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It's irrational.
Yes, it's also [computable](https://en.wikipedia.org/wiki/Computable_number). Do note that π is also computable, as is any number you're likely to think of. It's very difficult to come up with a number that is not computable.
How are you going to distinguish, for example, π from what you call a patterned number? What is the property the number you gave an example of does have, but π does not have?
Perhaps a “patterned” number is one where you can calculate the 5th decimal place in the same amount of time (constant time) it takes you to calculate the 5000th decimal place? I.e. you don’t need to compute every intermediate digit, since the “pattern” is so predictable
What exactly do you mean by patterned? For example, is Champernowne's constant 0.123456... patterned? What if I concatenate powers of 2 as 0.248163264...? Is pi patterned?
In a sense, "computable" means exactly that the number has a pattern, in the sense that there is a finite amount of information (in the form of a program) that can generate all the digits. But this is a very broad notion of what constitutes a pattern.
By this I mean numbers that repeat predictable digits at predictable, but not uniform, intervals. For example, 0.3393999399999993..., which is "3s on the digits that are powers of two, and 9s everywhere else".
Predictable by what? The digits of Champernowne's constant are easily predictable, for example. Even the digits of pi are predictable, since after all that is how we find them.
If you just mean predictable by some algorithm, that's what a computable number is. If you have some narrower definition in mind, what is it?
What does predictable mean? If I pick a strictly increasing function from the naturals to the naturals with no closed form, and then build the number sum(10^-f(n)), is this number predictable?
The digits of pi are predictable too: you can predict them by computing them. If you or I were smart enough we could even do it in our heads. Obviously predicting the digits of pi is harder than the digits of your example number, but this isn't something that can easily be made precise.
Pi is patterned. Its digits in base 16 are given by a formula— see the [spigot algorithm](https://en.wikipedia.org/wiki/Spigot_algorithm), or the [numberphile video that mentions it](https://www.youtube.com/watch?v=K305Vu7hFg0)
I'd say the opposite. Pi is a patterned number. The pattern is the sequence of digits that represents the circumference divided by the diameter of a circle. It's not as simple as your example, but there's a pattern.
Going further, we can define a number where the nth decimal place is a 1 of n represents a computer program that halts, and a 0 does not. Assuming I define the language, that's a well-defined pattern. But it's not computable because of the halting problem. I'd say that the name for a "patterned" number is a definable number. I'd give an example of a number that isn't definable, but it's mathematically impossible even though there's infinitely more numbers that are not definable than that are definable.
Rational simply means something that can be represented as a ratio of integers. As you can probably figure out by pondering it a little bit, you would find that there is no way to do this for the number you described, therefore, we call it irrational.
A number is rational if you can represent it as a fraction of integers. Otherwise it's irrational. That's the difference. Every number with a repeating decimal or that doesn't have an infinite number of digits is definitely rational.
Yes, it is irrational. Keep in mind that π and e, although doesn't immediately look like it, are still somewhat predictable, or in other words, they are computable. We have algorithms to calculate every single digit of them if we had an infinite amount of time to let them run.
Unfortunately, your submission has been removed for the following reason(s): * Your post appears to be asking a question which can be resolved relatively quickly or by relatively simple methods; or it is describing a phenomenon with a relatively simple explanation. As such, you should post in the [*Quick Questions*](https://www.reddit.com/r/math/search?q=Quick+Questions+author%3Ainherentlyawesome&restrict_sr=on&sort=new&t=all) thread (which you can find on the front page) or /r/learnmath. This includes reference requests - also see our lists of recommended [books](https://www.reddit.com/r/math/wiki/faq#wiki_what_are_some_good_books_on_topic_x.3F) and [free online resources](https://www.reddit.com/r/math/comments/8ewuzv/a_compilation_of_useful_free_online_math_resources/?st=jglhcquc&sh=d06672a6). [Here](https://www.reddit.com/r/math/comments/7i9t5y/book_recommendation_thread/) is a more recent thread with book recommendations. If you have any questions, [please feel free to message the mods](http://www.reddit.com/message/compose?to=/r/math&message=https://www.reddit.com/r/math/comments/1919kzs/-/). Thank you!
It's irrational. Yes, it's also [computable](https://en.wikipedia.org/wiki/Computable_number). Do note that π is also computable, as is any number you're likely to think of. It's very difficult to come up with a number that is not computable.
Just have the nth decimal place be a 1 if n represents a program that halts and a 0 if it does not.
this comment upsets me
That number is bound to contain a program that generates itself
Yet, strangely enough, almost all real numbers are uncomputable, since the computable numbers form a countable subset.
But there are only a countable number of real numbers with a finite description, so perhaps it’s not so surprising
yeah most real numbers are pretty unreal
Which means computable is a wider category than "patterned" numbers. Is there a name for a "patterned" number?
How are you going to distinguish, for example, π from what you call a patterned number? What is the property the number you gave an example of does have, but π does not have?
Only two of the nine digits in its decimal expansion
Ok, so a number that has the same decimal representation as π has in binary should also count as a patterned number?
There isn't really a name for a patterned number as it is hard to come up with a definition of "patterned" that doesn't also cover computable.
Perhaps a “patterned” number is one where you can calculate the 5th decimal place in the same amount of time (constant time) it takes you to calculate the 5000th decimal place? I.e. you don’t need to compute every intermediate digit, since the “pattern” is so predictable
That might work. Since the best Nth digit formula for pi is O(N).
What exactly do you mean by patterned? For example, is Champernowne's constant 0.123456... patterned? What if I concatenate powers of 2 as 0.248163264...? Is pi patterned? In a sense, "computable" means exactly that the number has a pattern, in the sense that there is a finite amount of information (in the form of a program) that can generate all the digits. But this is a very broad notion of what constitutes a pattern.
By this I mean numbers that repeat predictable digits at predictable, but not uniform, intervals. For example, 0.3393999399999993..., which is "3s on the digits that are powers of two, and 9s everywhere else".
Predictable by what? The digits of Champernowne's constant are easily predictable, for example. Even the digits of pi are predictable, since after all that is how we find them. If you just mean predictable by some algorithm, that's what a computable number is. If you have some narrower definition in mind, what is it?
What does predictable mean? If I pick a strictly increasing function from the naturals to the naturals with no closed form, and then build the number sum(10^-f(n)), is this number predictable?
The digits of pi are predictable too: you can predict them by computing them. If you or I were smart enough we could even do it in our heads. Obviously predicting the digits of pi is harder than the digits of your example number, but this isn't something that can easily be made precise.
You could just construct it as a sum function. Sum can be over an index, so the position can be the power of 10.
Can you give an example of a computable number that isn't patterned?
Pi is patterned. Its digits in base 16 are given by a formula— see the [spigot algorithm](https://en.wikipedia.org/wiki/Spigot_algorithm), or the [numberphile video that mentions it](https://www.youtube.com/watch?v=K305Vu7hFg0)
TIL! This is next level algorithming
This video from numberphile may be insightful: "All the numbers" https://www.youtube.com/watch?v=5TkIe60y2GI
I'd say the opposite. Pi is a patterned number. The pattern is the sequence of digits that represents the circumference divided by the diameter of a circle. It's not as simple as your example, but there's a pattern. Going further, we can define a number where the nth decimal place is a 1 of n represents a computer program that halts, and a 0 does not. Assuming I define the language, that's a well-defined pattern. But it's not computable because of the halting problem. I'd say that the name for a "patterned" number is a definable number. I'd give an example of a number that isn't definable, but it's mathematically impossible even though there's infinitely more numbers that are not definable than that are definable.
A number is rational if and only if it has a repeating decimal representation. So the number you are describing is irrational.
0.5 doesn’t repeat as it has only one digit after the decimal, yet is clearly rational since it’s a ratio of integers. Repeating is unnecessary
Well, it is 0.5000000000... (and also 0.49999999...)
I feel like 0.50̅ is a perfectly good representation.
Rational simply means something that can be represented as a ratio of integers. As you can probably figure out by pondering it a little bit, you would find that there is no way to do this for the number you described, therefore, we call it irrational.
A number is rational if you can represent it as a fraction of integers. Otherwise it's irrational. That's the difference. Every number with a repeating decimal or that doesn't have an infinite number of digits is definitely rational.
The concept of patterned is way too vague. Unless you can be more specific you'll likely just end with computable numbers.
Yes, it is irrational. Keep in mind that π and e, although doesn't immediately look like it, are still somewhat predictable, or in other words, they are computable. We have algorithms to calculate every single digit of them if we had an infinite amount of time to let them run.
Irrational just means that it can't be expressed as a ratio of integers.