Patterns in irrational numbers

Question

I understand that an irrational number has no periodic numerical pattern. I was wondering, however, if we could find logical patterns instead and if they would even be useful.

For example, let's say we have a number g, approximately: g=5.123768322988 and let's assume this is irrational. Let's say that I was able to prove that the numbers following the decimal in g follow a particular logical pattern: "three numbers less than 4 followed by 3 numbers greater than 5". Is this something that could be proven for irrational numbers like pi?

If so, would it even be useful for example in computing precise approximations in a cheaper way?

Answer 1

I was wondering, however, if we could find logical patterns instead and if they would even be useful.

Sorry but no there are no proved logical pattern for many irrational numbers.

Any irrational number has a non-terminating, non-repeating sequence of digits in its decimal representation (or the representation in any whole number base). This is easy to prove by contradiction. Any terminating decimal is obviously rational. Any repeating sequence with period n can be converted to a terminating decimal by multiplying by $(10^n−1)$ .

On a side note let me tell you that irrational numbers like $0.1010010001...$ are irrational having pattern.

Answer 2

Is this something that could be proven for irrational numbers like pi?

If this were proved for pi it would be a major accomplishment. To date, the majority view is that the decimal for pi is normal (in all bases), and therefore does not have the pattern you suggest.

Even if you proved it for $\sqrt{2}$ it would be major news (to mathematicians).

Answer 3

I think what you may be looking for is a continued fraction expansion for irrational numbers. This does help you find the pattern that you are suggesting for many irrational numbers (such as sqrt 2, the golden ratio, e, etc). However, to my knowledge, there is no pattern in the continued fraction for pi.