In 1867, HERMANN HANKEL formatted the  principle of preserving the formal laws of calculation  . It says that when a number range is expanded, the arithmetic laws of the starting range should also apply in the expanded range, if possible. This requirement is called the principle of permanence -  homework helper geometry  (  permanence  (Latin) - persistence). This is neither an axiom nor a proposition with evidential value, but a methodological principle. Nor can it achieve that all laws of the starting range apply unchanged in the extended range of numbers.

In the crowd  N the natural numbers follows for  c  ≠  0  the end  a  <  b  always
a  ⋅  c  <  b  ⋅  c (  Monotony of  the multiplication with regard to the smaller relation). 
In the crowd Z As is well known, this no longer applies to whole numbers. 
the end a  <  b  follows  a  ⋅  c  <  b  ⋅  c  only if  c  >  0 (positive) is; 
for c < 0 (negative) results a  ⋅  c  >  b  ⋅  c  (Reversal of the relation sign).

At the time of Hankel, a set-theoretical structure of the number ranges was still unknown -  calculus homework solver  . Today one would formulate subset the principle of permanence as a requirement that the new (extended) number range must contain the old (starting range) as a.