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 \(z_{2}\)

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

Complex Numbers Class 11 Examples

Complex Numbers Class 11
Complex Numbers Class 11
Complex Numbers Class 11
Complex Numbers Class 11
Complex Numbers Class 11
Complex Numbers Class 11
Complex Numbers Class 11


Practise This Question

Which of the following is a set?