Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

Some important identities of Complex Numbers

(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}

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