It’s often said that because pi is an infinite, non-repeating decimal it must contain all possible lengths of digits and then can be transformed into jpegs of the Mona Lisa or whatever.
A common response is that that isn’t true because infinite and non-repeating doesn’t imply that the digits are equiprobable. Pi is believed to be “normal” but it hasn’t been proven.
Even that is too weak though, yes? 0.1234567891122334455… is a normal number but will never contain “2234.”
Is there a name for the property that every possible string of digits will appear in a decimal representation of a number?
A disjunctive sequence contains every finite string. I’m not sure if there’s a specific name for a number formed by adding a radix point to a disjunctive sequence.
The definition of a Normal number that limits it to equiprobable single digits is not strictly correct. Equiprobability of digits is a property of normal numbers but it is insufficient to establish normality.
The example given, for example, 0.123456789112233445566… is not normal and not just because there are no 0’s. It will not contain all finite length strings of possible digits.
All normal numbers are disjunctive, i.e. they will contain every finite string, but not all disjunctive numbers will be normal.
So, the answer to the OP is normal numbers is the name of such numbers, at least for finite length subsequences. Clearly, containing every possible infinite length decimal expansion would be a different beast.
Edit:
Also, it is worth noting there is a difference between being absolutely normal and being simply normal in a particular base. There are numbers that are not normal in base 10 but are normal in other bases. Certain numbers (like e or pi) are suspected of being normal across all integer bases but this hasn’t been proven.
If we think of a (infinite) sequence of digits expressed in some base as a coded message, and containing certain specific information, wouldn’t the information still be present even if the message was expressed in a different base/code? Are only certain bases capable of containing (or encoding) all the information? Or is being normal not really ‘information’.
Phrased differently, if we can convert an expression from one base to another, isn’t the information they contain the same, including any property of normality?
As my Information Theory professor once said, it depends on how one defines “information”, which has always been a bit nebulous. Numbers are numbers, so, in one sense, they contain the same “information” no matter the base in which they are represented. But the concept or normality concerns that representation itself.
In this particular case, the OP is mixing a few things up.
There is the concept of a “simply” or “weakly” normal number. This is simply a number for which single digits can appear equiprobably. The thing is, rational numbers can be simply normal. For example 0.012345678901234567890123456789… (0123456789 repeating) is simply normal in base 10, even though it is a rational number.
Irrational numbers can be simply normal without being ‘regular’ normal in a particular base. They can also be simply normal in one base without being normal at all in another base. For example, the binary number 0.10101010… (2/3 in base 10) is both simply normal in base 2 but not normal in base 10.
Then there are generally normal numbers, which contain all finite possible integer subsequences. For base 10, a well known example is Champernowne’s number, i.e. 0.01234567891011121314… (i.e. just tag on each successive positive integer to the last). By construction, this number is fully normal in base 10 but it is not known if it is normal in any other base. Perhaps it is, but there is no proof.
One of the more interesting things is (and it is always a bit wonky when discussing infinite sets) that ‘almost every’ real number is normal. It can be proved that the set of non-normal real numbers has Lebesgue measure 0. What that means - as an example, the subset of real numbers that consists of only integers (or any countable subset of the reals) has Lebesgue measure 0, i.e. there are “more” normal reals than non-normal reals (whatever “more” means when comparing infinite sets).
In fact, most real numbers are truly normal, not just in any one base, but in every base. Practically speaking, what this means is that every number is normal, unless there’s some good reason for it to not be normal. Rational numbers have good reason not to be normal. A number of specially-constructed numbers have good reason not to be normal. But nobody knows of any good reason for pi to not be normal, so we guess that it is. Maybe we’re wrong: Maybe there’s some good reason why pi wouldn’t be normal, and we just haven’t discovered it yet. We don’t know for sure, and probably can’t know for sure.
This sentence makes it appear that you are arguing that normality and containing every sequence of digits are equivalent. This is not true (and given your subsequent posts you probably know this and just didn’t express it well in this post.) The property described in the OP is an necessary condition for normality but not a sufficient condition. For example we could write a sequence of digits that first goes through every combination of single digits, and then goes through every combination of 2 digits, then every combination of 3 digits etc. and so eventually hit every combination of N digits for any finite N. If we then placed separators between each of these sequences containing a number of 9’s equal the the digits in the sequence being separated, then it would still contain every sequence of numbers but wouldn’t be normal, since about half of the digits would be 9.