Tuesday, October 2, 2012

Distribution of Digits for π, e, Euler's Constant, and √2

Here are four famous numerical constants, extended to 50 decimal places: π, e, γ, and √2. For each constant, the approximation will be listed and a histogram of each of the 50 decimal places and the integer part (51 digits in total) will be presented.

Pi
π ≈ 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510

Digit Distribution: (digit: number of occurrences)
0: 2
1: 5
2: 5
3: 9
4: 4
5: 5
6: 4
7: 4
8: 5
9: 8

e

e ≈ 2.71828 18284 59045 23536 02874 71352 66249 77572 47093 69995

Digit Distribution:
0: 3
1: 3
2: 8
3: 4
4: 5
5: 6
6: 4
7: 7
8: 5
9: 6

γ (Euler's Constant)

γ ≈ 0.57721 56649 01532 86060 65120 90082 40243 10421 59335 93992

Digit Distribution:
0: 9
1: 5
2: 7
3: 5
4: 4
5: 6
6: 5
7: 2
8: 2
9: 6

√2

√2 ≈ 1.41421 35623 73095 04880 16887 24209 69807 85696 71875 37694

Digit Distribution:
0: 5
1: 5
2: 4
3: 4
4: 5
5: 4
6: 6
7: 6
8: 7
9: 5

Of the four constants presented, √2 has the most even distribution, at far as the first 50 decimal points are concerned.

Fun Fact: 99/70 gives √2 accurate to four decimal places.
99/70 = 1.41428 571428...

Resources

π, e, γ

Zwillinger, Daniel. CRC - Standard Mathematical Tables and Formulae 32nd Edition. CRC Press, Boca Raton, FL. 2012

√2

√2 Wikipedia: Retrieved October 2, 2012.


Enjoy! This is something I wanted to do for a while, as I am fascinated by numerical constants.

Eddie


This blog is property of Edward Shore, 2012.

4 comments:

  1. This comment has been removed by the author.

    ReplyDelete
  2. The distribution changes as you increase precision, though. For precision 10^6, for example, the histogram looks fairly flat:

    Histogram[First@RealDigits[N[Pi, 10^6]], {1}]

    (which is what one would expect)

    ReplyDelete
  3. Bhuvanesh:

    The distributions should flatten out for all the constants as more decimal places are considered.

    Eddie

    ReplyDelete
  4. I couldn't help but notice that γ looks a lot like √3/3! The first three digits of γ are 0.577 and the first three digits of √3/3 are also 0.577.

    ReplyDelete

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