses from other grid points can be observed.
probability
asked Nov 14 at 20:33
Narasimham
20.4k52158
Wondering if it makes sense to ask:
In $(x,y)$ plane what probability has an arbitrary straight line $y=mx+c$ to have a rational slope $m$? I.e., referencing to an infinitely large sheet of standard grid lines (with given origin, coordinate axes horizontal and vertical) made up of unit squares, what percentage of lines pass through rational points?
If a random straight line is drawn through a fixed point of rational coordinates then many near misses from other grid points can be observed.
probability
probability
asked Nov 14 at 20:33
Narasimham
20.4k52158
asked Nov 14 at 20:33
Narasimham
20.4k52158
edited Nov 14 at 21:44
asked Nov 14 at 20:33
Narasimham
20.4k52158
asked Nov 14 at 20:33
Narasimham
20.4k52158
asked Nov 14 at 20:33
Narasimham
20.4k52158
20.4k52158
It's actually not sufficient for $m$ to be rational to have the line pass through rational points. Regardless of whether $m$ is rational, you'll need either $c$ to also be rational or you will require $frac{c}{m}$ to be rational. Interestingly, if $m$ is irrational you can still pass through rational points provided that $c$ is a rational multiple of $m$.
– Michael Stachowsky
Nov 14 at 21:50
add a comment |
It's actually not sufficient for $m$ to be rational to have the line pass through rational points. Regardless of whether $m$ is rational, you'll need either $c$ to also be rational or you will require $frac{c}{m}$ to be rational. Interestingly, if $m$ is irrational you can still pass through rational points provided that $c$ is a rational multiple of $m$.
– Michael Stachowsky
Nov 14 at 21:50
It's actually not sufficient for $m$ to be rational to have the line pass through rational points. Regardless of whether $m$ is rational, you'll need either $c$ to also be rational or you will require $frac{c}{m}$ to be rational. Interestingly, if $m$ is irrational you can still pass through rational points provided that $c$ is a rational multiple of $m$.
– Michael Stachowsky
Nov 14 at 21:50
It's actually not sufficient for $m$ to be rational to have the line pass through rational points. Regardless of whether $m$ is rational, you'll need either $c$ to also be rational or you will require $frac{c}{m}$ to be rational. Interestingly, if $m$ is irrational you can still pass through rational points provided that $c$ is a rational multiple of $m$.
– Michael Stachowsky
Nov 14 at 21:50
add a comment |
1 Answer
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The answer to the question is unsatisfying: 0. The percentage of rational numbers in any given interval is zero, as seen in Percentage of rational numbers on an interval
answered Nov 14 at 20:52
Michael Stachowsky
1,208417
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
The answer to the question is unsatisfying: 0. The percentage of rational numbers in any given interval is zero, as seen in Percentage of rational numbers on an interval
answered Nov 14 at 20:52
Michael Stachowsky
1,208417
add a comment |
up vote
0
down vote
The answer to the question is unsatisfying: 0. The percentage of rational numbers in any given interval is zero, as seen in Percentage of rational numbers on an interval
answered Nov 14 at 20:52
Michael Stachowsky
1,208417
add a comment |
up vote
0
down vote
up vote
0
down vote
The answer to the question is unsatisfying: 0. The percentage of rational numbers in any given interval is zero, as seen in Percentage of rational numbers on an interval
answered Nov 14 at 20:52
Michael Stachowsky
1,208417
The answer to the question is unsatisfying: 0. The percentage of rational numbers in any given interval is zero, as seen in Percentage of rational numbers on an interval
answered Nov 14 at 20:52
Michael Stachowsky
1,208417
answered Nov 14 at 20:52
Michael Stachowsky
1,208417
answered Nov 14 at 20:52
Michael Stachowsky
1,208417
answered Nov 14 at 20:52
Michael Stachowsky
1,208417
1,208417
add a comment |
add a comment |
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It's actually not sufficient for $m$ to be rational to have the line pass through rational points. Regardless of whether $m$ is rational, you'll need either $c$ to also be rational or you will require $frac{c}{m}$ to be rational. Interestingly, if $m$ is irrational you can still pass through rational points provided that $c$ is a rational multiple of $m$.
– Michael Stachowsky
Nov 14 at 21:50