Fooling around with complex numbers! A complex number has two parts: a real part and an imaginery part. Complex numbers allow us to obtain soulutions for systems of equations, matrice and polynomials when we encounter taking the square root of a negative number. For example the following equations do not have a conventional solution, but they do have a complex solution.

> factor(x^2+6.0*x +13);

x^2+6.000000000*x+13.00000000

> solve(x^2+6.0*x +13);

-3.-2.*I   -3.+2.*I

> plot(x^2+6.0*x +13,x=-10..10);

Note that the plot of the equation does not cross the X-axis at any point; howeverr the function has a minimum around x=-3.0, which is the closest it ever gets to the x axis. Let us try another example.

> solve(x^4-6*x^3-22*x^2+56*x-64.0,x);

-4.   8.   1.-1.*I   1.+1.*I

> plot(x^4-6*x^3-22*x^2+56*x-64.0,x=-8..10);

Note that the function has two real solutions at (x=-4, x=8) and two imaginery solutions.

> plot(-x^2+6.0*x -13,x=-10..10);

> solve(-x^2+6.0*x -13,x);

3.-2.*I   3.+2.*I

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