Wyniki wyszukiwania

Sortuj według:

Ogranicz wyniki do:

Cauchy Mean Theorem

100%

Grabowski A.

Formalized Mathematics

2014

tom 22

nr 2

157-166

The purpose of this paper was to prove formally, using the Mizar language, Arithmetic Mean/Geometric Mean theorem known maybe better under the name of AM-GM inequality or Cauchy mean theorem. It states that the arithmetic mean of a list of a non-negative real numbers is greater than or equal to the geometric mean of the same list. The formalization was tempting for at least two reasons: one of them, perhaps the strongest, was that the proof of this theorem seemed to be relatively easy to formalize (e.g. the weaker variant of this was proven in [13]). Also Jensen’s inequality is already present in the Mizar Mathematical Library. We were impressed by the beauty and elegance of the simple proof by induction and so we decided to follow this specific way. The proof follows similar lines as that written in Isabelle [18]; the comparison of both could be really interesting as it seems that in both systems the number of lines needed to prove this are really close. This theorem is item #38 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/.

Calculus using proximities: a mathematical approach in which students can actually prove theorems

68%

O’Donovan R.

Open Mathematics

2017

tom 15

nr 1

30-36

Teaching and learning calculus are notoriously difficult and the didactic solutions may involve resorting to intuitive but vague definitions or informal gestures offered as proofs. The teaching literature is rife with examples of metaphors, adverb manipulations and descriptions of what happens “just before” the limit. It is then difficult to leave the domain of the mental image, thus losing the training in rigour. The author (with Karel Hrbacek and Olivier Lessmann) has endeavoured a radically different approach with the objective of training students to prove theorems while preserving both intuition and mathematical rigour. Hence we change the mathematical setting rather than the didactic setting. The result (which is a by-product of nonstandard analysis) has been used in several high schools in Geneva – Switzerland – for over ten years.

Ograniczanie wyników

1 Formalized Mathematics

1 Open Mathematics

1 Grabowski A.

1 O’Donovan R.

1 2017

1 2015

Wyniki wyszukiwania

Cauchy Mean Theorem

Calculus using proximities: a mathematical approach in which students can actually prove theorems