normaaliluku

normaaliluku is a Finnish term that translates to "normal number" in English. It is a concept primarily used in the field of number theory and mathematics, specifically in relation to the properties of real numbers. A number is considered normaaliluku if its digits in any base are uniformly distributed in its base-representation. This means that over a large number of digits, each digit appears with approximately equal frequency.

The concept of normal numbers was introduced by Émile Borel in 1909. He proved that almost all real numbers are normal in every base. "Almost all" in this mathematical context means that the set of non-normal numbers has Lebesgue measure zero. However, proving that a specific number is normal can be extremely difficult.

For example, the number known as Champernowne constant, formed by concatenating the positive integers in order (0.123456789101112...), is proven to be normal in base 10. Another example is the irrational number pi (π), which is conjectured to be normal in base 10, but this has not been mathematically proven. The distribution of digits in numbers like pi is a subject of ongoing research and fascination for mathematicians. The notion of normality provides a quantitative way to describe the randomness of a number's decimal expansion.