Digit Occurrences...
...in decimal expansions of irrational numbers.
In these tables, δ (delta) is the largest deficit in the number of occurrences of any digit – how many below an equal share any one digit falls – as a percentage of that equal share.
Unsurprisingly, in all these three cases, there are digits that don’t occur in the first ten digits of the expansions! That is, δ = 100%. For this not to be the case for some particular irrational number, each digit would have to occur exactly once. There must be such numbers, but I haven’t found any example. They’re certainly rare.
In all three expansions, every digit occurs at least five times in the first hundred. There must be examples of irrational numbers for which some digit does not occur in the first hundred, but again, they’re rare and I’ve not found any examples. Very much rarer still would be irrational numbers for which some digit didn’t occur in the first thousand.
Notice how δ gets smaller and smaller the longer the expansion. It seems to asymptotically approach zero. But does it continue to do so? Unproven.
But actually, that’s not what we need to prove: what we need to prove is that it isn’t 100% forever...which it obviously isn’t for these three numbers. But is it ever, for any number? Or more to the point, for any base-b expansion of any number?
| √2 | Occurrences in first n digits | ||||||
| Digit | n = 10 | n = 100 | n = 1,000 | n = 10,000 | n = 100,000 | n = 1,000,000 | n = 5,000,000 |
| 0 | 0 | 10 | 108 | 952 | 9959 | 99814 | 499479 |
| 1 | 2 | 8 | 99 | 1005 | 10107 | 98924 | 499237 |
| 2 | 2 | 8 | 108 | 1004 | 9876 | 100436 | 500545 |
| 3 | 2 | 11 | 82 | 980 | 10057 | 100191 | 499995 |
| 4 | 2 | 9 | 100 | 1016 | 10100 | 100024 | 500108 |
| 5 | 1 | 7 | 104 | 1001 | 10002 | 100155 | 499218 |
| 6 | 1 | 10 | 90 | 1032 | 9939 | 99886 | 501393 |
| 7 | 0 | 17 | 104 | 964 | 10008 | 100008 | 500047 |
| 8 | 0 | 12 | 113 | 1027 | 10007 | 100441 | 499600 |
| 9 | 0 | 8 | 92 | 1019 | 9945 | 100121 | 500376 |
| δ | 100% | 30% | 18% | 4.8% | 0.61% | 0.441% | 0.1564% |
| π | Occurrences in first n digits | |||||
| Digit | n = 10 | n = 100 | n = 1,000 | n = 10,000 | n = 100,000 | n = 1,000,000 |
| 0 | 0 | 8 | 93 | 968 | 9999 | 99959 |
| 1 | 2 | 8 | 116 | 1026 | 10137 | 99757 |
| 2 | 1 | 12 | 103 | 1021 | 9908 | 100026 |
| 3 | 1 | 12 | 103 | 975 | 10026 | 100229 |
| 4 | 1 | 10 | 93 | 1012 | 9971 | 100230 |
| 5 | 3 | 8 | 97 | 1046 | 10026 | 100358 |
| 6 | 1 | 9 | 94 | 1021 | 10028 | 99548 |
| 7 | 0 | 8 | 95 | 970 | 10025 | 99800 |
| 8 | 0 | 12 | 101 | 947 | 9978 | 99985 |
| 9 | 1 | 13 | 105 | 1014 | 9902 | 100106 |
| δ | 100% | 20% | 7% | 5.3% | 0.98% | 0.452% |
| e | Occurrences in first n digits | |||||
| Digit | n = 10 | n = 100 | n = 1,000 | n = 10,000 | n = 100,000 | n = 1,000,000 |
| 0 | 0 | 5 | 100 | 974 | 9885 | 99425 |
| 1 | 2 | 6 | 96 | 989 | 10264 | 100132 |
| 2 | 2 | 13 | 98 | 1005 | 9856 | 99845 |
| 3 | 0 | 8 | 109 | 1008 | 10035 | 100228 |
| 4 | 1 | 10 | 99 | 982 | 10039 | 100389 |
| 5 | 0 | 13 | 85 | 992 | 10034 | 100087 |
| 6 | 0 | 12 | 99 | 1079 | 10183 | 100479 |
| 7 | 1 | 16 | 99 | 1008 | 9875 | 99910 |
| 8 | 4 | 7 | 103 | 995 | 9966 | 99812 |
| 9 | 0 | 10 | 112 | 968 | 9863 | 99691 | δ | 100% | 50% | 15% | 2.6% | 1.44% | 0.575% |