# 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

#### Practise This Question

If |z|=1, then (1+z1+¯z)n+(1+¯z1+z)n is equal to