Some important identities of Complex Numbers

Mathematics Class XI Chapter 2: Complex Numbers and Quadratic Equations Some important identities of Complex Numbers

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(i) $(z_1+z_2)^2=z_1^2+z_2^2+2z_1z_2,~ \forall z_1,z_2 \in \mathbb {C}$

(ii) $(z_1-z_2)^2=z_1^2-2z_1z_2+z^2_2, ~~\forall z_1, z_2 \in \mathbb{C}$

(iii) $(z_1+z_2)^3=z_1^3+3z_1^2z_2+3z_1z_2^2+z_3^3$

(iv) $(z_1-z_2)^3=z_1^3-3z_1^2z_2+3z_1z_2^2-z_2^3$

(v) $z_1^2-z_2^2=(z_1+z_2)(z_1-z_2)$

Power of $i$

$i^2=-1$

$i^3=-i$

$i^4=1$

$i^5=-i$

In general, for any $k \in Z$ , $~~i^{4k}=1,~~ i^{4k+1}=i, ~~i^{4k+2}=-1,~~ i^{4k+3}=-i$

Here, $i,~ -i,~1,~-1$ are roots of the equation $x^4=1$ , where we get two real roots, $1$ and $-1$ and two imaginary roots $i$ and $- i$ , by solving:

$x^4=1 \implies x^4-1=0 \implies (x^2+1)(x^2-1)= 0$

$\implies x^2+1=0,~~ x^2-1=0 \implies x=\pm \sqrt{-1}, ~~ x=\pm 1$

$\implies x=i,-i, ~~ x=1,-1$

The Modulus of a Complex Number

Let $z=a+ib$ be a complex number. Then, the modulus of $z$ , denoted by $|z|$ , is defined to be the non-negative real number $\sqrt{a^2+b^2}$ i.e., $|z|=\sqrt{a^2+b^2}$

Example:

Let $z=3+4i$

$\implies |z|=\sqrt{3^2+4^2}=5$

Conjugate of a Complex Number

Let $z=a+ib$ , be a complex number. Then, the complex conjugate of $z$ , denoted by $\overline {z}$ , is a complex number $a-ib$ , i.e., $\overline{z}=a-ib$ .

Example:

If $z=3-4i$

$\implies \overline{z}=3-4i=3-(-4)i=3+4i$

NOTE:

(i) Let $z=a+i0$ , be a complex number which is purely real, it has no imaginary part.

i.e., $ReZ=a,~~ImZ=0$

(ii) Let $z=0+i$ , be a complex number which is called purely imaginary, it have no real part.

i.e, $ReZ=0, ~~ ImZ=1$

Some Important Properties:

(i) If $z=a+ib, \quad\overline{z}=a-ib$ then we get,

$\begin{aligned} \to ReZ&=\dfrac{z+\overline{z}}{2} =\dfrac{(a+ib)+(a-ib)}{2}\\&=\dfrac{2a}{2}=a \end{aligned}$

$\begin{aligned}\to ImZ&=\dfrac{z-\overline{z}}{2i}=\dfrac{(a+ib)-(a-ib)}{2i} \\&=\dfrac{2ib}{2i} \\&=b\end{aligned}$