Although physical reality is best modeled by the use of real numbers of infinite decimal places called irrational numbers (e.g. transcendental number like the ratio of the circumference over the diameter of an arbitrary circle and the square root index of radical with the radicand of 2), every digital computer can store only a truncated approximation of these numbers. These numbers are irrational numbers to begin with but after truncated inside the computer register they all become rational numbers. The largest number that can be input into a computer is limited by the size of its register. This register can hold the significant digits of any real number expressed as the product of this number and it base-10 exponent.


It is justifiably asserted that the best model of reality is limited by a truncated rationality. This imply that whenever someone calculated the area of a circle with given radius as an exact whole number, say the radius is equal to unity then the area is simply calculated as the number p square units, which is approximated by the rational number given as decimal 3.141592654 or as an improper fraction 314159254/1000000000. In this case, the significant digits are 314159254 and either the decimal or the denominator of 1 billion gives the equivalent base-10 exponent. Therefore, the calculated area is always less than the actual area of the circle. This is true for all physical calculations involving any factor of p including volume calculation. On the other hand, if p becomes a factor of the divisor then the calculated value is always greater than the actual value. For example, calculating the quantum number expressed as Planck’s constant of action divided by 4p. In all these cases, it is important to note that both p and 1/p must be expressed as an infinite series whose limit exists in order to improve the accuracy of the calculated value of this particular truncated rationality. The convergence of any infinite series used thus justifies the subsequent values of truncated rationality.