The RealPart and ImaginaryPart functions are placeholders for representing the real or imaginary parts, respectively, of an algebraic number given in RootOf notation. It is used in conjunction with evala.

The call evala(RealPart(RootOf(p))) computes $\mathrm{RootOf}\left(q\right)$, where $q$ is a squarefree polynomial with rational coefficients of smallest possible degree whose roots include the real parts of all the roots of $p$. Note that in general $q$ will have additional (real or complex) roots that do not correspond to real parts of roots of $p$.

The call evala(ImaginaryPart(RootOf(p))) works analogously.

The other calling sequences return a RootOf expression or a rational number encoding the real/imaginary part of the given $\mathrm{RootOf}$. Usually, the 1st operand of the $\mathrm{RootOf}$ will be an irreducible polynomial, and the 2nd operand (selector) will be of the same type as the selector of the input.

$\mathrm{evala}\left(\mathrm{RealPart}\left(\mathrm{RootOf}\left(x^4+x^2+1\right)\right)\right)$

$\mathrm{RootOf}\left(4{\mathrm{\_Z}}^2-1\right)$

$\mathrm{evala}\left(\mathrm{ImaginaryPart}\left(\mathrm{RootOf}\left(x^4+x^2+1\right)\right)\right)$

$\mathrm{RootOf}\left(4{\mathrm{\_Z}}^2-3\right)$

$f≔\mathrm{RootOf}\left(x^4+x^2+1\,\mathrm{index}=1\right)\:$

$\mathrm{re},\mathrm{im}≔\mathrm{evala}\left(\mathrm{RealPart}\left(f\right)\right),\mathrm{evala}\left(\mathrm{ImaginaryPart}\left(f\right)\right)$

$\mathrm{re},\mathrm{im}≔\frac{1}{2},\mathrm{RootOf}\left(4{\mathrm{\_Z}}^2-3\,\mathrm{index}={\mathrm{real}}_2\right)$

$\mathrm{evala}\left(\mathrm{Normal}\left(\mathrm{re}+\mathrm{I}\mathrm{im}-f\right)\right)$

$0$

$g≔\mathrm{RootOf}\left(x^4+x^2+1\,-1-\mathrm{I}..0\right)\:$

$\mathrm{evalf}\left(g\right)$

$\mathrm{-0.5000000000}-0.8660254038\mathrm{I}$

$\mathrm{re},\mathrm{im}≔\mathrm{evala}\left(\mathrm{RealPart}\left(g\right)\right),\mathrm{evala}\left(\mathrm{ImaginaryPart}\left(g\right)\right)$

$\mathrm{re},\mathrm{im}≔-\frac{1}{2},\mathrm{RootOf}\left(4{\mathrm{\_Z}}^2-3\,\mathrm{-1}..0\right)$

$\mathrm{evalf}\left(\mathrm{re}\right),\mathrm{evalf}\left(\mathrm{im}\right)$

$\mathrm{-0.5000000000},\mathrm{-0.8660254038}$

$\mathrm{evala}\left(\mathrm{RealPart}\left(\mathrm{RootOf}\left(x^{10}+1\right)\right)\right)=\mathrm{RootOf}\left(\mathrm{expand}\left(\mathrm{ChebyshevT}\left(5\,x\right)\right)\right)$

$\mathrm{RootOf}\left(16{\mathrm{\_Z}}^5-20{\mathrm{\_Z}}^3+5\mathrm{\_Z}\right)=\mathrm{RootOf}\left(16{\mathrm{\_Z}}^5-20{\mathrm{\_Z}}^3+5\mathrm{\_Z}\right)$

$f≔\mathrm{RootOf}\left(x^3-\mathrm{I}\right)\:$

$\mathrm{evala}\left(\mathrm{RealPart}\left(f\right)\right),\mathrm{evala}\left(\mathrm{ImaginaryPart}\left(f\right)\right)$

$\mathrm{RootOf}\left(4{\mathrm{\_Z}}^3-3\mathrm{\_Z}\right),\mathrm{RootOf}\left(2{\mathrm{\_Z}}^2+\mathrm{\_Z}-1\right)$

$g≔\mathrm{RootOf}\left(x^3-\mathrm{I}\,0..1+\mathrm{I}\right)\:$

$\mathrm{evala}\left(\mathrm{RealPart}\left(g\right)\right),\mathrm{evala}\left(\mathrm{ImaginaryPart}\left(g\right)\right)$

$\mathrm{RootOf}\left(4{\mathrm{\_Z}}^2-3\,0..1\right),\frac{1}{2}$

$p≔x^4+x-1$

$p≔x^4+x-1$

$\mathrm{fsolve}\left(p\,'\mathrm{complex}'\right)$

$\mathrm{-1.220744085},0.2481260628-1.033982061\mathrm{I},0.2481260628+1.033982061\mathrm{I},0.7244919590$

$r≔\mathrm{evala}\left(\mathrm{RealPart}\left(\mathrm{RootOf}\left(p\right)\right)\right)$

$r≔\mathrm{RootOf}\left(64{\mathrm{\_Z}}^{10}+64{\mathrm{\_Z}}^7-48{\mathrm{\_Z}}^6-{\mathrm{\_Z}}^4+16{\mathrm{\_Z}}^3-16{\mathrm{\_Z}}^2-\mathrm{\_Z}+1\right)$

$\mathrm{fsolve}\left(\mathrm{op}\left(r\right)\right)$

$\mathrm{-1.220744085},\mathrm{-0.2481260628},0.2481260628,0.7244919590$

$\mathrm{factor}\left(\mathrm{op}\left(r\right)\right)$

$\left({\mathrm{\_Z}}^4+\mathrm{\_Z}-1\right)\left(64{\mathrm{\_Z}}^6+16{\mathrm{\_Z}}^2-1\right)$

$$\begin{gathered} \mathbf{for}i\mathbf{to}4\mathbf{do} \\ r≔\mathrm{evala}\left(\mathrm{RealPart}\left(\mathrm{RootOf}\left(p\,\mathrm{index}=i\right)\right)\right); \\ \mathrm{print}\left(r\,\mathrm{evalf}\left(r\right)\right) \\ \mathbf{end}\mathbf{do}\: \end{gathered}$$

$\mathrm{RootOf}\left({\mathrm{\_Z}}^4+\mathrm{\_Z}-1\,\mathrm{index}={\mathrm{real}}_2\right),0.7244919590$

$\mathrm{RootOf}\left(64{\mathrm{\_Z}}^6+16{\mathrm{\_Z}}^2-1\,\mathrm{index}={\mathrm{real}}_2\right),0.2481260628$

$\mathrm{RootOf}\left({\mathrm{\_Z}}^4+\mathrm{\_Z}-1\,\mathrm{index}={\mathrm{real}}_1\right),\mathrm{-1.220744085}$

$\mathrm{RootOf}\left(64{\mathrm{\_Z}}^6+16{\mathrm{\_Z}}^2-1\,\mathrm{index}={\mathrm{real}}_2\right),0.2481260628$

$p≔x^4-3x^2+1$

$p≔x^4-3x^2+1$

$\mathrm{fsolve}\left(p\right)$

$\mathrm{-1.618033989},\mathrm{-0.6180339887},0.6180339887,1.618033989$

$\mathrm{evala}\left(\mathrm{RealPart}\left(\mathrm{RootOf}\left(p\right)\right)\right),\mathrm{evala}\left(\mathrm{ImaginaryPart}\left(\mathrm{RootOf}\left(p\right)\right)\right)$

$\mathrm{RootOf}\left({\mathrm{\_Z}}^4-3{\mathrm{\_Z}}^2+1\right),0$

The evala/RealPart and evala/ImaginaryPart commands were introduced in Maple 2025.

For more information on Maple 2025 changes, see Updates in Maple 2025.