%% Approximate Value of π
% Approximate the value of \pi using a rational representation of the quantity pi.
% The mathematical quantity \pi is not a rational number, but the quantity pi that approximates it is a rational number since all floating-point numbers are rational.
% Find the rational representation of pi.

format rat
pi

% The resulting expression is a character vector. You also can use rats(pi) to get the same answer.
%
% Use rat to see the continued fractional expansion of pi.

R = rat(pi)

% The result is an approximation by continued fractional expansion. If you consider the first two terms of the expansion, you get the approximation 3 + \frac{1}{7} = \frac{22}{7}, which only agrees with pi to 2 decimals.
% However, if you consider all three terms printed by rat, you can recover the value 355/113, which agrees with pi to 6 decimals.
% 3 + \frac{1}{7 + \frac{1}{16}} = \frac{355}{113}
%
% Specify a tolerance for additional accuracy in the approximation.

R = rat(pi,1e-7)

% The resulting approximation, 104348/33215, agrees with pi to 9 decimals.
% Copyright 2012 The MathWorks, Inc.