The canonical representation of natural numbers can be extended to positive rational numbers, i.e. numbers of the form \frac ab,~~~~~ a,b\text{ both being positive integers.} This can be done by allowing the exponents to be negative integers as follows:
Let a,b be natural numbers, both \ge 1, and let a=\prod_{i=1}^\infty p_i^{e_i}\text { and }b=\prod_{i=1}^\infty p_i^{f_i} be their canonical representations of a and b. By setting \alpha_i:=e_i - f_i we get \frac ab=\prod_{i=1}^\infty p_i^{\alpha_i} as the unique canonical representation of the positive rational number \frac ab > 0.
Please note that with each rational number \frac ab > 0 we can associate a unique series of the integer exponents \alpha_i\in\mathbb Z of is canonical representation. For instance, we have
\begin{array}{ccrrrrrrrr} 10=2^1\cdot 5^1&\text{ can be associated with the series }&1,&0,&1,&0,&0,&0,&0,&\ldots\\ \frac1{10}=\frac1{2^1\cdot 5^1}&\text{ can be associated with the series }&-1,&0,&-1,&0,&0,&0,&0,&\ldots\\ \frac{25}{1188}=\frac{5^2}{2^2\cdot 3^3\cdot 11^1}&\text{ can be associated with the series }&-2,&-3,&2,&0,&-1,&0,&0,&\ldots\\ \end{array}
Please also note that the multiplication of any two positive rational numbers results in the addition of their associated series of exponents in their canonical representations.
