cragar said:How would I figure out the Kolmogorov complexity of an irrational number?
Could I just look at the shortest formula to compute that number.
If it is an uncomputable real, does it have infinite complexity?
SteveL27 said:I saw an interesting example online the other day. Take a noncomputable number and interperse 0's in every other digit position. Then for each finite-length initial segment, it takes just a little more than half as much information to compute the number with the 0's in every other position as it does to compute the original number. So not all noncomputable numbers are equally random. It's tricker than I'd realized.
An irrational number is a real number that cannot be expressed as a fraction of two integers. It is a number that goes on infinitely without repeating and cannot be written in decimal form as a terminating or repeating decimal.
Rational numbers can be expressed as a ratio of two integers, while irrational numbers cannot. Additionally, irrational numbers have an infinite number of non-repeating decimals, while rational numbers have a finite number of digits in their decimal representation.
No, irrational numbers cannot be calculated exactly because they have an infinite number of digits and do not follow a repeating pattern. They can only be approximated to a certain degree of accuracy.
Irrational numbers play a crucial role in mathematics as they make up the majority of the real numbers. They are essential in many mathematical concepts and calculations, and their complexity adds depth and richness to the field of mathematics.
Yes, irrational numbers have practical applications in fields such as engineering, physics, and computer science. They are used in calculations involving circles, waves, and exponential growth, among other things.