Most modern computers with very few exceptions operate on the binary number system, This paper suggests a new generation of computer systems capable of performing computations in many number systems. At the present, a modern computer handles numbers other than binary through encoding prior to performing the arithmetic operations. The converting processes of encoding and decoding are both wasteful and time consuming. The proposed system is able to carry out computations in a given radix without conversions. In order to operate on a specific number system a radix control can be preset to the desired value, the computer is then converted instantly to be of that radix. In general, a number system is formed basically from counting with proper carry or carries to be generated when a desired radix is reached. High speed pulse counting with a controllable and variable carry producing mechanism is thus the most direct and efficient approach for the implementation of a variable radix computer system. In essence, a high speed electronic variable radix computer is a counterpart of a mechanical calculator which would consist of essentially wheels with number of teeth variable. This paper attempts to lay the basic and the most essential foundation necessary for such a versatile system. Similar to the conventional systems, the addition of numhers in an arbitrary radix can be performed by the use of a variable radix adder ; and the subtraction by the addition of the (R -l)'s complement, where R is the radix. Multiplication can be carried out by repeated additions and the division by the cyclic subtractions. The chosen number representation will be presented in the paper first and then the formuli, which are given either in simple expressions or in the compact series forms, for the basic arithmetic operations. In addition, numerical examples in various radices are given for the readers to ponder and enjoy.