query: What is the meaning of $S^1$ or $S^2$?
positive_passages.docid: 141991
positive_passages.text: What does the notation $S^n$ mean?
negative_passages.docid: 40534
negative_passages.text: What is $(-1)^{\frac{2}{3}}$?
negative_passages.docid: 135059
negative_passages.text: Prove that the unit circle, $S^1:=\{(x,y)\in \mathbb{R}^2: x^2+y^2=1\}$, Is a one-dimensional manifold
negative_passages.docid: 135212
negative_passages.text: The square of an integer has form $3n$ or $3n+1$
negative_passages.docid: 114736
negative_passages.text: What does 1 with an inequality in the subscript mean?
negative_passages.docid: 5250
negative_passages.text: Are $T\mathbb{S}^2$ and $\mathbb{S}^2 \times \mathbb{R}^2$ different?
negative_passages.docid: 105997
negative_passages.text: Geometric interpretation of $\frac {\partial^2} {\partial x \partial y} f(x,y)$
negative_passages.docid: 98921
negative_passages.text: Does $y(y+1) \leq (x+1)^2$ imply $y(y-1) \leq x^2$?
negative_passages.docid: 64454
negative_passages.text: $L^2([0,1])$ is a set of first category in $L^1([0,1])$?
negative_passages.docid: 57967
negative_passages.text: Difference between $\mathbb C$ and $\mathbb R^2$
negative_passages.docid: 102138
negative_passages.text: Why does $\sqrt{x^2}$ seem to equal $x$ and not $|x|$ when you multiply the exponents?
negative_passages.docid: 14633
negative_passages.text: What's the thing with $\sqrt{-1} = i$
negative_passages.docid: 48598
negative_passages.text: What's the difference between $\mathbb{R}^2$ and the complex plane?
negative_passages.docid: 146432
negative_passages.text: If $2^{\aleph_0}=\aleph_2$, what is known about $2^{\aleph_1}$?
negative_passages.docid: 158356
negative_passages.text: Need a proofreading why all the units are satisfied $a^2-2b^2 =\pm1$ for $\mathbf{Z}[\sqrt{2}]$
negative_passages.docid: 5689
negative_passages.text: Geometric interpretation of the multiplication of complex numbers?
negative_passages.docid: 134927
negative_passages.text: How can a square root be defined since it has two answers?