User:Eli355/sandbox

mercury (planet)

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In abstract algebra, the Quaternionic roots of polynomials are the roots of a polynomial in the quaternions, which are the quaternions that would make a polynomial equal to zero when substituted in place of the variable. Quadratic function For a quadratic polynomial $ax^{2}+bx+c$, the roots can be found by substituting a quaternion $t+ui+vj+wk$ into the variable, expanding it, and then separating the real and imaginary parts and setting all of them equal to zero in a system of equations. $a(t+ui+vj+wk)^{2}+b(t+ui+vj+wk)+c=0$ $a(t^{2}-u^{2}-v^{2}-w^{2}+2tui+2tvj+2twk)+b(t+ui+vj+wk)+c=0$ $at^{2}-au^{2}-av^{2}-aw^{2}+2atui+2atvj+2atwk+bt+bui+bvj+bwk+c=0$ ${\begin{cases}&at^{2}-au^{2}-av^{2}-aw^{2}+bt+c=0\\&2atui+bui=0\\&2atvj+bvj=0\\&2atwk+bwk=0\end{cases}}$ Taking one of the last three equations and cancelling out the common factor shows that $t=-{\frac {b}{2a}}.$ This value obtained for $t$ can be substituted into the first equation, which is then simplified. $a\left(-{\frac {b}{2a}}\right)^{2}-au^{2}-av^{2}-aw^{2}+b\left(-{\frac {b}{2a}}\right)+c=0\quad \Rightarrow \quad u^{2}+v^{2}+w^{2}={\frac {4ac-b^{2}}{4a^{2}}}$ This shows that the roots of a quadratic $ax^{2}+bx+c$ will be in the form $t+ui+vj+wk$ where $t=-{\frac {b}{2a}}$ and $u^{2}+v^{2}+w^{2}={\frac {4ac-b^{2}}{4a^{2}}}.$ This applies to quadratics with negative discriminants. For quadratics with positive discriminants, the hyperbolic quaternions can be used and this time the sum of the squares of the imaginary parts will be equal to ${\frac {b^{2}-4ac}{4a^{2}}}.$ Other degrees This can be generalized to higher degree polynomials since all polynomials can be factored into quadratics, with an additional linear factor if the degree is odd. Pages to create Quaternionic roots of polynomials Inverse Pythagorean theorem Illogicopedia Pierce expansion Cotangent expansion Orders of magnitude (angle) Orders of magnitude (surface tension) Orders of magnitude (current density) Orders of magnitude (linear charge density) Orders of magnitude (area charge density) Orders of magnitude (volume charge density) Capitol Green Apartments

List of derived Planck units

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Derived Planck units are units of measurement derived from the five base Planck units. They can be expressed as a product of one or more of the base units, possibly scaled by an appropriate power of exponentiation. Kinematic units The following table lists various kinematic derived Planck units. Table 3: Derived Planck units Name Dimension Expression Approximate SI equivalent Reference Planck acceleration Acceleration (LT−2) $a_{\text{P}}={\frac {c}{t_{\text{P}}}}={\sqrt {\frac {c^{7}}{\hbar G}}}$ 5.560815 × 1051 m/s2 Planck angular frequency Angular frequency (T−1) $\omega _{\text{P}}={\frac {1}{t_{\text{P}}}}={\sqrt {\frac {c^{5}}{\hbar G}}}$ 1.85487 × 1043 rad/s Planck Volumetric flow rate Volumetric flow rate $Q_{\text{P}}=l_{\text{P}}^{2}c={\frac {l_{\text{P}}^{3}}{t_{\text{P}}}}={\frac {\hbar G}{c^{2}}}$ 7.83 x 10-62 m3/s Mechanical units The following table lists various mechanical derived Planck units. Table 3: Derived Planck units Name Dimension Expression Approximate SI equivalent Reference Planck area Area (L2) $l_{\text{P}}^{2}={\frac {\hbar G}{c^{3}}}$ 2.6121 × 10−70 m2 Planck volume Volume (L3) $l_{\text{P}}^{3}=\left({\frac {\hbar G}{c^{3}}}\right)^{\frac {3}{2}}={\sqrt {\frac {(\hbar G)^{3}}{c^{9}}}}$ 4.2217 × 10−105 m3 Planck momentum Momentum (LMT−1) $m_{\text{P}}c={\frac {\hbar }{l_{\text{P}}}}={\sqrt {\frac {\hbar c^{3}}{G}}}$ 6.52485 kg⋅m/s Planck energy Energy (L2MT−2) $E_{\text{P}}=m_{\text{P}}c^{2}={\frac {\hbar }{t_{\text{P}}}}={\sqrt {\frac {\hbar c^{5}}{G}}}$ 1.9561 × 109 J Planck force Force (LMT−2) $F_{\text{P}}={\frac {E_{\text{P}}}{l_{\text{P}}}}={\frac {\hbar }{l_{\text{P}}t_{\text{P}}}}={\frac {c^{4}}{G}}$ 1.21027 × 1044 N Planck power Power (L2MT−3) $P_{\text{P}}={\frac {E_{\text{P}}}{t_{\text{P}}}}={\frac {\hbar }{t_{\text{P}}^{2}}}={\frac {c^{5}}{G}}$ 3.62831 × 1052 W Planck density Density (L−3M) $\rho _{\text{P}}={\frac {m_{\text{P}}}{l_{\text{P}}^{3}}}={\frac {\hbar t_{\text{P}}}{l_{\text{P}}^{5}}}={\frac {c^{5}}{\hbar G^{2}}}$ 5.15500 × 1096 kg/m3 Planck energy density Energy density (L−1MT−2) $\rho _{\text{P}}^{E}={\frac {E_{\text{P}}}{l_{\text{P}}^{3}}}={\frac {c^{7}}{\hbar G^{2}}}$ 4.633 × 10113 J/m3 Planck intensity Intensity (MT−3) $I_{\text{P}}=\rho _{\text{P}}^{E}c={\frac {P_{\text{P}}}{l_{\text{P}}^{2}}}={\frac {c^{8}}{\hbar G^{2}}}$ 1.38893 × 10122 W/m2 Planck pressure Pressure (L−1MT−2) $p_{\text{P}}={\frac {F_{\text{P}}}{l_{\text{P}}^{2}}}={\frac {\hbar }{l_{\text{P}}^{3}t_{\text{P}}}}={\frac {c^{7}}{\hbar G^{2}}}$ 4.633 × 10113 Pa Electromagnetic units The following table lists various electromagnetic derived Planck units. Table 3: Derived Planck units Name Dimension Expression Approximate SI equivalent Reference Planck current Electric current (QT−1) $I_{\text{P}}={\frac {q_{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {4\pi \epsilon _{0}c^{6}}{G}}}$ 3.4789 × 1025 A Planck voltage Voltage (L2MT−2Q−1) $V_{\text{P}}={\frac {E_{\text{P}}}{q_{\text{P}}}}={\frac {\hbar }{t_{\text{P}}q_{\text{P}}}}={\sqrt {\frac {c^{4}}{4\pi \epsilon _{0}G}}}$ 1.04295 × 1027 V Planck impedance Resistance (L2MT−1Q−2) $Z_{\text{P}}={\frac {V_{\text{P}}}{I_{\text{P}}}}={\frac {\hbar }{q_{\text{P}}^{2}}}={\frac {1}{4\pi \epsilon _{0}c}}={\frac {Z_{0}}{4\pi }}$ 29.9792458 Ω Planck magnetic inductance Magnetic induction (MQ−1T−1) $B_{\text{P}}={\frac {F_{\text{P}}}{l_{\text{P}}I_{\text{P}}}}={\sqrt {\frac {c^{5}}{\hbar G^{2}4\pi \epsilon _{0}}}}$ 2.1526 x 1053 T Planck electrical inductance Inductance (ML2Q−2) $L_{\text{P}}={\frac {E_{\text{P}}}{I_{\text{P}}}}={\frac {m_{\text{P}}l_{\text{P}}^{2}}{q_{\text{P}}}}={\sqrt {\frac {G\hbar }{c^{7}(4\pi \epsilon _{0})^{2}}}}$ 1.616 x 10−42 H Thermodynamic units The following table lists various thermodynamic derived Planck units. Table 3: Derived Planck units Name Dimension Expression Approximate SI equivalent Reference

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