normaalilukujia

Normaalilukujia, known in English as normal numbers, are real numbers that are "normal" in a specific statistical sense. A number is considered normal if its base-b expansion exhibits a uniform distribution of digits. This means that each digit from 0 to b-1 appears with equal probability, 1/b, and every possible sequence of digits of a given length appears with a frequency proportional to its probability. For example, in base 10, a normal number would have each digit 0 through 9 appearing approximately 10% of the time in its decimal expansion. Furthermore, pairs of digits like '00', '01', ..., '99' would each appear about 1% of the time, and so on for longer sequences.

The concept of normality was introduced by mathematician Émile Borel in 1909. He proved that almost all real numbers are normal in any given base. "Almost all" in this context refers to the set of normal numbers having Lebesgue measure equal to 1 in the interval [0, 1]. This implies that if you were to pick a real number at random, the probability of it being normal is 1.

Despite the theoretical prevalence of normal numbers, finding specific examples is surprisingly difficult. Many common mathematical constants, such as pi, e, and the square root of 2, are conjectured to be normal, but these conjectures remain unproven. The number known as Champernowne constant, formed by concatenating all positive integers in order (0.123456789101112...), is proven to be normal in base 10. However, proving normality in other bases for such numbers can be more complex. The study of normal numbers is a branch of number theory and ergodic theory, exploring the statistical properties of number expansions.