User:Ankitpatel715

A different perspective on irrational numbers
Let a positive integer $$n$$ be represented by its unique prime factorization


 * $$ n = p_1^{n_1} p_2^{n_2} \cdots p_r^{n_r} = \mathbb{P}^{\vec n} $$

where $$\mathbb{P}$$ is the set of primes and the exponentiation is elementwise. The list of exponents $$\vec n$$ is composed of positive integers, each strictly smaller than $$n$$ itself. Since the prime factorization is unique, there is an isomorphism between $$(\mathbb{N}_{+},\times)$$ and $$(\mathbb{N}^\mathbb{P},+)$$. Thus, integers can be represented in either the usual unary or decimal representation or in the new log-space representation.

In log-space, arithmetical rules are quite similar to $$\ln(x)$$. Let $$p,q,k \in \N$$.