Complex Numbers...

Add and express in the form of a complex number $$a+bi$$
$$(2+3i)+(-4+5i)-\dfrac {(9-3i)}{3}$$

If $$\alpha $$ and $$\beta$$ are two different complex numbers with $$|\beta|=1$$, then $$\left | \dfrac{\beta -\alpha}{1-\bar{\alpha }\beta } \right |$$ is equal to.

If $$z_ 1 = 2 \sqrt 2 (1 + i)$$ and $$z = 1 + i \sqrt 3$$, then $$z_1^2 z_2^3$$ is equal to

Evaluate in standard form: $$\dfrac {(2-3i)}{(2-2i)}$$, where $${i}^{2}=-1$$.

Express in the form of a complex number $$a+bi$$
$$-(7-i)(-4-2i)(2-i)$$

If $$z$$ is a complex number such that $$|z|\geq 2$$ then the minimum value of $$\left |z + \dfrac {1}{2}\right |$$ is

What is $${ i }^{ 1000 }+{ i }^{ 1001 }+{ i }^{ 1002 }+{ i }^{ 1003 }$$ equal to (where $$i=\sqrt { -1 } $$)?

A complex number z is said to be unimodular if $$|z| =1. $$. Suppose $$z_1$$ and $$z_2$$ are complex numbers such that $$\frac{z_1-2z_2}{2-z_1\overline {z}_2}$$ is unimolecolar and $$z_2$$ is not unimodular. Then the point $$z_1$$ lies on a:

Let $$z_1$$ = 18 + 83i, $$z_2$$ = 18 + 39i, ana $$z_3 $$= 78 + 99i. where i = $$\sqrt-1$$. Let z be a unique comlpex number with the properties that $$\dfrac{z_3 - z_1}{z_2 - z_1}$$ $$\cdot$$ $$\dfrac{z - z_2}{z - z_3}$$ is a real number and the imaginary part of the size z is the greatest possible.

If $$z=\sqrt{20i-21}+\sqrt{21+20i}$$, then the principal value of arg 'z' can be