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Algorithm of grey numbers and grey algebraic system is the theoretical basis of grey system theory. It is possessed of very important significance in the developing course of grey system theory. A definition for grey "Kernel" has been put forward. The axioms for algorithm of grey numbers and a grey algebraic system is built based on grey "Kernel" and the degree of greyness of grey numbers. And the properties of the algorithm are studied. Up to now, the algorithm of grey numbers has been transformed to algorithm of real numbers. So, the difficult problem for setting up algorithm of grey numbers and grey algebraic system has been solved to a certain degree. The algorithm of grey numbers could be extended to the case of algorithm of grey algebraic equation, grey differential equation and grey matrix. The algorithm system of grey numbers is also with great value for research on grey input-output and grey programming, etc.