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This question was a part of a discussion at an interview.

QUESTION: How can you determine rigorously if $e$ or $\pi$ are points on the real line?

MY OPINION: They should be, since they are defined to be real and irrational in nature. But then again they are transcendental numbers. I have read about the density of rational and irrational numbers in the real line, not about transcendental numbers. Also $e$ and $\pi$ have their values as per strict definitions imposed on them.

Can anyone state a theorem or some lemma which can solve this problem? Or any other idea is also welcome.

EDIT: For further clarification of my question, I wanted to actually ask that $e$ or $\pi$ are real numbers, no doubt but are there distinct points on the line which represent them? I mean, can you pick out those points that this and this are the points. The given answers have shown that they are there on the line. But is it just that their presence on the real line can be established but the exact location of the points cannot be proved?

Aloizio Macedo
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SchrodingersCat
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    What do you mean by "points on the real line"? All real numbers are points on the real line. That's why it's the real line – Stella Biderman Dec 25 '15 at 07:28
  • A point on the real number line. – SchrodingersCat Dec 25 '15 at 07:28
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    What are your definitions of $e$ and $\pi$? Pretty much every definition I've ever seen defines them to be real numbers having certain properties (then you have to prove that such a real number exists, but with such a definition it is meaningless to talk about them existing as anything other than real numbers). – Eric Wofsey Dec 25 '15 at 07:28
  • For $e$, the very definition of $e=\lim_n(1+\dfrac{1}{n})^n$ shows that $e$ is a real number. Because the real line is closed, and the limit of the sequence exists, hence $e$ is in $\mathbb R$. – Landon Carter Dec 25 '15 at 07:29
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    @SchrodingersCat In your view, what would it mean for $e$ to not be on the real line? What are examples of numbers that you know that aren't? Also, $\pi$ and $e$ are typically never defined to be irrational; their irrationality is proven following their definitions. – Erick Wong Dec 25 '15 at 08:03
  • @SchrodingersCat Could you tell us the purpose of the interview? This information can help us to think about what would be an expected answer. – Pedro Dec 25 '15 at 10:28
  • @ErickWong Your question wasn't addressed to me, but complex numbers with non zero imaginary part are examples of numbers that aren't on the real line. – Pedro Dec 25 '15 at 10:35
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    What is your definition of "the real line"? –  Dec 25 '15 at 17:38
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    @SchrodingersCat After the edit, your question still makes no sense. The location of the point $\pi$ is at exactly $\pi$. You should ask yourself what sort of numbers you think this question applies to: can $1$ be located "exactly"? What about $1/3$? What about $\sqrt{2}$? What about $\sqrt[3]{2}$? What is it precisely that makes $\pi$ and $e$ less certain in your mind? – Erick Wong Dec 26 '15 at 08:36
  • The question was not "Can you rigorously determine what their values are" but "Can you rigorously determine they are real". To which the answer is "No. Rigor isn't necessary (or even possible).." – fleablood Dec 26 '15 at 16:53
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    @fleablood This was quite the thing I was asking. I don't know what made others feel that the question is unclear. The rigor part was perhaps overlooked by all. – SchrodingersCat Dec 26 '15 at 16:58
  • @pedro. The complex numbers are not ordered. pi is the radius of a real circle's (one with a positive radius) circumference to radius. These are are orderable values and thus the result is a measurable quantity. The reals have the lower bound property and thus all orderable values are reals, so pi is. The only issue is whether pi is well defined and constant. The axioms of geometry (strengthened by calculus) confirm this. Similar with e. – fleablood Dec 26 '15 at 17:02
  • It's a frustrating question because it's either subtle and sly about whether you know your theory and definitions backwards and forward or it's a ham-fisted meaningless question. – fleablood Dec 26 '15 at 17:08

5 Answers5

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The series $$ \pi=\lim_{n\to\infty}4\sum_{k=0}^n\frac{(-1)^k}{2k+1} $$ converges by the Alternating Series Test. Furthermore, as shown in this answer $$ e=\lim_{n\to\infty}\left(1+\frac1n\right)^n $$ is an increasing sequence bounded above by $4$.

Therefore, both numbers are limits of rational numbers, and as such are real numbers.

Community
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robjohn
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The real numbers are defined to be an extension of the rationals that has the least upper bound property. Thus for something to be determined to be a real number you need to show it is consistently definined and can be approximated. To be glib all you have to do is show something exists at all.

This is the gist of the question and should be able to be expanded upon by the interviewee. Why are the rationals arbitrarily close so that"tacking on" the least upper bound problem makes the resulting reals complete; what does that actuall mean, anyway?

$e = lim (1 + 1/n)^n$ can therefore be verified to be real simply by noting {$(1+ 1/n)^n$} is increasing and bounded. That's it. That's all you need to show. Such a sequence has a real least upper bound. That's $e$. End of story.

Showing $\pi$ is real is a matter of showing $pi$ exists at all., that the ratio between a circle's circumfirence and its diameter is consistant and constant for all circles, is much harder. But if it is, it has to be a real number because there isn't anything else it can be. It's certainly not a yellow elephant, after all. But you do have to show it actually exists (as oposed to , say, "the ratio of rectangles diagonal to it's base"). I'd actually have a real tough time if I were put on the spot to sure that is a well defined concept.

fleablood
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$\pi$ and $e$ are Real Numbers. Since the Real Number Line is defined to be a line made up of the Real Numbers under the usual ordering, then of course they are on it. This is only possibly an interesting question if your definition of a number doesn't necessitate it being in $\mathbb{R}$. In that case, proving it is in $\mathbb{R}$ proves it is a point on the Real Line.

Stella Biderman
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  • What about "the number which squared equals -1" or "the largest natural number" or "the ratio between the two legs of a right triangle"? – fleablood Dec 25 '15 at 18:59
  • Those labels don't refer to anything, in the sense that the set ${x\in\mathbb{R}:P(x)}$ is empty when $P$ is any of those properties. The definition of $\pi$ and $e$ make it clear (IMO) that they are real numbers that exist. The ratio of a circle's diameter to it's radius is approximated by a sequence of rationals. The limit definition of $e$ is a finite limit of real numbers. – Stella Biderman Dec 25 '15 at 20:29
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    That's partially my point. In a way the question has very little to do with $\pi$ or $e$ but to do with whether the interviewee knows (and more importantly can express) what a real number actually is. That R is complete means anything that "should" be a real number is a real number. Which is subtlety that makes this question intriguing. Except the use of the word "rigorous" indicates that possibly the interviewer didn't know what s/he was talking about. There is nothing rigorous in showing $/pi$ or $e$ are real. The mere fact they exist, proves that. – fleablood Dec 26 '15 at 16:04
  • ${z \in \mathbb R: P(x)}$ being empty: that's true for root(-1) and largest natural, but ratio between legs of a right triangle is far from empty. It's infinite. That's the problem. To be a number such a set must have only one unique element. (although what I was trying to get at with the largest natural was such a conditions was inconsistent; maybe I didn't succeed) – fleablood Dec 26 '15 at 16:10
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Because of your edit, I will consider the following question:

Can we determine the exact location of the points $\pi$ and $e$ in the real line?

Assuming we know where the number zero is in the line, we can determine the exact location of a number $x$ if we can construct a segment with length $x$.

enter image description here

For example, we can determine the exact location of $\sqrt{2}$ because we can construct a line segment with length $\sqrt{2}$. For this, draw a square with side $1$ and then draw its diagonal. The diagonal is a segment with length $\sqrt{2}$.

enter image description here

So, the answer for our question is: No, we can't determine the exact location of $\pi$ and $e$ because we can't draw segments with these lengths.

But why not? Because $\pi$ and $e$ are not algebraic numbers while a constructible number must be algebraic.

Pedro
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This is a great interview question since it's a question that sounds obvious, but requires some fairly deep thinking to answer in a fully satisfactory way.

Take $\pi$ for instance. Usually it's defined as the ratio of a circle's circumference to its diameter, for a circle in the Euclidean plane. Since the quotient of two (nonzero) real numbers is real, $\pi$ must be real.

But hold on. How do we know that the ratio is the same for any two circles in the plane? Maybe if I draw bigger circles, or center them at different places, I get different numbers, so that $\pi$ is not well-defined. How do I prove that the size and location of the circle doesn't matter?

Maybe I take a different tack: I remember that $\tan \pi/4 = 1$ and define $\pi = 4\arctan 1$. And $\arctan(x)$ I can define in terms of a series that I can prove converges.

But wait, how do I know that $\arctan$, as given by this series, has anything at all to do with circles and circumferences? Am I sure that establishing the relationship between this formal series and circles won't need, somewhere, to assume that $\pi$ (the ratio) is a well-defined real number? It's not so obvious.

Maybe I go back to circles and take a second approach. I can underestimate the circumference of a circle by inscribing a polygon, and overestimate it by circumscribing a polygon (that this is true is maybe itself not so obvious). Then I take the limit as the number of sides increases of both estimates and show that $\pi$ gets squeezed towards a well-defined real number.

user7530
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    The details of the calculation of circumference of a circle don't matter, as long as it is done in a way that is invariant under translation and scales linearly with the radius. Since any circle can be mapped to any other circle by a suitable translation and scaling, their ratios of circumference to diameter are the same. – Robert Israel Dec 25 '15 at 08:05
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    @RobertIsrael Sure. How do you know the circumference is invariant under translations and scales linearly with radius? – user7530 Dec 25 '15 at 08:45
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    Because you define it in a way that is invariant under translations and scales linearly... – Robert Israel Dec 25 '15 at 20:50
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    Wow I have no idea where all of these downvotes are coming from. Sure, if you assume you have access to the full machinery of differential geometry and analysis the question is trivial. The point of the question is to justify how you know the "obvious" fact that $\pi$ is well-defined and real. – user7530 Dec 26 '15 at 16:10
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    I think this is a good question, but that the interviewer used the word "rigorous" makes me wonder if s/he know what s/he was talking about. Real numbers are complete. Anything quantifiable and well defined is real. Even to show $\pi$ is well defined isn''t rigorous. It's a matter of matter of identifying one's axiomatic hypothesis. Euclid's axioms say "parts of the same are the same" which circles with same radii have same circumference (else a single circle could have multiple circumferences). Principals of .... to be continued... – fleablood Dec 26 '15 at 16:22
  • ...continued... calculus and infinitisimals says we can measure a circumference as the limit of the perimeter of a regular polygon. That isn't "rigorous" and can't be, nor is arguing why we believe it "rigorous". – fleablood Dec 26 '15 at 16:24