% By default, Matlab and Octave display only a limited
% number of digits. To display approximately 16 decimal
% digits, use 'format long'. To return to the default
% display format, use 'format'.

format long

% Functions in Matlab and Octave are called directly by
% their names; no package names or namespaces are required.

% Mathematically, these three expressions are equal:
%
%   (5 + 2*sqrt(7))(5 - 2*sqrt(7))
%       = 5^2 - (2*sqrt(7))^2
%       = 25 - 28
%       = -3.
%
% Numerically, however, the first result may differ slightly
% from -3 because sqrt(7) cannot be represented exactly as a
% floating-point number.

(5 + 2*sqrt(7))*(5 - 2*sqrt(7))
5*5 - 2*2*7
5^2 - 2^2*7       % ^ denotes exponentiation


printf( '\nRoots of a polynomial with real roots:\n' )

% Find the roots of the polynomial
%
%        2
%    3.8x  + 4.2x - 1.7.
%
% For
%
%       2
%     ax + bx + c = 0,
%
% the quadratic formula gives
%
%              2
%        -b +/- sqrt(b - 4ac)
%   x = ---------------------.
%                 2a

a =  3.8
b =  4.2
c = -1.7

% Be sure to put parentheses around the denominator.
% For example,
%
%     (-b - sqrt(b^2 - 4*a*c))/2*a
%
% is interpreted as
%
%     ((-b - sqrt(b^2 - 4*a*c))/2)*a,
%
% which is not the quadratic formula.

(-b - sqrt(b^2 - 4*a*c))/(2*a)
(-b + sqrt(b^2 - 4*a*c))/(2*a)


printf( '\nArithmetic with the imaginary unit:\n' )

% Matlab and Octave use both 'i' and 'j' for the imaginary
% unit. However, either name can be reassigned as a variable.
% Writing 1j explicitly represents the imaginary unit even
% if the variable j has been assigned another value.
%
% Parentheses are required when raising 1j to a power:
% write (1j)^5 rather than 1j^5.

(1j)^5
(5 + 2j)*(5 - 2j)
5*5 - 2*2*(1j)*(1j)
5^2 - (2j)^2


printf( '\nRoots of a polynomial with complex roots:\n' )

% Now find the roots of the polynomial
%
%        2
%    3.8x  + 4.2x + 1.7.
%
% Here the discriminant is negative, so the two roots
% are a complex-conjugate pair.

c = 1.7

(-b - sqrt(b^2 - 4*a*c))/(2*a)
(-b + sqrt(b^2 - 4*a*c))/(2*a)