Algebra of Complex Numbers

If $\arg \left(z\right)<0,$ then $\arg \left(-z\right)-\arg \left(z\right)=$

The value of the sum  $\sum _{n=1}^{13}\left(i^n\hspace{0.17em}+i^{n+1}\right),$ where  $i=\sqrt{-1}$ , equals

For positive integers  $n_1, n_2$  the value of expression ${\left(1+i\right)}^{n_1}+{\left(1+i^3\right)}^{n_1}+$${\left(1+i^5\right)}^{n_2}+{\left(1+i^7\right)}^{n_2}$ where  $i=\sqrt{-1}$ , is a real number if and only if

Let $A=\left\{\frac{1967+1686i \sin \theta }{7-3i \cos \theta }: \theta \in R\right\}$. If $A$ contains exactly one positive integer $n$, then the value of $n$ is

Let $\overline{z}$ denote the complex conjugate of a complex number $z$ and let $i=\sqrt{-1}$. In the set of complex numbers, the number of distinct roots of the equation $\overline{z}-z^2=i\left(\overline{z}+z^2\right)$ is _______.

Let $f\left(x\right)=\frac{1}{x-i}$, where $x\in R$ and let $f\left(\alpha \right), f\left(\beta \right), f\left(\gamma \right), f\left(\delta \right)$ be four points on the Argand plane. Then the maximum value of $\left|f\left(\alpha \right)-f\left(\beta \right)\right|$ is

Calculate $\log z$ for $z=-1-\sqrt{3}i$..

Subtract the complex number $2+4i$ from the complex number $5+i$ in the vector form.

Subtract the complex number $2+4i$ from the complex number $5+2i$ in the vector form.

Subtract the complex number $2+3i$ from the complex number $5+2i$ in the vector form.

Subtract the complex number $1+3i$ from the complex number $5+2i$ in the vector form.

Subtract the complex number $1+3i$ from the complex number $4+2i$ in the vector form.

Subtract the complex number $-9-2i$ from the complex number $2+3i$.

Find the addition of two complex numbers $\left(2+3i\right)$, and $\left(-9-2i\right)$.

Find the conjugate of the complex number $\frac{1+2i}{1-3i}$ in polar form.

Write the conjugate of the complex number $1-i$ in polar form.

Write the conjugate of the complex number $-1-i$ in polar form.

Write the conjugate of the complex number $1+i$ in polar form.

Write the conjugate of the complex number $-3\sqrt{2}+3\sqrt{2}i$ in polar form.

For the two complex numbers $z_1=10+4i$ and $z_2=-7+5i$, find $z_1-z_2$.