This thesis presents a study of the understanding of real, rational, and irrational numbers by 30 fourth semester college science students in the Montreal region.  The written answers to a set of seven questions were analyzed to determine the students’ interpretations of mathematical signs according to C. S. Peirce’s classifications and to describe their modes of thought according to Vygotsky’s theory of concept development.  From these interpretations, we are able to reconstruct a facsimile of what the students’ concept images are as they pertain to the sets in question.  Finding the concept images to be idiosyncratic and rarely in agreement with what conventional mathematics holds to be true, we examine the way the number systems are approached in school and in the field of mathematics and use this, along with the analyses, to make pedagogical recommendations.