02-14-2017, 03:03 PM
Useless trivia for the day...
Let \( n \) be in decimal representation. To find an (\( a + \sqrt{b} \))-approximation of \( n \), i.e. find integers \(a\) and \( b \) such that \( n = a+ \sqrt{b} \), we can use the following simple idea.
Since \( (n-a)^2 = b \), we simply check whether any of the values among
\[ n^2, (n-1)^2, (n-2)^2, \dotsm, (n-\lfloor n \rfloor)^2 \]
are integers.
Example:
X:=1+8^.5;
algn(X); // -----> { 1, 8 } representing \( 1 + \sqrt{8} \)
Let \( n \) be in decimal representation. To find an (\( a + \sqrt{b} \))-approximation of \( n \), i.e. find integers \(a\) and \( b \) such that \( n = a+ \sqrt{b} \), we can use the following simple idea.
Since \( (n-a)^2 = b \), we simply check whether any of the values among
\[ n^2, (n-1)^2, (n-2)^2, \dotsm, (n-\lfloor n \rfloor)^2 \]
are integers.
Code:
export algn(x)
begin
local fn:=ip(x);
local a, b, r;
local DIGIT:=9;
for a from 0 to fn do
b:=round((x-a)^2, DIGIT);
r:=b-ip(b);
if (abs(r) < 10^(-DIGIT+1)) then
return({a,b});
end;
end;
return({x,0});
end;Example:
X:=1+8^.5;
algn(X); // -----> { 1, 8 } representing \( 1 + \sqrt{8} \)