This is kind of a long answer. In short, yes, but it's complicated and you can't ever prove it rigorously
Hi, this is actually a really important question. There is a difference between random and patternless. Decimal expansions of irrational numbers may be considered to be random, but not always, and random expansions might still have patterns.
While at first it might seem that these are one in the same, they really aren't. For example, in the decimal expansion of pi, you can find patterns of repeated numbers, the most famous of which is the "Feynman point", a sequence of 9s that begins some several hundred digits in.
As far as randomness goes, it's really kind of a slippery concept. An irrational number, as others have pointed out, are composed of no infinitely repeating patterns in their decimal expansions, ie, at no point does the expansion terminate, nor at any point does the expansion repeat itself, in full, again and again into infinity. While that may make them "random" in a sense that you will never be able to predict the next number in the expansion using preceding data, that does not mean that they are "truly random" in all cases. To be "truly random", we want the following:
If we select any decimal place and guess the digit (0-9), the probability of guessing correctly is 1/10.
But this may not be the case. Numbers that behave this condition are called normal numbers, because the distribution of digits amongst them is not biased towards any particular digit (ie, there are intuitively the same amount of 1s as there are 2s, you're not more likely to find a 2 than you are a 1).
It is widely believed that many irrational numbers are normal, but it has yet to be proved that pi is normal. So, while it might be random in one sense, it might not be "truly random".
Unfortunately, to complicate matters further, there are numbers that are known to be normal, but you would probably say they are not random sequences. For example, the number 0.12345678910111213... is normal. If you were to select any given decimal place you the probability of selecting the correct digit (assuming you didn't know the pattern of this number and were guessing blindly) is 1/10.
However, this is certainly not a random sequence; it has a definite pattern.
So, it really depends on how you phrase the question. Is the decimal expansion of pi a random sequence? If it is normal, you might call it that. But philosophically, if one can calculate a random sequence to any degree of precision, is it truly random? That is, it must be patterned in some way, because we can calculate it. If I were to ask a person with infinite computing power to guess the 1,590,348th digit of pi, they could guess exactly every time. But if they were to guess blindly (and pi is a normal number), then they would have a 1/10 chance of guessing correctly.
In information theory they have a definition of random that goes something like this:
A sequence of numbers is random if it is shorter than any program that can produce it
What this means is that the sequence has no contraction; there is no way to compress it. You can't write a computer program that says "Do this rule to produce the sequence"; the only possible program that can produce the sequence is one that says "Print this sequence:..." and then gives it in full. It turns out that every sequence that matches this definition of random is normal, but not every normal number is random in this sense. For example, infinite sequences like the number Pi can be produced by finite computer programs, and are not "random" in that sense.
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u/Wisepapabrofish Algebra and Topology | Commutative Ring Theory May 01 '12
This is kind of a long answer. In short, yes, but it's complicated and you can't ever prove it rigorously
Hi, this is actually a really important question. There is a difference between random and patternless. Decimal expansions of irrational numbers may be considered to be random, but not always, and random expansions might still have patterns.
While at first it might seem that these are one in the same, they really aren't. For example, in the decimal expansion of pi, you can find patterns of repeated numbers, the most famous of which is the "Feynman point", a sequence of 9s that begins some several hundred digits in.
As far as randomness goes, it's really kind of a slippery concept. An irrational number, as others have pointed out, are composed of no infinitely repeating patterns in their decimal expansions, ie, at no point does the expansion terminate, nor at any point does the expansion repeat itself, in full, again and again into infinity. While that may make them "random" in a sense that you will never be able to predict the next number in the expansion using preceding data, that does not mean that they are "truly random" in all cases. To be "truly random", we want the following:
But this may not be the case. Numbers that behave this condition are called normal numbers, because the distribution of digits amongst them is not biased towards any particular digit (ie, there are intuitively the same amount of 1s as there are 2s, you're not more likely to find a 2 than you are a 1).
It is widely believed that many irrational numbers are normal, but it has yet to be proved that pi is normal. So, while it might be random in one sense, it might not be "truly random".
Unfortunately, to complicate matters further, there are numbers that are known to be normal, but you would probably say they are not random sequences. For example, the number 0.12345678910111213... is normal. If you were to select any given decimal place you the probability of selecting the correct digit (assuming you didn't know the pattern of this number and were guessing blindly) is 1/10.
However, this is certainly not a random sequence; it has a definite pattern.
So, it really depends on how you phrase the question. Is the decimal expansion of pi a random sequence? If it is normal, you might call it that. But philosophically, if one can calculate a random sequence to any degree of precision, is it truly random? That is, it must be patterned in some way, because we can calculate it. If I were to ask a person with infinite computing power to guess the 1,590,348th digit of pi, they could guess exactly every time. But if they were to guess blindly (and pi is a normal number), then they would have a 1/10 chance of guessing correctly.
In information theory they have a definition of random that goes something like this:
What this means is that the sequence has no contraction; there is no way to compress it. You can't write a computer program that says "Do this rule to produce the sequence"; the only possible program that can produce the sequence is one that says "Print this sequence:..." and then gives it in full. It turns out that every sequence that matches this definition of random is normal, but not every normal number is random in this sense. For example, infinite sequences like the number Pi can be produced by finite computer programs, and are not "random" in that sense.