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Since there are holes in the $\mathbb{Q}$ we have constructed $\mathbb{R}$ in order to fill in these holes. But I was wondering why we don't use $\mathbb{C}$ or some other number system that is even larger as the main number system. I was also wondering what are the implications of discovering a larger number system. Could someone briefly explain it to me?

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    $\begingroup$ What's the 'main number system'? $\endgroup$
    – Stanley
    Jun 19 '15 at 14:16
  • $\begingroup$ What I mean by "main number system" is the one that most people are familiar with. $\endgroup$
    – user112267
    Jun 19 '15 at 14:19
  • $\begingroup$ Ultimately there are systems of "numbers" going in different directions in which none are naturally contained in any other, such as the $p$-adic numbers for different primes $p$ or the finite fields $\mathbf Z/(p)$ for different primes $p$. (And finite fields are very useful in coding theory, so it's not as if these constructions are pure abstractions.) That you only conceive of number systems inside the complex numbers is because the kind of education in math that many people receive only operates within them. Not by a long shot is the path "up to the complex numbers" the only way you can go. $\endgroup$
    – KCd
    Jun 19 '15 at 19:37
  • $\begingroup$ You define "main number system" as "the one that most people are familiar with." Do you think most people are familiar with complex numbers? $\endgroup$ Jul 16 '15 at 23:18
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There is no "main" number system. We study the reals to say things about the reals, we study the complex numbers to say things about the complex numbers, we study the integers to say things about the integers, and so on. All of these sets are interesting and worth studying in their own right.

The reals enjoy a large application in engineering, along with the complex, but that doesn't make them "special" or the "main number system."

A larger number system would not be "discovered", it would be invented. An example of such a number system are the real Hamilton quaternions.

But again, there is no "main number system." There are many well defined sets of objects, and they are studied. That's it.

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  • $\begingroup$ Then what is special about the rationals containing the integers as a subfield and the reals containing the rationals as a subfield and so on? For some reason it leads me to believe that there is some sort of superior or better number system if it's really large. I'm just wondering if there's some sort of unifying number system that we can use for everything. $\endgroup$
    – user112267
    Jun 19 '15 at 14:27
  • $\begingroup$ What do you mean "use for everything?" $\endgroup$ Jun 19 '15 at 14:34
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    $\begingroup$ The discovered or invented matter depends on your math philosophy (+1 though) $\endgroup$
    – wythagoras
    Jun 19 '15 at 18:10
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    $\begingroup$ @user112267 in pure math there can't be a system that we can use for everything because every different set has it particularities which are interest of study. In Engineering we use Real or Complex depending on what branch we are dealing with and in theoretical psychics we use hypercomplex numbers and so on. Real are just the one with more applications and the most easy to explain to a large public, that's why it is "the main number system". $\endgroup$ Jun 19 '15 at 19:13
  • $\begingroup$ @wythagoras Yes, you're right. I didn't mean to stir up that issue, I was really just trying to drive home the fact that new number sets being (invented/discovered) is limited only by our imagination and consistent definitions, and the newly (invented/discovered) set may or may not have any connection to the natural world. $\endgroup$ Jun 23 '15 at 17:04
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As the previous answer said, it all depends on what properties you care about. The integers are best for modelling discrete things; the rationals are great for doing arithmetic; the reals and/or complex numbers are needed when you want to do analysis/calculus.

The reals vs. the complex numbers have different benefits and hence different applications. Since $\mathbb{C}$ is algebraically closed (every polynomial has a root), algebra is really easy. The rich structure also lets you do complex analysis which enjoys a number of nice properties absent in the real case (see spaces with complex coordinates for some remarkable examples).

So, why do we bother with $\mathbb{R}$ at all? Well the obvious answer is that real numbers are everywhere in physics and engineering and so forth. The real line fits an intuitive idea of the world we live in in a way that the complex does not. Perhaps this is why most of differential geometry and related fields focuses on spaces with real coordinates (although people do study spaces with complex coordinates ). Mathematically, perhaps the most important advantage that the real numbers have over complex numbers is that they are ordered. This lets them be useful as a measurement of magnitude which is vital in many of their real world and mathematical applications. In fact, they are the unique complete ordered field, where complete means "no holes", unlike the rational numbers, and ordered field means that the order plays nicely with the algebraic operations (for example, if $a>b$ then $a+c > b+c$).

Edit: as Sam brought up in his comment, the complex numbers fail the criterion of being ordered in a strong sense: not only is there no obvious or canonical order on them, but an ordering that plays nicely with the field operations is impossible. The linked Wikipedia article on ordered fields explains this, but it is instructive to work it out on your own. Try to find a contradiction by assuming $i > 0$ or $i<0$ and proving somehow that $-1 > 0$.

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    $\begingroup$ Just wanted to say that this answer is the correct answer, not the accepted one. The reason that we use them is that they are the unique complete ordered field - in particular, this explains why they are often used rather than the complex numbers which have many great properties but unfortunately are not ordered. $\endgroup$ Jun 19 '15 at 18:50
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Note that larger number systems are known, for example the surreal numbers (as well as the superreal and the hyperreal numbers, which are subsets of the first), which also include the infinitesimal numbers as well as the transfinite ordinal numbers, as well as variations of them, for example $\omega+1.56$.

Why do we almost never use them?

  • In elementary algebra, we don't use them since $\mathbb{C}$ is algebraically closed so they won't appear as solution.
  • In calculus infinitesimal numbers are sometimes used, but they shouldn't be treated as a (sur)real number. As well as $\pm \infty$ which shouldn't be treated as the surreal number $\pm \omega$.
  • In set theory, where ordinal and cardinal numbers are frequently used, the infinitesimal are not used and things like $\omega^4+2.34$ aren't used either. Therefore it is not useful to consider surreal numbers, and they behave differently (see comments).
  • In other fields of mathematics where I am aware of, it surely doesn't make sense to talk about this, because infinitesimal numbers or transfinite or infinite numbers are not used.

I believe that the superreal and the hyperreal numbers are used in abstract algebra.

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  • $\begingroup$ To say that the surreals contain the ordinals is a wee bit misleading, since they do not agree with the ordinals on neither the arithmetic, nor the order (the field has cofinality $\rm Ord$, sure, but it's not well-ordered by itself). The same can be said about cardinals, although you made no such claim here, so I won't say it here. $\endgroup$
    – Asaf Karagila
    Jun 23 '15 at 18:43
  • $\begingroup$ The statement about infinitesimals is a bit misleading; standard analysis doesn't have them, while nonstandard analysis has tons of them. $\endgroup$
    – Ian
    Jun 23 '15 at 18:44
  • $\begingroup$ @AsafKaragila Can you clarify what you meant? Do you mean that $\omega+1$ as ordinal is not the same as $\omega+1$ as surreal number? $\endgroup$
    – wythagoras
    Jun 23 '15 at 18:59
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    $\begingroup$ I mean that, I agree, of course, that there is a continuous embedding of $\rm Ord$ into the Surreals. But $1+\omega=\omega+1$ for the surreals, and that is not true in the usual ordinal arithmetics. So saying that the field includes all the ordinals is a bit misleading, since "the ordinals" come with addition, multiplication and exponentiation, and the surreals disagree with the ordinals on all three operations. $\endgroup$
    – Asaf Karagila
    Jun 23 '15 at 19:14
  • $\begingroup$ @AsafKaragila Yes, okay, I see. $\endgroup$
    – wythagoras
    Jun 23 '15 at 19:15

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