A lattice point on the xy-plane is any point (p,q) where both p and q are integers.
Suppose you have a line passing through the origin with irrational slopeExamples of lattice points are the points (1,2) (1,3) (1,4) (2,2) (2,3) (-2,3) (45,56) (10000, 10003) (-34,55). Examples of points which are not lattice points are (1/2, 3) (4,5/6) (√2, 5) (pi,4). Explain why such a line can never pass through a lattice point.
This is one of my math questions but i cant seem to figure it out?!
2007-10-22 06:32:19 · 3 answers · asked by i 1 in Science & Mathematics ➔ Mathematics
y= mx.........line passing through the origin
m = irrational no.
=> y = (irrational no.) x .........not possible
2007-10-22 06:41:14 · answer #1 · answered by harry m 6 · 0⤊ 0⤋
If the slope is irrational then it can not be expressed as a/b
Where a and b are real numbers. Since all lattice points are (a,b) where a and b are integers, the line cannot pass through a lattice point because the slope could then be expressed as a/b which cotradicts the definition of an irrational number.
2007-10-22 13:42:49 · answer #2 · answered by ironduke8159 7 · 0⤊ 0⤋
But that's not true!
y = ax passes through the origin, which is a lattice point.
But your line can't pass through any other lattice
point. For then you would have a = y/x.
Since y and x are integers, that would imply
a is rational.
2007-10-22 13:40:17 · answer #3 · answered by steiner1745 7 · 0⤊ 0⤋