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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. 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. – balddraz 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. – balddraz 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. – balddraz 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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