query: What is the definition of real numbers?
positive_passages.docid: 48028
positive_passages.text: True Definition of the Real Numbers
negative_passages.docid: 47871
negative_passages.text: Why did mathematician construct extended real number systems, $\mathbb R \cup\{+\infty,-\infty\}$?
negative_passages.docid: 87410
negative_passages.text: Are there real numbers that are neither rational nor irrational?
negative_passages.docid: 137564
negative_passages.text: What exactly is meant by infinity?
negative_passages.docid: 78294
negative_passages.text: How does one denote the set of  all positive real numbers?
negative_passages.docid: 21606
negative_passages.text: Can all real numbers be represented by the sum of a convergent series?
negative_passages.docid: 130149
negative_passages.text: Computablility of a real number
negative_passages.docid: 2742
negative_passages.text: Is infinity a number?
negative_passages.docid: 138977
negative_passages.text: Infinitely many rational numbers between two different real numbers
negative_passages.docid: 77957
negative_passages.text: How is $\mathbb N$ actually defined?
negative_passages.docid: 11586
negative_passages.text: Sequences of Real Numbers in a closed bounded interval
negative_passages.docid: 114043
negative_passages.text: Prove the existence of a real number satisfying a property
negative_passages.docid: 102307
negative_passages.text: Discrete Math Proofs Involving Real Numbers
negative_passages.docid: 70332
negative_passages.text: Is there a third dimension of numbers?
negative_passages.docid: 142781
negative_passages.text: What exactly is a number?
negative_passages.docid: 86914
negative_passages.text: Cardinality of the set of hyperreal numbers
negative_passages.docid: 33705
negative_passages.text: Is $0$ a natural number?