g infinite, the TM would also never halt in the task of performing its infinite computation, and this points to the same problem of truncation. It is then possible to devise a TM program that outputs the decimal places of IT or e, and stores them on the TM tape, but the machine would never halt unless the program included an algorithmic counter to set a limit to the output's size. The same restriction applies to physical computers, with the difference that the limit is intrinsically defined by memory size and execution time. It would appear from the above analysis and conclusions that any real number having an infinite number of decimal places, including transcendental or irrational numbers, such as IT, e, 2 , is TM computable, at least up to some truncation level. But it turns out to our surprise that this is not the case! 7 Meaning numbers that are not a solution of any polynomial equation; not to be confused with irrational numbers (such as 2) which apply to this case (namely, here, the solution of x2 - 2 = 0). See for instance http://en.wikipedia.org/wiki/Transcendental_number 7.3 Universal Turing machine Universal Turing machine Indeed, Turing showed that for some numbers there exists no table of instructions that is capable of generating the nth digit of its representation for any arbitrary n. Such numbers are said to be incomputable numbers. Now we are entering the core of the subject of "algorithmic complexity," where our mind is going to be further challenged! For starters, we shall note the following: The number of computable numbers is relatively small compared with the number of incomputable ones; Computable numbers with an infinite set of decimal places can be computed by a TM program of finite size. See further for a discussion regarding computable numbers. We have seen that the TM needs an action table, call it a program, to perform computations. But where would such a program be physically located? The answer is that we can use the input tape as a way to store this program. It may be written in th