Copy of an email sent to nic villar (currently xmasing in south america lucky ########) by db






Hi Nic
I got to thinking that

To show that any addition to the circumference of a circle will give the same overall difference in radius for all radius greater than 1unit measure.
i.e. 1 metre added to the circumference of a 3m rope will make the size of its described circle differ in radius by the same amount as 1 metre added to a large circular star.

c(1) = n
c(2) = n + m
where c is circumference and n , m ><1 ( dont have greater than or equal to, greater than less than 1 means the same)
we need radius (r) in terms if n and m

c(1);
2(pi)r = n
so r = n / 2(pi)
c(2);
2(pi)r' = n + m
so r' = (n+m)/2(pi) or n/2(pi) + m/2(pi)
now since r = n/2(pi) this gives
r' = r + m/2(pi) i.e. the radius r' of the larger circle is the radius of the first circle r plus m/(2pi), the addition of m/2(pi) is clearly independant of r so will be the same
for all r ><1 as required

happy christmas proof for you

POST SCRIPT On reflection it works for all real values and also all imaginary numbers

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