query: If $z$ lies on the circle $|z-1|=1$ then$\frac {z-2} z$ is a purely imaginary number
positive_passages.docid: 146530
positive_passages.text: If $z$ lies on the circle $|z-1|=1,$ then the value of $\frac{z-2}{z}$ is
negative_passages.docid: 84374
negative_passages.text: Rational function with absolute value $1$ on unit circle
negative_passages.docid: 40824
negative_passages.text: Complex Equation Of a Circle $|z-z_1|^2+|z-z_2|^2=k$.
negative_passages.docid: 88887
negative_passages.text: Using the properties of real numbers, verify commutativity on complex numbers.
negative_passages.docid: 11935
negative_passages.text: Is $\mathbb{Z} \left[ \frac{1+i \sqrt{7}}{2} \right]$ euclidean?
negative_passages.docid: 139168
negative_passages.text: Prove that there is an unique $z$ s.t. $f(z) = z$ where $z$ is a complex number
negative_passages.docid: 70522
negative_passages.text: $y^2 = \frac{x^5 - 1}{x-1}$ & $x,y \in \mathbb{Z}$
negative_passages.docid: 113993
negative_passages.text: Complex power of a complex number
negative_passages.docid: 58636
negative_passages.text: Do the real numbers and the complex numbers have the same cardinality?
negative_passages.docid: 34709
negative_passages.text: Complex numbers - roots of unity
negative_passages.docid: 86197
negative_passages.text: Let $z_1 = x_1 +iy_1 $ and $z_2 = x_2 +iy_2$ be two complex numbers. The dot product of $z_1$ and $z_2$ is defined by $ = x_1x_2+y_1y_2$
negative_passages.docid: 136287
negative_passages.text: Separate real and imaginary part of $\arccos(z)$
negative_passages.docid: 43834
negative_passages.text: Simple Complex Number Problem: $1 = -1$
negative_passages.docid: 158524
negative_passages.text: Complex numbers $z$ such that $|z|= 1$
negative_passages.docid: 92205
negative_passages.text: Does Euler formula work for complex $z$?
negative_passages.docid: 91687
negative_passages.text: If $\frac{z^2_{1}}{z_{2}z_{3}}+\frac{z^2_{2}}{z_{3}z_{1}}+\frac{z^2_{3}}{z_{1}z_{2}} = -1.$Then $|z_{1}+z_{2}+z_{3}|$
negative_passages.docid: 52178
negative_passages.text: Prove the roots of $z^7+7z^4+4z+1=0$ lie inside a circle of radius $2$.