The common approach to real arithmetic on computers is floating point arithmetic, which can produce erroneous results due to roundoff errors. An alternative is exact real arithmetic and in this project such arithmetic is implemented in the well-known computer system Mathematica by the use ofconstructive real numbers. All basic operations are implemented as well as the common elementary functions and limits of general convergent sequences of real numbers. Also, as an application to ordinary differential equations, Euler's method for solving initial value problems is implemented.