Teenagers use social networks on a daily basis; they provide communication and information services and offer the chance to establish new relationships or maintain existing ones. Self-esteem, the esteem we have for our abilities and for what we can achieve, is affected by the comparison with others and by whether we are accepted by them or not. Since social networks enable their users to accept people or not, and they are also spaces highlighting social comparisons, it is important to investigate whether they have impact on teenagers' self-esteem. This paper examines the perceptions of teenagers about social networks and investigates whether Facebook, as a representative of social networks, has impact on teenagers' self-esteem. The survey was conducted among students in Athens, aged 16 (N = 71) and the Self Perception Profile for Adolescents (Harter, 1988) questionnaire was used, in its Greek version PATEM IV, adding questions regarding the opinions of teenagers for the use of social networks and their acceptance and popularity, which were correlated with the PATEM IV sub-scales. The study showed that self-esteem is not connected with the frequency with which students use Facebook or the level of acceptance or popularity of the users. It appears that teenagers primarily seek recognition and establishing relationships with their peers through social networks. The closer the relationships, the more they use social networks to communicate and less for self-promotion. The number of "likes" is positively correlated with relationships with the opposite sex. Teenagers with more "likes" on their profile pictures have developed better relationships with the opposite sex. Percentage of 51.6% expresses the need of higher acceptance and social recognition by others and uses "tags" in order to increase the "likes" received. 87.1% uses social networks on a daily basis and 57.1% two hours and more daily, but the majority finds enough time to go out. It is alarming that 53% of the teenagers sacrifice their sleeping and studying time to find time to

For every real number x, we define as integer part the biggest integer k so that k ≤ x and is expressed [x]. The difference of the number from its integral part is defined as decimal part of x and expressed with () [) 0,1 x ∈. Consequently, for every x, the Kronecker's orbit is defined, namely the set () { } nx n ∈ . Through Kronecker's orbit, rational numbers are characterized as the numbers whose orbit is a bounded set, while irrational numbers are characterized as the numbers whose orbit is a dense set. Using this fundamental theoretical result and utilizing a computer, a didactic approach was established, initially referring to the definition of rational numbers as fraction equivalence classes and basically to the segregation of rational and irrational numbers. This didactic approach also incorporates elements of ancient Greek mathematics history. The proposition was applied to students and was evaluated.