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- https://phys.libretexts.org/Bookshelves/Waves_and_Acoustics/The_Physics_of_Waves_(Goergi)/01%3A_Harmonic_Oscillation/1.04%3A_Complex_NumbersTo divide a complex number \(z\) by a real number \(r\) is easy, just divide both the real and the imaginary parts by \(r\) to get \(z/r = a/r + ib/r.\) To divide by a complex number, \(z'\), we can u...To divide a complex number \(z\) by a real number \(r\) is easy, just divide both the real and the imaginary parts by \(r\) to get \(z/r = a/r + ib/r.\) To divide by a complex number, \(z'\), we can use the fact that \(z'^* z' = |z'|^2\) is real.
- https://phys.libretexts.org/Courses/Prince_Georges_Community_College/PHY_2030%3A_General_Physics_II/19%3A_Electric_Current_and_Resistance/19.5%3A_Alternating_CurrentsPhasors are used to analyze electrical systems in sinusoidal steady state and with a uniform angular frequency.
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Classical_Mechanics_(Dourmashkin)/23%3A_Simple_Harmonic_Motion/23.09%3A_Appendix_23B_-_Complex_Numbers\[z_{3}=z_{1} z_{2}=\left(x_{1}+i y_{1}\right)\left(x_{2}+i y_{2}\right)=\left(x_{1} x_{2}-y_{1} y_{2}\right)+i\left(x_{1} y_{2}+x_{2} y_{1}\right)=x_{3}+i y_{3} \nonumber \] \[z_{1}+z_{2}=\left|z_{1}...\[z_{3}=z_{1} z_{2}=\left(x_{1}+i y_{1}\right)\left(x_{2}+i y_{2}\right)=\left(x_{1} x_{2}-y_{1} y_{2}\right)+i\left(x_{1} y_{2}+x_{2} y_{1}\right)=x_{3}+i y_{3} \nonumber \] \[z_{1}+z_{2}=\left|z_{1}\right| e^{i \phi_{1}}+\left|z_{2}\right| e^{i \phi_{1}}=\left(\left|z_{1}\right|+\left|z_{2}\right|\right) e^{i \phi_{1}} \nonumber \] \[z_{1} z_{2}=\left|z_{1}\right| e^{i \phi_{1}}\left|z_{2}\right| e^{i \phi_{2}}=\left|z_{1} \| z_{2}\right| e^{i \phi_{1}+\phi_{2}} \nonumber \]
- https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Electromagnetics_and_Applications_(Staelin)/14%3A_Appendices/14.02%3A_Complex_Numbers_and_Sinusoidal_RepresentationThis page explains that energy-storing linear systems are more responsive to sinusoidal inputs, represented using complex notation. It emphasizes the relationship between amplitude, frequency, and pha...This page explains that energy-storing linear systems are more responsive to sinusoidal inputs, represented using complex notation. It emphasizes the relationship between amplitude, frequency, and phase while illustrating the geometric interpretation of complex numbers as vectors.
- https://phys.libretexts.org/Bookshelves/University_Physics/Physics_(Boundless)/19%3A_Electric_Current_and_Resistance/19.5%3A_Alternating_CurrentsPhasors are used to analyze electrical systems in sinusoidal steady state and with a uniform angular frequency.
