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I have just begun my study of complex numbers and I learned where imaginary numbers came from and their importance. However there's one thing that I need to clarify and that is the properties of real numbers and their proofs.

  1. Closure Laws
    For all $a,b \in \mathbb{R}$, $a+b$, $a-b$, $ab$, $a/b$ are real numbers. Thus $\mathbb{R}$ is closed under four fundamental operations.

  2. Commutative Laws
    For all $a,b \in \mathbb{R}$ $a+b = b+a$ and $ab = ba$.

  3. Associative Laws
    For all $a,b,c \in \mathbb{R}$ $a+(b+c) = (a+b)+c$ and $a(bc) = (ab)c$.

  4. Additive Identity
    For all $a \in \mathbb{R}$ there exists $0\in \mathbb{R}$ such that $a+0 = 0+a = a$.

  5. Additive inverse
    For all $a \in \mathbb{R}$ there exists a $b \in \mathbb{R}$ such that $a+b = b+a = 0$, the additive identity $b = -a$ is called the additive inverse or the negative of $a$.

and similarly Multiplicative Identity, Multiplicative inverse, Distributive Law, Trichotomy Law, Transitivity of order, Monotone Law of Addition, Monotone law of multiplication.

I understand that the above laws hold good throughout mathematics. Should these laws be accepted as being true "on faith" or are there proofs? If yes, I am curious to know the proofs. As per my understanding no textbook has ever talked about proofs for these.

Kazark
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alok
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  • I editied your question to be moderately more readable, but there are still several statements that are poorly formulated. E.g. the existence of 0. – kahen Sep 02 '11 at 09:11
  • You can accept these laws or axioms as defining the real numbers. An alternate approach is to start with only the natural numbers, and "build" the rational numbers and real numbers in certain ways (this is where the Dedekind cuts approach is relevant). If you adopt the second approach, then you will need to prove that all the stated properties hold, so that you can convince yourself that what you ended up constructing is really what you wanted to construct. – Srivatsan Sep 02 '11 at 17:40
  • The Real Numbers are not closed under division – Anshuman Agrawal Feb 16 '22 at 20:09

7 Answers7

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One accessible account is in Michael Spivak's textbook Calculus. Here a real number is defined to be a subset $\alpha$ of the rational numbers that is non-empty, bounded above, and satisfies $x\in\alpha\mathrm{\ and\ }y\leq x\Rightarrow y\in\alpha$. (There's a natural way to identify these real numbers with real numbers as we usually think of them containing the rationals as a subset: for instance $\sqrt{2}$ in this sense is the following set $\{x\in\mathbb{Q}:x<\sqrt{2}\}$.) You can then define the field operations and the order, and patiently show that this gives you a complete ordered field (so proofs of the properties you ask about are given). The detailed proofs take several pages.

As far as I recall, Spivak gives a pretty full account of this, including the uniqueness of a complete ordered field.

Shane O Rourke
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  • Francis Su from Harvey Mudd College gives a (sketch of) the construction of the real numbers as Dedekind Cuts (that's the technical name for what Shane described above) in the first three videos of this playlist: http://www.youtube.com/watch?v=sqEyWLGvvdw&feature=results_main&playnext=1&list=PL04BA7A9EB907EDAF – kahen Sep 02 '11 at 09:36
  • I should have mentioned in the beginning of this post that i'a m a beginner .reading Spivak's Calculus is not my cup of tea eventhough i have a good grasp of derivatives and integrals. – alok Sep 02 '11 at 12:36
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    @alok: It may be surprising, but rigorously constructing the real numbers and thereby proving their basic properties is beyond the level of a true beginner in this subject. My guess is that you will have a hard time finding a presentation much more elementary / easier to read than the one in Spivak's text. – Pete L. Clark Sep 02 '11 at 17:34
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There are indeed proofs of the existence and uniqueness (up to unique isomorphism) of a complete ordered field.

Take a look at this question for example: Completion of rational numbers via Cauchy sequences (that I gave an answer to)

Community
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kahen
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I would suggest Landau's Foundations of Analysis. Beginning with the axioms of the natural numbers, the author develops the rational, real and complex number systems. Some of the notation is a little outdated but the treatment is clear and logical and every step is included.

ItsNotObvious
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  • The level of math knowledge that i posess is limited to derivatives and integrals with little bit of partial derivates and ODE.I have tried reading books on analysis from spivak's and apostol's text but i find it very difficuilt. – alok Sep 02 '11 at 12:38
  • @alok I think you should at least give Landau's text a quick glance; it has essentially no prerequisites. Everything you need to know to understand the book is in the book itself. – ItsNotObvious Sep 02 '11 at 15:55
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If you 're really curious for this, it has a lot of require of knowledge.Firstly, you have to understand the word "axiom".(in school level it has just meant "it is a self evident truth).Actually "axiom" has wide meaning.you should read this source(completely):std. 9th, ncert mathematics, appendix 1(proof in mathematics). After this you should learn old ncert textbook of class xi, ch. 14(mathematical reasoning).These are basic. Now you are able to understand the notion "axiomatic system".

Actually in the mathematics there are plenty of axiomatic systems.... The question which you 've asked is also themself are set of axioms. They are just axioms, you can't find any about of proof of this.Actually, these are the one of the axiomatic system. I cannot explain these in my answer but i give you here only model of this.

But there are many axiomatic systems. A very famous one is Peano--Dedekind axioms(Simply, peano axioms). In this axiomatic system you can find proof of your problem.Peano axioms implies commutative laws, associative laws, etc. but only for natural numbers.After this, you will able to construct the set Z of integers(Abstract algebra:Group)with help of concept of functions.Simillarly you will construct set of rational numbers and then real numbers and then you can prove the above properties of real numbers which you have asked. if you want to study them it requires very highly mathematical powers.Best book for study this (as my experience) is :"From numbers to analysis"(Author:Dr. Inder K Rana) And some pdf files for your study :: http://public.csusm.edu/aitken_html/m378/.

HHHH
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Most books on Calculus and Introductory Analysis state these properties as axioms for the real numbers. (Though the ones you listed only define ordered fields. The rational numbers also satisfy them. There is a crucially important property of the real numbers, completeness, which actually defines the reals uniquely.) On the other hand, there are constructions of the real numbers from the natural numbers and then these properties are theorems.

lhf
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  • I'll be glad if someone could advice me on the prerequisite needed to read books like spivak's and apostol's calculus text's.I'am more comfortable reading edward/larson's and stewart's calculs text's. – alok Sep 02 '11 at 12:40
  • @alok, just give Spivak or Apostol a try. If it makes sense to you, great. I've seen in the web a pdf version of Spivak's Prologue, which discusses the properties of real numbers. I think it was reading for a Calculus course. – lhf Sep 02 '11 at 12:44
  • I read the first chapter from spivak's text and attemted the very first exercise.I could just end up staring at it for an hour with no solution.Going by the fact that the first question itself is so tough to answer i don't think i can continue reading this "wonderfull" text! I've realised that i'am too dumb to read "hard text's like spivak,apostol,knuth,landau etc.May be i need to stick with standard texts like stewart's calculus for the time being {even that's hard for me at times} and then read tougher texts.What do you say? – alok Sep 02 '11 at 13:32
  • @alok, perhaps you could post a separate question asking for help with that exercise from Spivak. – lhf Sep 02 '11 at 14:07
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If some book states them like that, you should NOT take them on faith, NOR believe that they can get proven. The set of all real numbers is NOT closed under the operation of division, as the above statement of the closure laws indicates, since there does not exist division by 0 on the set of all real numbers.

Doug Spoonwood
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It is perfectly ok to take an axiomatic approach and not prove any of these axioms. You will never know then if the axioms are consistent or not (i.e., whether or not you can derive a contradiction), but that's ok since the best you can ever do is prove the axioms relative to some other model. Usually, one starts with a model of the rational numbers (or the integers) and constructs the reals, and then proves that the reals have the properties you state (and more). A survey of such constructions is given in the real numbers - a survey of constructions (published, Rocky Mountain Journal of Mathematics). You can then ask, well how do we know the rationals (for instance) have the properties we claim they do? well, you can construct the rationals from the integers and ask the same question of the integers, and so on until you reach the bottom.

Ittay Weiss
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  • Actually in this system, you can derive a contradiction. Just consider the conjunction of the statements "For all x, for all y, x + y belongs to the set of real numbers" and "for all x, for all y, x / y belongs to the set of real numbers." Thus, you do know that the set of axioms given by the OP is inconsistent. – Doug Spoonwood Oct 19 '15 at 12:28
  • I assumed OP was referring to the usual second order axiomatization of the reals where no contradiction is known. – Ittay Weiss Oct 19 '15 at 12:32