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Shoelace Thm.
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Given a circle centered at the origin, how can one prove that the limit of the quotient of the number of lattice points on the circle over the radius goes to zero as radius goes to infinity?
Lattice points on a circle are points that have integer coordinates and lie on the circumference of a circle. The coordinates of these points can be represented as (x,y) where x and y are both integers.
The number of lattice points on a circle depends on the radius of the circle. The general formula for finding the number of lattice points on a circle with radius r is 4r + 1. For example, a circle with a radius of 3 will have 13 lattice points.
To plot lattice points on a circle, you can use the general formula (x,y) = (r cosθ, r sinθ) where r is the radius and θ is the angle in radians. You can also use a graphing calculator or software to plot the points.
Lattice points on a circle have various applications in geometry, number theory, and physics. For example, they can be used to determine the circumference and area of a circle, to represent solutions to mathematical equations, and to study the distribution of energy levels in a two-dimensional plane.
No, lattice points on a circle can only have positive integer coordinates. This is because the coordinates represent the number of units away from the center of the circle, and negative distances are not possible on a circle.