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mercury (planet)

Extended content

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

Extended content

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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