- #1
daniel.e2718
- 10
- 0
I couldn't really think of a good title for this question, lol.
Is it possible that a real number exists that can only be expressed in exact form when that form must includes complex numbers?
For example, the equation
[itex]2 \, x^{3} - 6 \, x^{2} + 2 = 0[/itex]
has the following roots
[itex]x_1 = -\frac{1}{2} \, {\left(\frac{1}{2} i \, \sqrt{3} +
\frac{1}{2}\right)}^{\left(\frac{1}{3}\right)} {\left(i \, \sqrt{3} +
1\right)} - \frac{-i \, \sqrt{3} + 1}{2 \, {\left(\frac{1}{2} i \,
\sqrt{3} + \frac{1}{2}\right)}^{\left(\frac{1}{3}\right)}} + 1[/itex]
[itex]x_2 = -\frac{1}{2} \, {\left(-i \, \sqrt{3} + 1\right)} {\left(\frac{1}{2}
i \, \sqrt{3} + \frac{1}{2}\right)}^{\left(\frac{1}{3}\right)} - \frac{i
\, \sqrt{3} + 1}{2 \, {\left(\frac{1}{2} i \, \sqrt{3} +
\frac{1}{2}\right)}^{\left(\frac{1}{3}\right)}} + 1[/itex]
[itex]x_3 = {\left(\frac{1}{2} i \, \sqrt{3} +
\frac{1}{2}\right)}^{\left(\frac{1}{3}\right)} +
\frac{1}{{\left(\frac{1}{2} i \, \sqrt{3} +
\frac{1}{2}\right)}^{\frac{1}{3}}} + 1[/itex]
which have numerical approximations of
[itex]x_1 \approx 0.65270364[/itex]
[itex]x_2 \approx -0.53208889[/itex]
[itex]x_3 \approx 2.8793852[/itex]
When I run these through Sage's simplify_full() command (five times, even), they just become single fractions.
Is this a CAS simplification thing or are there real numbers that simply cannot exist without complex numbers?
Is it possible that a real number exists that can only be expressed in exact form when that form must includes complex numbers?
For example, the equation
[itex]2 \, x^{3} - 6 \, x^{2} + 2 = 0[/itex]
has the following roots
[itex]x_1 = -\frac{1}{2} \, {\left(\frac{1}{2} i \, \sqrt{3} +
\frac{1}{2}\right)}^{\left(\frac{1}{3}\right)} {\left(i \, \sqrt{3} +
1\right)} - \frac{-i \, \sqrt{3} + 1}{2 \, {\left(\frac{1}{2} i \,
\sqrt{3} + \frac{1}{2}\right)}^{\left(\frac{1}{3}\right)}} + 1[/itex]
[itex]x_2 = -\frac{1}{2} \, {\left(-i \, \sqrt{3} + 1\right)} {\left(\frac{1}{2}
i \, \sqrt{3} + \frac{1}{2}\right)}^{\left(\frac{1}{3}\right)} - \frac{i
\, \sqrt{3} + 1}{2 \, {\left(\frac{1}{2} i \, \sqrt{3} +
\frac{1}{2}\right)}^{\left(\frac{1}{3}\right)}} + 1[/itex]
[itex]x_3 = {\left(\frac{1}{2} i \, \sqrt{3} +
\frac{1}{2}\right)}^{\left(\frac{1}{3}\right)} +
\frac{1}{{\left(\frac{1}{2} i \, \sqrt{3} +
\frac{1}{2}\right)}^{\frac{1}{3}}} + 1[/itex]
which have numerical approximations of
[itex]x_1 \approx 0.65270364[/itex]
[itex]x_2 \approx -0.53208889[/itex]
[itex]x_3 \approx 2.8793852[/itex]
When I run these through Sage's simplify_full() command (five times, even), they just become single fractions.
Is this a CAS simplification thing or are there real numbers that simply cannot exist without complex numbers?