Number Theory T...

Let $$\displaystyle\ z_{1}= a+ib, z_{2}= p+iq$$ be two unimodular complex numbers such that $$\displaystyle\ Im(z_{1}z_{2})=1$$. If$$\displaystyle\ \omega_{1}= a+ip, \omega_{2}=b+iq$$ then

If $$z = x+iy$$ and $$w = \dfrac{(1-iz)}{(z-i)}$$, then $$|w| = 1$$ implies that, in the complex plane

If $$(\sqrt 3-i)^n=2^n, n\in N$$, then $$n$$ is a multiple of

Given that $$i = \sqrt {-1}$$, find the multiplicative inverse of $$5 - i$$.

Write the complete number $$- 2 - 2i$$ in polar form.

An example for twin primes is _____.

What is the product of the complex numbers $$\left( -3i+4 \right) $$ and $$\left( 3i+4 \right) $$?

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

What is the next-highest prime number after 67?

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.