A very beautiful number made of  99999  ⋯(99999999^n)

Daisuke Minematsu
Satoshi Hashiba
Ryohei Miyadera       Kwansei Gakuin high school.

Today we are going to introduce a very beautiful number.

When mathematicians talk about a beautiful number, they usually mean that the number has a beautiful mathematical property.

Here we are going to introduce a number that is beautiful from the view point of an artist.

Example 1. If you calculate the number (9999999999999999999999999^69)  express it as a matrix whose length of row is  25, then you get the following figure. The mathematical structure of this number is studied in Theorem 1.

Mathematica code for Example 1.

Clear[a, c, x, y, yy, n, m, 
        k];
n = 25; 
    yy = (10^n) - 1;
a = 
        IntegerPart[4000/(Log[yy] // N)];
c = 
        yy^a;
b = 
        IntegerDigits[c];
Show[Graphics[
          Evaluate[
Table[
              Table[{Hue[Part[b, (y + 25  x)]/9.7], 
                  Text[Part[
                      b, (y + 25  x)], {1.3  
                        y, -x}]}, 
{y, 1, 25}], {x, 0, 
                68}]]],AspectRatio -> Automatic,Background -> RGBColor[0, 0, 0],ImageSize ->{270, 540}]

Example 2.  The following is the graph of the function  f (x) = -((Subscript[xlog, 10]) 
                  x + (1 - x) (Subscript[log, 10]) (1 - x)).  Please compare this to the figure in Example 1. It is not difficult to see the similarity between these two.

Mathematica code for Example 2.

Plot[-(x*Log[10, x] + (1 - x)*Log[10, 1 - x]), {x, 0, 1}, 
      PlotRange -> {0, 1}, PlotStyle -> RGBColor[1, 0, 0], 
      Background -> RGBColor[0, 0, 0]]

Theorem 1.  We expand ((((10^k)-1))^n)and express it as a matrix whose length of row is k. If we make n ⟶ ∞, then the figure made of numbers 1,2,3,4,5,6,7,8 is getting nearer and nearer to the graph of the function y = – (x(Subscript[log, 10])x + (1–x)(Subscript[log, 10])(1–x)), where the x-coordinate is vertical.

For a proof see Theorem 1 in Miyadera [1].

Remark.  This result is closely related to the research about the expansion of (((a + b))^n). See Miyadera [1], [2] and [3].

Example 3. We can make a beautiful movie using the figure in Example 1. Please look at the following table. We make the number X = 999...999 bigger, and at the same time we make the number Y smaller. In this way we keep the size of the number (X^Y) almost the same.
When we make a movie, we cut some part of the digits of the number.

Mathematica calculation for the following table.

Clear[aa, n, X, Y];
aa = 
        Table[{x = (10^n) - 1, y = IntegerPart[4000/Log[x] // N], 
            Length[IntegerDigits[(x^y)]]}, {n, 1, 
            25}];
\(FrameBox[
        GridBox[Join[{{"number X", "number Y", \*""the number of Digits of (X^Y)""}}, aa], RowLines -> True, 
          ColumnLines -> True]] // DisplayForm\)

We make X bigger and Y smaller, and we make the graphics of the each number (X^Y). Then we get the following movie.

Mathematica code for Example 3. You can make a movie by the following code. We omit the output because of the size of it.

Clear[a, c, b, x, y, n, m, k];
Do[yy = (10^n) - 1;
a = IntegerPart[4000/(Log[yy] // N)];
c = yy^a;
b = IntegerDigits[c];
bb = Length[b];
Show[Graphics[{Text[
                    SuperscriptBox[yy " ", a] // DisplayForm, {90, 10}, 
                    TextStyle -> {FontSize -> 14}], 
                  Evaluate[
Table[
                      Table[{Hue[Part[b, (y + 25  x)]/9.7], 
                          Text[Part[
                              b, (y + 25  x)], {7  y, -6 
                                x}]}, 
{y, 1, 25}], {x, 0, 
                        64}]]}],AspectRatio -> Automatic,Background -> RGBColor[0, 0, 0],ImageSize ->{270, 540}],{n, 1, 25}]

Example 4 (An open problem). Let x =

99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999900000000000000000000000000000000000000000000000000000000000000000000000000000000899999999999999999999999999999999999999999999999999999999999999999999999999999, and

let y = 11. If you calculate the number (x^y)  and arrange it as the shape of a rectangle with the width of 80, then you get the following figure.
To study the mathematical structure of this number is an open problem.

Mathematica code for Example 4.

r = 8; s = 80; 
    n = 26;
yy = 
        10^(12  n + 2) - 10^(6  n + 2) + r*10^(3  n - 1) + 
          10^(3  n - 1) - 1;
a = 
        IntegerPart[8000/(Log[yy] // N)];
b = 
        IntegerDigits[yy^a];
bb = 
        Length[b];
Print[
        yy];
Print[a];
Show[Graphics[
          Evaluate[
Table[
              Table[{Hue[Part[b, (y + (s)*x)]/9.7], 
                  Text[Part[b, (y + (s)*x)], {y, -x}, 
                    TextStyle -> {FontSlant -> "Plain", 
                        FontSize -> 6}]}, 
{y, 1, 
                  s}], {x, 0, IntegerPart[bb/(s)] - 1}]]],AspectRatio -> Automatic,Background -> RGBColor[0, 0, 0],ImageSize ->{640, 350}]

Reference.

R.Miyadera, D.Minematsu and S.Hashiba [1], A beautiful figure made of 999...9, Visual Mathematics , Vol 6, No.2, 2006, Mathematical Institute of the Serbian Academy of Sciences and Arts.

http://www.mi.sanu.ac.yu/vismath/miyadera/999.html

R. Miyadera and Y. Kotera [2], Una Bella Curva Che Troviamo in Connessione con lo Sviluppo di  (((a + b))^n) Archimede 2/2005.

R. Miyadera [3], The curve of the expansion of (a + b)^n, Wolfram Information Center,

http://library.wolfram.com/infocenter/MathSource/5743/