Volume XVI , issue 2 ( 2013 ) | back |

Algebra as a tool for structuring number systems | 47$-$66 |

**Abstract**

Numbers in their natural dependence on sets of visible things are abstract products which result from ignoring nature of elements of these sets and any kind of their organization (the way how they are arranged, grouped, etc). We call formulation of this way of cognition {ıt Cantor principle of invariance of number\/} and, in this paper, we apply it to a coherent exposition of the first grade arithmetic. Reacting to the way how the elements of a set are grouped, one expression is written, regrouping these elements, another expression is written. Then, such two expressions are equated since they denote one and the same number. Thus, this interplay between meaning and symbolic expressing is the ground upon which the range of numbers up to 20 is structured. Two disjoint sets together with their union make an additive scheme to which an addition or a subtraction task may be attached. Dependently on such a task, sums and differences are written as expressions denoting numbers. To reach the unique decimal (in digits) denotation of a number some steps of transforming the corresponding expressions are made. Displaying these intermediate steps leads to a thorough understanding of the arithmetic procedures, which should precede their suppressing which leads to automatic performance. In particular, methods of adding and subtracting when the ten-line is crossed are treated in detail.

**Keywords:** Invariance of number; additive scheme; addition and subtraction tasks; number system up to 20.

**MSC Subject Classification:** 97H20

**MathEduc Subject Classification:** H22

A note on infinite descent principle | 67$-$78 |

**Abstract**

We provide some theoretical background for infinite descent principle and its relation to the principle of mathematical induction and the well ordering property. Also, we provide some interesting examples by applying the infinite descent principle as an extremal principle in several situations. At the end we prove several assertions which confirm the understanding that principle is very important for the students when solving various complex problems they are faced with.

**Keywords:** Infinite descent principle; principle of mathematical induction; well ordering property.

**MSC Subject Classification:** 97E60, 97F60, 97K20

**MathEduc Subject Classification:** E65, F65, K25

To be integer or not to be rational: that is the questio$\sqrt{N}$ | 79$-$81 |

**Abstract**

Another proof is given of the fact that the square root of a nonnegative integer is either an integer or an irrational. Bibliography on this theme is presented.

**Keywords:** Rational and irrational numbers.

**MSC Subject Classification:** 97F60

**MathEduc Subject Classification:** F64

New methods for calculation of some limits | 82$-$88 |

**Abstract**

The purpose of the present paper is to establish some properties of certain classes of sequences.

**Keywords:** Limits of sequences; problem solving.

**MSC Subject Classification:** 97D50, 97I30

**MathEduc Subject Classification:** D55, I35

ISDET\,2013 Conference | 89$-$98 |

**Abstract**

Abstracts of the plenary lecture and the four invited, introductory lectures in the section Mathematics with Informatics of ISDET\,2013 Conference are presented.

**MSC Subject Classification:** 97--06

**MathEduc Subject Classification:** A99