MaplePrimes - answers and comments on Question, Real Numbers vs. Complex Numbers

Real roots with complex expressions

Robert Israel — Thu, 21 Feb 2008 22:50:51 Z

Actually, the results of solve here are all real numbers, but their expressions in terms of radicals use complex numbers. This fact, discovered in the 16th century by Italian mathematicians (notably Cardano, Tartaglia and Bombelli), was actually the main reason for the development of complex numbers.

If you want real expressions for your real roots, you can get them in terms of trig functions using evalc:

> evalc([solve(x^3-6*x^2-7*x+58=0)]);

[2/3*57^(1/2)*sin(1/3*arctan(1/126*4701^(1/2))+1/6*Pi)+2, -1/3*57^(1/2)*sin(1/3*arctan(1/126*4701^(1/2))+1/6*Pi)+2-1/3*3^(1/2)*57^(1/2)*sin(-1/3*arctan(1/126*4701^(1/2))+1/3*Pi), -1/3*57^(1/2)*sin(1/3*arctan(1/126*4701^(1/2))+1/6*Pi)+2+1/3*3^(1/2)*57^(1/2)*sin(-1/3*arctan(1/126*4701^(1/2))+1/3*Pi)]

real complex root!

lemelinm — Fri, 22 Feb 2008 03:14:18 Z

This is very interesting. If you do solve(x^3-6*x^2-7*x+58 = 0);, you obtain square root of a complex number. I decided to take part by part the answer for the first root. Of course, since I do an evalf, I am sure that they will be an approximation, not the right answer. But as you can see, when I sum up part1+part2+2, I get a real number (with 0. as imaginary part).

What I am wondering is why Maple cannot simplify that fact immediately?

eq1 := (1/3)*(-378+(3*I)*sqrt(4701))^(1/3);

evalc(eq1);

part1 := evalf(%);

eq2 := 19/(-378+(3*I)*sqrt(4701))^(1/3);

evalc(eq2);

part2 := evalf(%);

tot := part1+part2+2;

Mario Lemelin

mario.lemelin@cgocable.ca

Maybe yes

lemelinm — Fri, 22 Feb 2008 07:45:20 Z

Yes I understsand your point. But at least it should be done if I ask to simplify symbolic. Instead I receive something even worst.

But if I do a plot in the region of the numerics solutions, I see that they will be 3 reals roots. Is there a way to have the exact solution after knowingthat in a inexpensive way?

plot(x^3-6*x^2-7*x+58, x = -4 .. 6);

Mario Lemelin

mario.lemelin@cgocable.ca

to simplify or not to simplify

acer — Fri, 22 Feb 2008 03:40:23 Z

If Maple always went to the expense of trying to do such simplifications then it would hurt the performance for anyone who didn't need or want it for a particular example.

How would one go about turning that simplification off?

And what if there were radicals in the coefficients of the cubic? It could become very expensive to try to determine whether the roots were purely real.

Would drawing the line at, say, rational coefficients appear inconsistent?

And what if one didn't want a result with trig in it, but wanted the explicit radicals only (when possible) from `solve`?

It may be difficult to justify an automatic and potentially expensive simplification which might not always suit everyone.

acer