Summary: The numerical algorithms for calculating Riemann-Liouville (RL) fractional-order integrations and derivatives are studied in this paper. Firstly, the definitions of RL fractional-order integrations and derivatives are analyzed. There is an integral flaw in the definition, so it is difficult to calculate RL fractional-order integrations and derivatives. Then, the method of removing the integral flaw is given, the numerical algorithms for calculating RL fractional-order integrations and derivatives are designed. It is proved that the numerical algorithms are first order accuracy. Finally, an example is given, the results show that the proposed algorithms are effective.