query: When are quantities outside of the real numbers considered equal, and when do they exist?
positive_passages.docid: 148644
positive_passages.text: Definition of complex number
negative_passages.docid: 2796
negative_passages.text: How to tell Mathematica that certain variables are real/imaginary, integer-valued, etc
negative_passages.docid: 154822
negative_passages.text: Properties of the composition law $a @ b= ab + b + a$ on the real numbers
negative_passages.docid: 126633
negative_passages.text: Is a contradiction enough to prove a set equality to $\varnothing$?
negative_passages.docid: 157733
negative_passages.text: Are positive real numbers $x,y$ allowed to be taken out during this proof?
negative_passages.docid: 138446
negative_passages.text: Proving that there exists an irrational number in between any given real numbers
negative_passages.docid: 102406
negative_passages.text: Why is it not true that if $\Phi \models \exists x \phi$ then there is a term $t$ with $\Phi \models \phi \frac{t}{x}$?
negative_passages.docid: 118175
negative_passages.text: Can we say that $\frac{0}{0}$ is every number?
negative_passages.docid: 98294
negative_passages.text: Can decimal numbers be considered "even" or "odd"?
negative_passages.docid: 93447
negative_passages.text: How is Russell’s Paradox equal to the universal set?
negative_passages.docid: 19655
negative_passages.text: Why don't we define "imaginary" numbers for every "impossibility"?
negative_passages.docid: 62473
negative_passages.text: Arguments for the existence of infinity
negative_passages.docid: 70332
negative_passages.text: Is there a third dimension of numbers?
negative_passages.docid: 149853
negative_passages.text: What infinity is greater than the continuum? Show with an example
negative_passages.docid: 70196
negative_passages.text: Is there any well-ordered uncountable set of real numbers under the original ordering?
negative_passages.docid: 57938
negative_passages.text: When do two functions become equal?
negative_passages.docid: 127052
negative_passages.text: Independence of non-constant real variables