Search in Decimals

The decimal digits of a number are the digits located after the decimal point in its base-10 representation.

A real number can have two types of decimal expansion:

A rational number (such as $ \frac{1}{2} = 0.5 $ or $ \frac{1}{3} = 0.333\ldots $) has either a finite number of decimal digits or a periodic expansion (a sequence of digits that repeats indefinitely).

An irrational number (such as $ \pi $ or $ \sqrt{2} $) has an infinite and non-periodic decimal expansion, meaning there is no repeating pattern.

Therefore, infinitely many numbers have infinitely many decimal digits, but only irrational numbers have infinite non-repeating decimals.

Finding a pattern in the decimal digits of a number depends on the nature of the number.

For a rational number, a pattern necessarily exists if the expansion is periodic, and it can be determined analytically.

For an irrational number, there is generally no repeating pattern. The digits appear to be distributed without apparent structure.

In some specific cases (such as $ \pi $ or $ e $), mathematicians conjecture that their digits behave like a random sequence (this is called a normal number).

In practice, to search for a given pattern (for example a sequence of digits), the user must scan the digits sequentially using a search algorithm.

A real number is said to be normal (sometimes incorrectly referred to as a universal number) if, in its decimal expansion, all finite sequences of digits appear with the expected frequency.

Thus, in a normal number in base 10: each digit from $ 0 $ to $ 9 $ appears about $ 10\% $ of the time, and each pair of digits appears about $ 1\% $ of the time, and so on.

In other words, every finite combination of digits appears somewhere in its decimal expansion.

Mathematicians conjecture that constants such as $ \pi $, $ e $, or $ \sqrt{2} $ are normal, but this has not yet been proven.