ude that for squaring an integer n, the number of TM computations (or transitions) is asymptotically n4: the result of squaring integer n is a string of n2 + 1 symbols, but it takes n4 head moves to write the result onto the tape. In the preceding five application examples, I have illustrated how the TM can perform various types of basic computations. In all cases, integers were used. But what about real numbers? Can Turing machines also perform computations with real numbers? These are the questions we address next. Considering real numbers, we ought first to distinguish two cases. First, a real number n is associated with a size In/, which ranges from zero to infinity. Second, some real numbers are nonintegers, and, therefore, are defined through a certain suite of decimal places. The suite can be of finite size, such as 5/4 = 1.25 (the other decimal places being zero), or of infinite size, such as 1/3 = 0.333333 or 13/11 = 1.181818 (decimals, or patterns of decimals, being infinitely repeated). We will consider later on the case of numbers whose infinite decimal suites do not exhibit patterns (irrational and transcendental numbers, whose decimals may or may not be computable through a polynomial equation). Just as with our physical computers, real numbers with finite size and finite numbers of decimal places can be handled by the TM through the exponent format n = Pn X 10th or more generally through any type of floating-point representation. For instance, 203.5 = 2.0350 X 102 = 20.350 X 101 = 2035.0 X 10-1 are three possibilities for the same real number 203.5. The case where the number of decimal places is infinite can also be processed by a TM, but only up to some truncation level, just as with any physical computer. For instance: 13/11 = 1.1181818 1.11818 x 101 ~ 1.1181 x 101 ~ 1.118 x 10 ¹ and SO on. Concerning transcendental numbers, 7 like IT or e, we may think that because of infinite memory size, the TM could have an advantage over a physical computer. Alas, the number of decimals for such numbers bein