Dark Calculator Cookie Settings

Dark Calculator uses cookie for smoother and better performance. Would you allow us to have cookies enabled on your browser?

Theory [Multiply Complex Numbers]

Two complex numbers \((a+ib)\) and \((c+id)\) when multiplied with each other results into: $$(a+ib)(c+id)=(ac-bd)+i(ad+bc)$$ According to Euler's theorem, any complex number can be written into exponential form, which helps us to write the imaginary numbers in the form of: $$a+ib=r_1e^{i\theta_1}$$ $$c+id=r_2e^{i\theta_2}$$ where \(r\) is the magnitude of the complex number (which has been indexed) and \theta is the polar angle the line joining the pole and the number creates with the real axis. Mathematically, they can be formulated as: $$r_1=\sqrt{a^2+b^2}$$ $$\theta_1=\tan^{-1}\left(\frac{b}{a}\right)$$ $$r_2=\sqrt{c^2+d^2}$$ $$\theta_2=\tan^{-1}\left(\frac{d}{c}\right)$$

Multiplying the complex numbers results in: $$(a+ib)(c+id)=r_1e^{i\theta_1}\cdot r_2e^{i\theta_2}=r_1r_2e^{i(\theta_1+\theta_2)}$$ Therefore, the magnitudes get multiplied upon multiplication of the numbers and the angles get added. There is another approach to this: $$(a+ib)(c+id)=(ac-bd)+i(ad+bc)$$