Plot a graphic of a function with domain of rational numbers?

Questions
,
1

I would like to present to my students of the functions Exp[x] near x=Sqrt[2], letting x vary only over the rational numbers. The reason is to be able to present to them the notion of limit and continous functions.

2

I’m not sure exactly what kind of output you would like to get here. The rational numbers are dense in the reals, so you would be plotting your function for 14/10 and 15/10, but then also for the midpoint 29/20, and the midpoint between that and the boundaries, and the midpoints between midpoints. Even so, that would only give a subset of the rationals equivalent to the set of binary fractions.

I guess you could use ListPlot, with the argument of ListPlot being some set of rational numbers, maybe even ordering them by where they appear in the famous snake-like path through the rationals

1/1 2/1 3/1
1/2 2/2 3/2
1/3 2/3 3/3

Then the code would be

ListPlot[ Table[{x,Exp[x]}, {x, {1,2,1/2,1/3,2/3,3/2,3}}] ]

Do you have a more detailed description of what you have in mind? Maybe with an example?

3

Explaining better: I want to give a sequence o rational numbers, x_n converging to sqrt[2] and I want to be able to plot in a nice visible way the correponding sequence of y_n=Exp[x_n];
Better yet a want to be able to plot y_n as a function of x_n in order to visualize where y_n is converging to.

4

Explaining better: I want to give a sequence o rational numbers, x_n converging to sqrt[2] and I want to be able to plot in a nice visible way the correponding sequence of y_n=Exp[x_n];
Better yet a want to be able to plot y_n as a function of x_n in order to visualize where y_n is converging to.

Your code above may work but I want to be able to get to a accuracy or precision level “as high as I wish” …

5

Okay, if you specifically want, not just rational numbers, but successively better rational approximations to Sqrt[2], then you could use

Table[Rationalize[N[Sqrt[2], n], 0], {n, 1, 20}]

To break this down

N[Sqrt[2], n]

returns a floating-point approximation of Sqrt[2] to ‘n’ digits.

Rationalize[..., 0]

returns the rational number closest to the input, with ‘0’ indicating the error tolerance.

Table[..., {n, 1, 20}]

returns the input for each value of ‘n’ from 1 to 20.

Tying it all together, we can put this list of rational numbers in as an argument to ListPlot:

vals=Table[Rationalize[N[Sqrt[2], n], 0], {n, 1, 10}];
ListPlot[
    Table[
        {x, Exp[x]}, 
        {x, vals}
    ]
]

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6

Thanks Duncan !
It kind of worked…but the problem is the the accuracy or the precision to make a visiable plotting with ListPlot…What do I have to do for the machine to distinguish number in 20th decimal place, in the plotting comand, I mean.

7

I am afraid I don’t really understand what you want. Perhaps this? The last points virtually coincide.

n = 5;
x[1] = 1;
x[n_] := 1/2 (x[n - 1] + 2/x[n - 1])
rn = Table[ x[j], {j, n}]  (* rational numbers converging to sqrt (2) *)
pp = {#, Exp[#]} & /@ rn
pp // N
ListPlot[pp, PlotStyle -> PointSize[.02]]
ListLinePlot[pp, Epilog -> {PointSize[.02], Point /@ pp}]
8

Thanks Hans.
Kind of it but I thought about using something like Manipulate to be able to zoom in near and near to be able to “see” the sequence of Exp[x_n] getting near and near the value of Exp[Sqrt[2]]

9

Perhaps your “zoom” can be achieved like this (I would be careful to make sure that a < b ; but in fact this seems not to be necessary )

p1[a_, b_] :=  ListLinePlot[pp, Epilog -> {PointSize[.02], Point /@ pp},   PlotRange -> {{a, b}, All}]

p1[1.4, 1.51]  (* check other values of a and b *)

or

Manipulate[
 p1[a, b],
 {a, 0, 1.4}, {b, .5, 2}]
10

Thanks for the hints guys.
I think my question really can be put as:
How can I Plot the graphic of f(x)=2^x, for instance, in the interval (N[Sqrt[2],n],N[Sqrt[2],n+1]) ?
take n=10 for example…

11

Perhaps like this

fx[n_] := Module[{a},
  a = Drop[RealDigits[N[Sqrt[2], n]][[1]], 1];
  1 + FromDigits[a]/10^Length[a]]

Then

n = 5;
x1 = fx[n]
x2 = fx[n + 1]
Plot[2^x, Evaluate[{x, x1, x2}]]
12

Thanks Hans,

Yet it does not work for n >12…
I need it to work for big n…20 or more…

13

As I understand your

fx[n_] equals to

Rationalize[N[Sqrt[2], n]]

except that you have a rational number…Yet, for big number the machine does not recognize
fx[n} and fx[n+1] as distinct numbers…

14

Yes, fx[n_] is basically equivalent to Rationalize, albeit a bit different.

What about

max=40;
fx[n_] := Module[{a},
  a = Take[RealDigits[SetPrecision[N[Sqrt[2]], max]][[1]], 2 ;; n];
  1 + FromDigits[a]/10^Length[a]]

But does that really make sense? For these small differences your function will be a straight line.