The mathematical analysis based on the classical infinite system　is called "classical mathematical analysis", and the mathematical analysis based on the new infinite system is called "new mathematical analysis (the fourth generation of mathematical analysis)".

1  similarities：

1 .1 Both kinds of mathematical analysis use limit idea and limit theory to make quantitative cognition to those "limit idea related infinite mathematical things" , otherwise no calculation is possible.

1 .2 In practical operations, both kinds of mathematical analysis treat those "x----> 0 infinite mathematical things" as number forms with "visible, concret and touchable quantitative nature" (although this is not admited theoratically in classical mathematical analysis), otherwise no calculation is possible.

1 .3  In practical operations, both kinds of mathematical analysis regard the quantitative cognizing process and result of those "X----> 0 infinite mathematical things" as "advanced approximate calculation" , otherwise the operation process of limit theory cannot be explained scientifically.

1 .4  Both kinds of mathematical analysis regard the inverse of differential and integral in mathematical analysis is a perfect "approximation countervailing theory" invented by human , otherwise the "approximation" cannot be explained scientifically.

     The similarities between new mathematical analysis and classical mathematical analysis in treating "X ---->0 infinite mathematical things" determine the new mathematical analysis has retained many  accumulated precious intellectual wealth in classic mathematical analysis since ancient times.

2  Differences：

2.1 Different "infinite ideas" and different "treating objects"

 In the foundation of classical mathematical analysis, "infinite" is composed by two concepts of "potential infinite and actual infinite" which can not be well defined and are inevitably confused. So, it is impossible to know clearly what on earth those "X----> 0 infinite mathematical things" are for people to treat (numbers or non-numbers? potential infinite mathematical things or actual infinite mathematical things? potential infinite numbers or actual infinite numbers? ...?)------they are non-number "variables" that are forever in change.

 In new mathematical analysis, "infinite" is composed by two concepts of "abstract infinite law" and "the carriers of 'abstract infinite law'", which can be well defined and cannot be confused. According to the new number system, all of those "X----> 0 infinite mathematical things" for people to treat are clearly known as "the mathematical carriers of 'abstract infinite law'" ------they are numbers.

2.2 Different "infinite number theories and infinite number forms"

 In classical mathematical analysis, people can not get rid of the confusion and bondage of "potential infinite number and actual infinite number" and have had to define all the "infinite numbers" participating in any quantitative calculations as "non-number variables of both potential infinite number and actual infinite number that are forever in change" mixed by the "potential infinite number and actual infinite number" derived by the confused "potential infinite and actual infinite" concepts. But, there is never such "variables" in existing classical number system at all; so, whenever the "infinite numbers" are calculated mistakenly as "potential infinite numbers in forever change", paradoxes have been inevitably produced (as a members of zeno's paradox family, harmonic series paradox is the most typical case).

In new mathematical analysis, all the "infinite numbers" participating in any quantitative calculations are "mathematical carriers of abstract infinite law" with characteristics of number sense representing the existence of abstract infinite law, which are clearly defined as numbers with quantitative nature and within the new number system. And, "abstract infinite law in forever change" will never be taken as number in any quantitative calculations. In new number system, those "X----> 0 infinite mathematical things" belong to “inter--number” and is called “intersimal”

2.3 Different processing theory and operations to "infinite mathematical things"

 In classical mathematical analysis, the defects of existing classic number system unable people to understand the nature of "variables", making its quantitative cognitive theory of "infinite mathematical things" can be nothing to do with existing classical number system and nothing to do with the specific quantitative properties of "infinite mathematical things". So, theoraticaly or operationaly, it is impossible to established essential rules of "mathematical analysis" to know exactly what "X----> 0 infinite mathematical things" being processed are and what quantitative properties they have (because there is no such number form in the existing classic number system), some kinds of formal languages and pipeline operations are ok (such as the completely equivalent languages and pipeline operations in three generations of classical mathematical analysis: "let infinitesimal be 0" in pre--standard analysis, "take the limit" in standard analysis and "take the standard number" in non-standard analysis) . In many situations, some unified pipelining operation theories can be used for the processing of all the "infinite mathematical things" (the most typical situation is: whether in calculus or in harmonic series, all the "X----> 0 infinite mathematical things" are with exactly the same quantitative nature without any differences). And such pipeline operation theories, processes and results in the "potential infinite--actual infinite" based three generations of classical mathematical analysis (pre--standard analysis, standard analysis, non-standard analysis) are completely equivalent, inevitably producing exactly the same "infinitesimal--infinity quantitative cognition paradox families" .

 In new mathematical analysis, the quantitative cognitive theory of "infinite mathematical things" is a  targeted operation theory closely related to the new number system as well as the specific quantitative properties of "infinite mathematical cariers". Before calculating, all the "X----> 0 infinite mathematical things"  participating in any quantitative calculations must be truly "mathematical analyzed" according to the "theory of infinite law carrier" to know what quantitative meanings and properties they have so as to perform the targeted operations to them. And, such targeted operation theories, processes and results are nonequivalent to any of the three generations of classical mathematical analysis in many situations--------it is impossible to have those suspended "infinitesimal--infinity related paradox families" .