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 = **1.4142**8 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.

This comment has been removed by the author.

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

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

(which is what one would expect)

Bhuvanesh:

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

Eddie

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