The following table illustrates the nature of errors caused by base 2 representation of fractions. The table shows the value of 0.1 approximated in base 2, using increasing numbers of bits.

Table 8: Errors caused by base 2 representation of fractions

The values in the table are not exact, since they were generated using the built-in base 2 floating point types and have been printed to a limited precision. As you can see, the size of the error decreases as a larger number of bits are used. Unfortunately, it will never be zero for a finite number of bits. As a result, calculations performed using a base 2 representation produce counter-intuitive and problematic results in many circumstances. For example, the following code fragment will result in an error message:

double hundredth = 0.01; double sum = 0.0; for(int j=0; j<100; j++) sum += hundredth; sum -= 1.0; if(sum != 0.0) cout << "error!\n";