Learning Peano Axioms helps in calculus because of number concept

Calculus is not just about functions, it is also about numbers.  In calculus, we end up finding the need for having real numbers so that more sequences converge to a number.  Sequences of rationals often don’t converge to a rational number.   But we can define a concept of internal convergence for a sequence.

For example, the infinite decimal is a sequence that internally converges in the sense that the distance between elements gets smaller the farther we are in the expansion, i.e. smaller decimal places.  So two truncations of the number expanded until after the 1/100 place only differ by 1/100. Two truncations of the number expanded beyond the 1/1000 place differ only by a 1/1000.

Such truncation convergence tells us we have an internal convergence criterion.   So the sequence itself should converge to something.   We can define that something a variety of ways, e.g. as the decimal sequence itself.

The number concept has to be expanded beyond rational numbers when we have an infinite decimal expansion that does not converge to a rational number.

Trouble in learning calculus comes from a sudden confronting in the limitations of our number concept as we learn a convergence concept for sequences and for functions.

If we learn the natural numbers from the Peano Axioms including proving the laws of addition, then we have learned the structure of a type of number.

The same applies to learning rational numbers as functions on the naturals.  We learn a structure for thinking of the rationals.

When we get to real numbers as internally convergent sequences of rationals, we have a new structure to learn for numbers.  If we have not learned the structure for naturals from the Peano Axioms nor rationals as functions on the naturals, then we will not be ready for the structure of convergent sequences.

So calculus and its convergence concepts will be too much for us to grasp properly. So we will then fall back to rules on limits, etc. that we can use to pass the test. What limit and continuity mean, we will be vague on.   Quickly, we forget the test rules and we are back to not understanding calculus.

 

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About New Math Done Right

Author of Pre-Algebra New Math Done Right Peano Axioms. A below college level self study book on the Peano Axioms and proofs of the associative and commutative laws of addition. President of Mathematical Finance Company. Provides economic scenario generators to financial institutions.
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