Introduction to Complex Numbers
What is a Complex Number?
Last Updated: January 15, 2025Definition: Let \(\bf z=(a,b)\in \reals \times \reals \) be an ordered pair. \(\bf z\) is said to be \(\bf complex number\) if and only if it satisfies the following conditions. Definition: Let \(\bf z=(a,b)\in \reals \times \reals \) be an ordered pair. \(\bf z\) is said to be \(\bf complex number\) if and only if it satisfies...
Modulus-Amplitude form
Last Updated: July 4, 2024Modulus-Amplitude form Definition: Let \(\bf z=a+ib\) be a non-complex number and \(\bf |z|=r\). Then \(\bf z\) can represented as \(\bf z=r(\cos\theta +i \sin\theta)\). And this form of \(\bf z\) is said to be the \(\fcolorbox{red}{white}{\bf modulus-amplitude}\) or \(\fcolorbox{red}{white}{\bf polar}\) form of \(\bf z\). The \(\fcolorbox{red}{white}{\bf \( \theta\)}\) is said to be a amplitude or argument...
Problems on Modulus-Amplitude form
Last Updated: July 4, 2024\(\bf z=1+i\cot\theta\) where \(\bf \pi~ \text{\textless}~\theta~\text{\textless} ~\frac{3\pi}{2} \) Problem-1: Find the modulus amplitude form of \(\bf z=1+i\cot\theta\) where \(\bf \pi~ \text{\textless}~\theta~\text{\textless} ~\frac{3\pi}{2} \) .\( \\ \) Solution: Given that \(\bf z=1+i\cot\theta\) where \(\bf \pi~ \text{\textless}~\theta~\text{\textless} ~\frac{3\pi}{2} \) .\( \\ \) Let \(\bf z=r(\cos\alpha +i \sin\alpha)\) where \(\alpha\) be the principal amplitude .\( \\...
De Moivre’s Theorem
Last Updated: July 4, 2024De Moivre’s Theorem Statement: \( \bf \left(\cos\theta+i\sin\theta\right)^{n}=\cos n\theta+i\sin n\theta \) when \(\bf n\in \Z \) \(\bf \cos n\theta+i\sin n\theta \) is one of the value of \( \bf \left(\cos\theta+i\sin\theta\right)^{n}\) when \(\bf n=\frac{p}{q} \) where \(\bf q\ne 1 \) Proof:\( \\ \) Case-1: Let \(\bf n=0 \) Then the \begin{align*} \bf L.H.S=\left(\cos\theta+i\sin\theta\right)^{0}=1 \end{align*} And the...
nth Root of a Complex Number
Last Updated: July 4, 2024N-th Root of a Complex Number \( \bf (z)\) Statement: Let \(\bf z=r(\cos\theta +i \sin\theta)\), where \(\bf -\pi~\text{\textless}~\theta~\le~\pi \), is a non-complex number. If \(\bf n\) be a positive integer then there exists \(\bf n\) distinct values of \(\bf z^{\frac{1}{n}}\) . Proof: Given that \(\bf z=r(\cos\theta +i \sin\theta)\), where \(\bf -\pi~\text{\textless}~\theta~\le~\pi \), is a non-complex...
