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

Question

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?

Answer 1

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 $x$.

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 $\sqrt{2}$.

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.

Answer 2

$\pi$ 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.

Answer 3

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.

Answer 4

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.

Answer 5

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.