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Natural Numbers, Natural Shapes
Axiomathes, 2018We explain the general significance of integer-based descriptors for natural shapes and show that the evolution of two such descriptors, called mechanical descriptors (the number N(t) of static balance points and the Morse–Smale graph associated with the scalar distance function measured from the center of mass) appear to capture (unlike classical ...
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2017
This chapter starts by establishing the natural numbers with the help of Peano’s axioms. Using the axiom of mathematical induction, addition and multiplication of the natural numbers are introduced. In the second part of the chapter, we develop the concept of divisibility of natural numbers.
Jürg Kramer, Anna-Maria von Pippich
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This chapter starts by establishing the natural numbers with the help of Peano’s axioms. Using the axiom of mathematical induction, addition and multiplication of the natural numbers are introduced. In the second part of the chapter, we develop the concept of divisibility of natural numbers.
Jürg Kramer, Anna-Maria von Pippich
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Another Characterization of the Natural Numbers
Mathematical Logic Quarterly, 1989The natural numbers N together with the successor function s is the unary algebra (N,s) freely generated by 0. This note proves that it is possible to restrict the freeness property to ``B-algebras'' (B,r), where B is the power set of a set A and r is an extensive function from B into B: \(X\subseteq r(X)\) for every \(X\in B\).
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On Linear Equations in Natural Numbers
International Journal of Algebra and Computation, 1997This paper utilizes the theory of finitely generated commutative monoids for the study of linear equations on the set \(N\) of natural numbers. The equation \(c_1x_1+c_2x_2+\dots+c_nx_n=x\), where \(c_1,c_2,\dots,c_n\), \(x\) are elements of \(N\), is denoted by \(e(x)\). A general solution in \(N\) for the family of linear equations \(\{e(x)\}_{x\in N}
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A natural way to write non-Natural numbers
2020In the early 20th century, continued fractions used to be a regular ingredient of the school curriculum, but they have now all but disappeared. In this article we present an elementary treatment based on the Stern-Brocot tree, with emphasis on the fact that they provide a more natural way to represent irrational numbers than the usual decimal expansion.
Arora, Gurdial +2 more
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1996
Abstract Created by the human mind to count the objects in various assemblages, numbers have no reference to the individual characteristics of the objects counted. The number six is an abstraction from all actual collections containing six things; it does not depend on any specific qualities of these things or on the symbols used ...
Richard Courant, Herbert Robbins
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Abstract Created by the human mind to count the objects in various assemblages, numbers have no reference to the individual characteristics of the objects counted. The number six is an abstraction from all actual collections containing six things; it does not depend on any specific qualities of these things or on the symbols used ...
Richard Courant, Herbert Robbins
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1984
Here we encounter such basic concepts as integers, composites, and primes, and we learn the very fundamental fact that the composites can be represented in a unique way as a product of primes.
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Here we encounter such basic concepts as integers, composites, and primes, and we learn the very fundamental fact that the composites can be represented in a unique way as a product of primes.
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The Polykays of the Natural Numbers
Biometrika, 1960Barton, D. E., David, F. N., Fix, Evelyn
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1986
The object of this chapter is to define a set to represent the numbers 0, 1, 2, .... To be complete, we must also show how to add and multiply these numbers and prove all the usual laws: commutative, associative, etc. The most important idea contained in our construction is that of mathematical induction.
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The object of this chapter is to define a set to represent the numbers 0, 1, 2, .... To be complete, we must also show how to add and multiply these numbers and prove all the usual laws: commutative, associative, etc. The most important idea contained in our construction is that of mathematical induction.
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