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For example in my Analysis class the professor showed $\sqrt{2}$ exists using Archimedean properties of $\mathbb{R}$ and we showed $e$ exists. I want to know why it's important to show their existence?

Martin Sleziak
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Laid E. Luck
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    If you show $\sqrt{2}$ exists you actually mean, that you show that there exists $p$ such that $p^2=2$ and you will define $\sqrt{2}:=p$. – Max Oct 17 '15 at 22:33
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    What definition of real numbers are you using? – Tim Raczkowski Oct 17 '15 at 22:41
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    If it seems obvious that "$\sqrt{2}$ exists" in the sense that there exists a real number $a$ with $a^2=2$, why isn't it also obvious that there exists a real number $b$ with $b^2=-1$? – R.. GitHub STOP HELPING ICE Oct 18 '15 at 04:00
  • @Max: You can't just define $\sqrt{2}:=p$, you have to establish uniqueness first. In this case there are two numbers with that property, one negative and one positive, and we define $\sqrt{2}$ to be the positive one. – Stefan Oct 19 '15 at 12:50
  • IMO @Max's comment is the correct answer to this question. It is a misconception to think that existence is a property that numbers may or may not have. Before you even write $\sqrt{2}$ you must define what you mean by that. The proof shown by the professor has to be done before (or be a part of) this definition. – Stefan Oct 19 '15 at 12:56
  • @StefanWalter surely the solution is not unique and one has to decide (for example by convention or what ever) which solution to take, you are completely right about that detail. One could also use the notation "$p\in\sqrt{2}$" and call "$p=\sqrt{2}$" slight misuse of notation :-) ) – Max Oct 19 '15 at 14:54
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    You need to make sure that the definition is consistent. For instance, "the largest real $r$ such that $r^2<2$" does not exist. –  Oct 29 '15 at 20:09
  • @StefanWalter: Existence isn't automatic by simply writing the symbol $\sqrt{2}$. After having deliberated on this for a while, the assertion of the existence within $\mathbb{R}$ of $\sqrt{2}$ feels like what it's really claiming is the existence within $\mathbb{R}$ of an element with 'the $\sqrt{2}$ property', to wit, being positive and squaring to 2 (even though this is easily seen to be satisfied by at most one real number, properties in general need not (uniquely) define an object, tying in with Max's $p \in \sqrt{2}$ comment.).... – Vandermonde Nov 25 '15 at 01:55
  • ... It is a priori conceivable that no such real number exists, which is why proof of existence is important, but I view it as perfectly meaningful (and natural) to speak of, say, there being no $\sqrt{2}$ in $\mathbb{Q}$ or of the nonexistence of unicorns. So one needs to first define the property before meaningfully discussing it, but the existence of a corresponding tangible object is in no way a prerequisite to talking about the property and one can still mean something very definite by a statement containing $\sqrt{2}$ or similar. – Vandermonde Nov 25 '15 at 02:09
  • @Vandermonde: "the assertion of the existence within $\mathbb{R}$ of $\sqrt{2}$ feels like what it's really claiming is the existence within $\mathbb{R}$ of an element with 'the $\sqrt{2}$ property'" I agree, but I don't think the OP understood that. If it's clear what you mean, you can certainly talk of the existence of $\sqrt{2}$. I still don't like it, but that's a matter of taste. – Stefan Nov 25 '15 at 14:22

7 Answers7

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Before I eat a sandwich it's a good idea to make sure it exists, right? Well, it obviously exists if I can see it, so I may not be aware of the need to establish existence, but I had established existence.

In other situations this becomes less silly, particularly in cases where the existence of whatever it is you wish to study or to use is not evident. Situations in real life are plentiful: Before you set out to study the supernatural you better establish that a supernatural phenomenon exists (famously, lots of research went into studying the supernatural without any established supernatural phenomenon in existence), before you believe the claims a book makes about the wishes of that or other god, you had better establish if god at all exists, before you use the psychic abilities of a medium in order to locate the whereabouts of a missing person, you had better establish the existence of the medium's claimed abilities. (James Randi dedicates a large portion of his life debunking such nonsense, all resulting from not being careful with checking that something exists before using it.)

In mathematics then it's not different. Before you can speak of the square root of $2$, you'd better establish that it exists. Similarly for other numbers. It's as simple as that. It's not a good idea to just accept that some number exists. After all, I can claim that there exists a real number $x$ which solves the equation $x^2+1=\rm {monkey}$, would you accept that? Probably not. What about my claim that there exists a real number $x$ solving $x^2+1=0$, do you accept that? Why not? So finally, do you just accept that there exists a real number $x$ solving $x^2-2=0$? Do you do that based on blind faith? Do you see the added value now in the proof that such a number actually exists?

A more purely mathematical answer: Some constructions of the real numbers lend themselves more easily to constructing certain numbers. The ability to construct a certain number in a given system may serve to show the utility of the construction. Thus, other than the general importance of establishing that certain numbers we expect to exist actually do exist, a particular proof of the existence of a number may be related to claims of the usefulness of a particular construction of the reals.

David
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Ittay Weiss
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    $\pm\sqrt{\mathrm{monkey}-1}$. Seems legit. – David Richerby Oct 18 '15 at 00:15
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    @DavidRicherby: You discovered what number comes before $\text{monkey}$? Tell me please! I swore I had made good progress on counting backwards from $\text{potato}$ but my instructor says the slow ones will just never be cut out for academia. – Vandermonde Oct 18 '15 at 00:56
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    Before $monkey$ is $monkex$, of course. – user253751 Oct 18 '15 at 02:05
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    A minor correction: it's James Randi, not James Randy. (Though I admit your version is a little more funny...) – Potato Oct 18 '15 at 02:19
  • :D @Potato (thanks!) – Ittay Weiss Oct 18 '15 at 03:19
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    Before taking chemotherapy drugs, you should probably make sure that cancer cells actually exists in your body and that there are enough of them to merit chemotherapy. :-P – Asaf Karagila Oct 18 '15 at 05:19
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    @DavidRicherby: Only if your monkey lives in a bigger field than the one you usually play in. =) – user21820 Oct 18 '15 at 10:22
  • I'd contest the claim that existence of something should be shown before any discussion of said thing. Particularly in mathematics, there have been some very fruitful discussions of things before their existence could be shown. Consider properties of hyperbolic geometry (before its models were found), consequences of the axiom of choice (it's an axiom so proving it will be tough), consequences of P=NP (even though that seems unlikely). There were times when even imaginary numbers would have been considered non-existent, but considering them nevertheless led to the complex numbers we know. – MvG Oct 19 '15 at 09:34
  • @MvG quite true. I did not say that one should not discuss something before that something is shown to exist. Contemplating whether something exists or not is an important thing to do, and part of that contemplation is discussing what might potentially be the consequences of its existence or non existence. What the properties of such a thing may be. Sure, this is quite alright to do. But before you go ahead and 'use' something, it's wise to make sure it exists. Also, I'm saying this in the context of modern mathematics. The illnesses of pre-axiomatic mathematics are behind us now. – Ittay Weiss Oct 20 '15 at 23:45
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You believe $\sqrt{2}$ is a number... so the question is whether or not it's a real number.

If the real numbers didn't have a number whose square is 2, that would be a rather serious defect; it would mean that the real numbers cannot be used as the setting for the kinds of mathematics where one wants to take a square root of $2$.

Basically, things cut both ways; while you could use the argument as justifying the idea of taking the square root of 2 is a useful notion, the more important aspect is that it justifies the idea of the real numbers is a useful notion.

Also, the argument is useful as a demonstrates of how to use the completeness properties to prove things.

Furthermore, it reinforces the notion that this type of reasoning can be used to define specific things. Some people have a lot of difficulty with this type of argument; e.g. "you haven't defined $\sqrt{2}$, you've just defined a way to get arbitrarily close to a square root of 2 without ever reaching it". It's a lot easier to dispel such misconceptions when the subject is something as clearly understood as $\sqrt{2}$.

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    I like to think about this as: a proof is not so much establishing a fact, as establishing a relationship between your proof system and that fact. And "System X can prove fact Y" is sometimes a statement about fact Y and sometimes a statement about system X :) – Ben Millwood Oct 18 '15 at 14:48
  • "You believe $\sqrt{2}$ is a number... so the question is whether or not it's a real number." I don't understand that sentence. What kind of numbers are you referring to? Surely the OP doesn't know about complex numbers yet. – Stefan Oct 19 '15 at 12:43
  • @BenMillwood That's often true whether or not X is a formal system or just a new idea. – Akiva Weinberger Oct 19 '15 at 12:55
  • @Stefan: The kinds of numbers one learned about in an informal way in the course of learning arithmetic/algebra/geometry. If we provide some construction/definition of the real numbers, we should require some convincing that it really does capture this prior notion of arithmetic; even if we know that was the whole intent of the formalism, one should still verify that it actually succeeded. –  Oct 19 '15 at 16:47
  • Understood, thanks. I agree that all these "elementary" notions should be carefully defined with rigorous proofs. However, most students have never thought about if, why and in what manner numbers exist. That's just not an issue at school. So, instead of "we'll now prove something you've always known to be true", I would rather stress that the "existence" of $\sqrt{2}$ isn't some deep fact, but simply a way of talking about, let's say, approximate rational solutions of $x^2=2$. – Stefan Oct 19 '15 at 18:50
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Consider the geometric questions:

  1. Where does the line $y = x$ intersect the circle $x^{2} + y^{2} = 4$?

  2. Where does the line $y = -1$ intersect the parabola $y = x^{2}$?

In 1., the existence of intersection points is equivalent to existence of square roots of $2$ in your number system. If your "universe" uses rational coordinates (which are visually indistinguishable from real coordinates), the answer is nowhere. Particularly, a line through the center of a circle in the rational plane $\mathbf{Q}^{2}$ need not intersect the circle. This is vexing for Euclidean geometry.

In 2., you may be inclined to answer "nowhere", but that would be a limited viewed imposed by your number system. Existence of intersection points here amounts to existence of square roots of $-1$.

The real point (heh) is, these questions are deeply analogous. The types of geometric construction you can make, and the types of solutions you can expect for systems of equations, depend very much on properties of your number system. If you're going to make sweeping claims (the intermediate value theorem, there exists a differentiable function equal to its own derivative, ...), you'd better have some theorems to back you up.

Andrew D. Hwang
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    I think saying "the parabola" there actually does make the answer "none". If you're going to say that "Where the line $y=−1$ intersects the parabola $y=x^2$" has solutions, then to find those solutions, you're looking at something other than "the parabola". – Glen_b Oct 19 '15 at 01:48
  • The locus $y = x^{2}$ intersects $y = -1$ at the points $(\pm i, -1)$; if we don't specify our number system in advance, those two points do lie on "the parabola". The thing is, this type of discussion is generally (tacitly) predicated on using real numbers, leading the OP to ask why it was necessary to prove existance of certain real numbers. If we don't know in advance what number system we "should" use, existence of certain types of number ($\sqrt{2}$ and $e$, say) may be a powerful deciding factor (and non-existence of others, $i$ for example, may not be). – Andrew D. Hwang Oct 19 '15 at 10:45
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    My point relates to you characterizing the object as a parabola. – Glen_b Oct 19 '15 at 10:50
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    Ah, I see. In that case, for "parabola" please read "the conic (with one point at infinity)". :) – Andrew D. Hwang Oct 19 '15 at 21:30
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There is one thing the existing (good) answers don't mention, which is what really the numbers that you are referring to are all about. It starts from the natural numbers, which are supposed to model counting up from 0 indefinitely. We really have no choice but to accept both their existence and properties on faith some way or another (see https://math.stackexchange.com/a/1334753/21820 if interested). But we can build a lot of things using just natural numbers. Firstly we can use a pair of natural numbers to denote fractions. Now what on earth are fractions? They are what happens when you want to consider division of things into smaller pieces or decomposing actions into identical subactions. This we denote using rational numbers. But in the real world, things don't seem to be quite rational at all. The Greeks had an ideal model of geometry that they used to describe the world, where points were infinitely sharp and lines were infinitely thin, and in that model at least there was a length that could be constructed by (ideal) straightedge and compass that could not be expressed as a ratio of two integers. As you probably know, that was $\sqrt{2}$. If the 'tools of perfection' (straightedge and compass representing circle and line) already produces irrational lengths, what more if we go beyond?

But there is an 'easy' answer. In the real world we only care that we can obtain a sufficiently good approximation of quantities. To measure a length of a table, we could use a ruler or measuring tape, and get an approximation to within 1m, or 1cm, or even 1mm. 'Clearly', to get better approximations all we need to do is to subdivide the measuring instrument markings into finer parts. Note that each approximation we make in this way is a rational fraction of a metre. That really is nothing more than the definition of a real number using decimal expansions, where getting a better approximation is simply taking more significant digits. All you have to do now is to define what addition and multiplication mean for decimal expansions and you would have reconstructed the field of real numbers. It then remains to prove all the basic properties that we had all been taught to take for granted, some of which are actually not so easy to prove. (It is now accepted that we cannot even measure any quantity to arbitrary precision, but that only matters at the quantum scale, and is way off-topic.)

So far we can get everything from the natural numbers (plus a few other things, like the existence of sequences of objects, equivalently functions on natural numbers, and so on). Of course, we should now check whether we really have got a real number whose product with itself is $2$. That is what your professor would have been going through the trouble of proving. Similarly for other numbers like $e$ that we use throughout mathematics. It cannot be very well appreciated what these results mean unless we really grasp the amazing fact that you can 'construct' a decimal expansion that when you multiply really has $2$ before the decimal point but all zeros after the decimal point! Did you realize how incredible that was? =)

Community
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user21820
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  • Note that depending on the foundational system being used for your mathematics course, there may be other ways to construct something isomorphic to the real numbers, in other words a totally ordered field where every upper-bounded non-empty set has a supremum, which your professor probably axiomatically defined. That unfortunately is probably why you do not realize how incredible the real numbers are. An excellent exercise is to prove all these axiomatic properties using only the decimal expansion definition. – user21820 Oct 18 '15 at 11:16
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I don't think it's so much showing a number exists as it is showing the nature of real numbers is that all bounded sets have limits. If some numbers didn't exist we wouldn't really care whether $\sqrt 2$ did or not.

What we do care about is that if we have a set of rationals all of whose squares are less than $2$ and a set of rationals whose squares are greater than $2$, then it must be that some real has a square equal to $2$. That that must follow is what is of interest.

Glen_b
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fleablood
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  • For example, it's not in the least importance that .112123123412345123456... exists. But it's very important that I can know it must exists. – fleablood Oct 17 '15 at 23:23
  • This reminds me of a quote -- I might have seen it in a reply to the infamous question about 'Kalle-numbers', can't remember -- along the lines that what is important is not so much numbers/objects themselves as the relations they satisfy; in other words, no one waits or looks for the individual points such as $\sqrt 2$ to come up so they can find a use for them, but they are useful insofar as – Vandermonde Oct 24 '15 at 04:03
  • they form a complete (this axiom being what is actually of use, obviating worries about constructible lengths not existing and continuous functions changing sign but having no root) field and taking some away would lose this property, sort of resembling how one cannot ascribe significance to an individual interval advertised as having $p %$ confidence of covering some unknown or to evaluating a representative of an $L^p$ function at a point but can do so to the distribution from which they are rolled or to anything that isn't sensitive to behaviour on a particular set of measure zero. – Vandermonde Oct 24 '15 at 04:03
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It is important to be sure that things you are dealing with, like $\sqrt{2}$ or $e$, exist, otherwise correct manipulations with them will make no sense. Some examples:

  1. Let $x$ be a solution to $x+1=x$. Then $1=0$.
  2. Perron paradox: Let $N$ be the largest positive integer. If $N>1$ then $N^2>N$, which contradicts the assumption. Hence, $N=1$.

When you are ordering dinner in a restaurant, you better prove first that the wallet exists in your pocket, otherwise you may easily get into trouble.

A.Γ.
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You have to start with definitions of the real numbers and real number multiplication. Then you must use set theory and the rules of logic to construct a set $x$ and prove (a) that $x$ is a real number by your definition, and (b) that $x\cdot x = 2$ by your definition of multiplication. Only then are you entitled to say that there exists a real number $x$ such that $x\cdot x = 2$.

Dan Christensen
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