Prime numbers are positive integers that are only divisible by 1 and themselves. Examples include 2, 3, 5, 7, and 11.
An ortho-normal basis is a set of vectors that are both orthogonal (perpendicular) to each other and have a length of 1. In the context of prime numbers, this means that they are not only relatively prime (have no common factors), but also have a magnitude of 1 when used as coefficients in linear combinations.
A number can be expressed as a linear combination of prime numbers by finding the unique set of prime numbers that, when multiplied by their respective coefficients, add up to the original number. This is known as the Fundamental Theorem of Arithmetic. By using this representation, prime numbers can serve as a basis for all numbers since they are both relatively prime and have a magnitude of 1 as coefficients.
Using prime numbers as an ortho-normal basis allows for a unique and efficient representation of all numbers. It also has applications in cryptography and data encryption, as well as in number theory and other mathematical fields.
While prime numbers serve as a useful basis for many mathematical applications, there are some limitations to using them as an ortho-normal basis for all numbers. For example, some numbers may require a large number of prime numbers to be represented accurately, which can make calculations more complex.