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I have seen in a book that a number whose square is nonnegative is called real number. How can we explain what a real number is?

Asaf Karagila
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Waqar Ali Shah
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    From Wikipedia: The currently standard axiomatic definition is that real numbers form the unique Archimedean complete totally ordered field (R ; + ; · ; <), up to an isomorphism. – Zain Patel Jul 07 '15 at 22:28
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    That's a slightly perverse definition of the reals. It is the case though that the subset of the complex numbers with that property (when squared is a nonegative real number) are real numbers. However, it is circular! To define the reals from the ground up, there is a standard set of postulates/axioms. See for example, Spivak's Calculus for a careful discussion. – Simon S Jul 07 '15 at 22:29
  • Ok, but I need a simple answer – Waqar Ali Shah Jul 07 '15 at 22:29
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    You are trying to define "real numbers" using the term "real numbers"( or similar to that. At least you have to specify among what numbers.. Anyway it would cost a lot more than defining real numbers directly since you are trying to define it as a sub-object) It's nonsense.. – Rubertos Jul 07 '15 at 22:33
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    Perhaps you should read the answers [here](http://math.stackexchange.com/questions/62852/in-set-theory-how-are-real-numbers-represented-as-sets?rq=1). Adequately defining the reals is generally not going to be "simple", but even without knowing the exact definition, you should know how to manipulate them. – JMoravitz Jul 07 '15 at 22:35
  • Assuming from your tag and your title ("Set Theory"), I can only assume that you want to define real numbers using set theory. There is no particularly simple way to do it. – 0XLR Jul 07 '15 at 22:36
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    The "simplest" way to define real numbers in a single line is as **the complete ordered field**. – 0XLR Jul 07 '15 at 22:37
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    @ZeroXLR you need to add "with the archimedean property", otherwise the hyperreals and surreals (and presumably a lot of in-between fields) would also qualify. – Arthur Jul 07 '15 at 23:02
  • Yes. You are correct. Thank you for the fix. – 0XLR Jul 07 '15 at 23:05
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    It is not a good characterization. Many reasons. Note that the square of $3+i$ is non-negative. – André Nicolas Jul 07 '15 at 23:32
  • @AndreNicolas, you are wrong. The square of 3+iota is a complex number. – Waqar Ali Shah Jul 08 '15 at 14:58
  • @WaqarAliShah: Indeed it is complex and non-real. It is also non-negative, in the sense that it is not negative. – André Nicolas Jul 08 '15 at 15:01
  • @AndreNicolas, What do you mean by non-negative? If a real number is non-negative then it would either be positive or zero but we can't discuss that in complex numbers. – Waqar Ali Shah Jul 08 '15 at 15:12
  • I was objecting to the formulation. of the quoted "definition" of real number. If it had said "positive or $0$" it would have been OK, though not useful. But as I read it, the reasonable interpretation of "non-negative" is "not negative." – André Nicolas Jul 08 '15 at 15:20

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