In teaching primary teacher trainees, an awareness of the characteristic features, especially commutativity and associativity of basic operations play an important role. Owing to a deeply set automatism rooted in their primary and secondary education, teacher trainees think that such characteristics of addition are so trivial that they do not need to be proved. It does not cause a difficulty in applying mathematical knowledge in everyday situations but primary teachers must have a deeper insight. That is why it is reasonable to show these characteristic features to primary teacher trainees in a different algebraic structure. An example for that could be the algebra of vectors. In this paper the algebraic structure of exploded numbers containing the set of real numbers as a subset is selected as an example. With the help of super-operations (super-addition, super-multiplication, super-subtraction and super-division) introduced for exploded numbers, we try to extend addition for exploded numbers as well. The question of the method of extension and the examination of the characteristics of the extended addition arises. While seeking for the answer, surprising facts emerge, such as the phenomenon that each real number will have one and only one addition incompetent pair among exploded numbers. In this paper we introduce the quasi-extended addition for exploded real numbers which is essentially different from super-addition. On the other hand, the quasi-extended addition is the (traditional) addition for real numbers. Moreover, we investigate some properties (for example commutativity, associativity) of quasi-extended addition. Finally, we find some similarity between the countable infinity and the exploded of 1. The quasi-extension of addition is useful for students to observe different kinds of algebraic properties, too.