In our school mathmatics, the irrational numbers and the real numbers are defined and instructed on the basis of decimal fractions. In relation to this fact, we identified the essences of the real number and the irrational number defined by the decimal fractions through the historical analysis. It is revealed that the formation of real numbers means the numerical measurements of all magnitudes and the formation of irrational numbers means the numerical measurements of incommensurable magnitudes. Finally, we suggest instructional plan for the meaninful understanding of the real number concept.