Complex Numbers Class 11

Complex Numbers Class 11 – A number that can be represented in form p + iq is defined as a complex number. [Where, p and q are real numbers and \(i=\sqrt{-1}\)]. For a complex number z = p + iq, p is known as the real part, represented by Re z and q is known as the imaginary part represented by Im z of complex number z.

Consider two complex numbers \(z_{1}\) = p + iq and \(z_{2}\) = m + in

Now, \(z_{1}\) + \(z_{2}\) = (p + m) + i(q + n)

\(z_{1}\) – \(z_{2}\) = (p – m) + i (q – n)

\(z_{1}\times z_{2}\) = pn + mn + i (pq + mq)

\(\frac{z_{1}}{z_{2}}\) = \(\frac{p+iq}{m+in}=\frac{p+iq}{m+in}\times \frac{m-in}{m-in}=\frac{pm+qn+i(qm-pn)}{m^{2}+n^{2}}\)

Identities of Complex number \(z_{2}\) and \(z_{2}\)

\(\left ( z_{1} +z_{2}\right )^{2}=z_{1}^{2}+z_{2}^{2}+2z_{1}z_{2}\)
\(\left ( z_{1} -z_{2}\right )^{2}=z_{1}^{2}+z_{2}^{2}-2z_{1}z_{2}\)
\(\left ( z_{1} +z_{2}\right )^{3}=z_{1}^{3}+z_{2}^{3}+3z_{1}^{2}z_{2}+3z_{1}z_{2}^{2}\)
\(\left ( z_{1} -z_{2}\right )^{3}=z_{1}^{3}-z_{2}^{3}-3z_{1}^{2}z_{2}+3z_{1}z_{2}^{2}\)
\(\left ( z_{1}^{2} -z_{2}^{2}\right )=(z_{1}+z_{2})(z_{1}-z_{2})\)