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:

Definition: Canonical Representation of Positive Rational Numbers

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.

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