Surprisingly, using these representations even simple operations such as addition and subtraction are not computable. It is not clear whether the first digit of the result here should be a one or a zero. If, for example, the numbers continued as shown here until 100 th digit of a , but at this point the next digit of a was greater than three then the result of the calculation would be greater than one. If the same digit in fact turned out to be less than three, the result would be less than one. In general it is not possible to compute an output digit without examining an infinite number of digits.
Similar examples show that all integer bases suffer from the same problem.
In order to find algorithms for exact real number computation using infinite objects, we must use different representations. The representations used in this work are described next, and we also mention a number of other possible representations. Note that all the representations used for exact real arithmetic are equivalent to one another in the sense that it is possible to convert between them in a computable way. Some representations may be more efficient than others.