An animation of numbers 999...999^{n}.

Daisuke Minematsu

Satoshi Hashiba

Ryohei Miyadera Kwansei Gakuin University.

We have presented a beautiful figure made of 999...9^{n} and its theoretical background in Miyadera [1] ( Visual Mathematics , Vol 6, No.2, 2006 ).

Here we are going to show a beautiful movie made of these numbers.

You can appreciate the beauty of these numbers very much when we make a movie using them.

Example 1. We are going to study the numbers with the form of x^{y} with x = 99...99 and an integer y.

We are going to make x bigger and y smaller while keeping the size of the number x^{y} almost the same as the following table shows.

We are going to make a movie using these numbers x^{y} in this table.

We express these numbers as matrixes whose length of row is 40, and color each numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 0 with different colors. We express all these numbers x^{y} as 40×38 matrix by cutting off some part of the digits of these numbers. Then we can easily make a movie using these numbers.

The movie starts with Picture 1, and ends with Picture 2. Please click the underlined text below to start the movie.

Picture 1.

V

Picture 2.

Remark.

The figure in Picture 2 is almost identical to the graph of the function

y = - (Log_{10}x + (1-x)Log_{10}(1-x)), where the x-coordinate is vertical.

As to this fact see Theorem 1 in Miyadera, Minematsu and Hashiba [1].

The movie looks more beautiful when we use bigger numbers.

Example 2. This time we are going to make a movie using bigger numbers. We start with 99999^{902}, and end with 999999999999999999999999999999999999999999999999999999999999999999^{68}.

The movie starts with Picture 3 and ends with Picture 4. Please click the underlined text below to start the movie.

Picture 3.

Picture 4.

Remark.

The figure in Picture 4 is almost identical to the graph of the function

y = - (Log_{10}x + (1-x)Log_{10}(1-x)), where the x-coordinate is vertical.

References.

[1] R.Miyadera, D.Minematsu and S.Hashiba, A beautiful figure made of 999...9^{n}, 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

[2] R.Miyadera, D.Minematsu and S.Hashiba, A beautiful figure made of 9999...999^n, Wolfram Information Center,

http://library.wolfram.com/infocenter/MathSource/6194/