In mathematics, there is a kind of number carriers with cognizable quantitative properties (such as imaginary number, fuzzy number, infinitesimal variables and monads … in present mathematics)"; these “mathematical carriers of abstract concepts and laws” can join any quantitative calculation process with finite number forms (number forms with Archimedean Property) but their exact values are unknown. They are defined as “number forms with Half Archimedean Property". The history of our mathematics has proved that people need to carry out various necessary qualitative and quantitative cognitions and studies on those “mathematical carriers of abstract concepts and laws with “Archimedean Property” or “Half Archimedean Property".
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