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 properties of complex numbers

If  z_1, z_2   are two complex numbers , then

(i) |z_1z_2|=|z_1||z_2|

(ii)  \left| \dfrac{z_1}{z_2} \right|=\dfrac{|z_1|}{|z_2|},~~~ |z_2| \neq 0

(iii)  \overline{z_1z_2}=\overline{z_1}~~\overline{z_2}

(iv) \overline{z_1\pm z_2}= \overline{z_1} \pm \overline{z_2}

(v) \left( \dfrac{\overline{z_1}}{z_2} \right)=\dfrac{\overline{z_1}}{z_2},~~ z_2 \neq 0

(vi)  z \overline{z}=a^2+b^2=\overline{z} \cdot z

(vii)  z \overline{z}=0 \text { iff } z=0

(viii) Triangle Inequality:

(A)  |z_1+z_2| \leq |z_1|+|z_2|

(B) \left| |z_1|-|z_2| \right|\leq \left| z_1+z_2 \right|

(ix) z \cdot \overline{z}=1

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