Another well-known non-positional system is the Roman system. Also here, the letters indicate values, not according to their position: I for 1, V for 5, X for 10, L for 50, C for 100, D for 500 and M for 1000. They can be combined according to rules for yielding other numbers, such as VIII for 8. Certain rules imply subtraction rather than addition: XL, for instance, stands for 40, since X is subtracted from a greater value, L, that comes immediately thereafter. The value 222, for instance, requires in this system a rather long string: CCXXII. Also here, a decimal orientation clearly underlies a nonpositional system. One of the interesting features of the decimal–positional system is that it works equally well for both integer and fractional values: the fraction 12 , for instance, can be written as 0.5. We attach a value to a decimal fraction according to the same principle we use for integers. Indeed, the decimal fraction 0.531, for instance, represents the following sum of decimal powers:
0.531 = 5·10–1 + 3·10–2 + 1·10–3 . Or, in other words,
0.531 = 5·
1 1 1 + 3· + 1· . 10 100 1000
Or, to take a different example,
725.531 = 7·100 + 2·10 + 5·1 + 5·
1 1 1 + 3· + 1· . 10 100 1000
And here we have another fundamental property of the decimal–positional system that we usually take for granted, namely, the ability to recognize, on the basis of representation alone, what kind of number we are dealing with. An integer, for example, is a number whose decimal representation has no digits after the point. In the decimal representation of a rational number, to take another example, the digits after the point 138 eventually repeat themselves in a certain order (such as in 101 = 1.366336633 . . . or 1 = 0.2). In the decimal representation of an irrational number, on the contrary, there 5 is no such repetition (for instance, π = 3.14159 . . . ). The insight that integers and fractions can be written according to the same principles played a major historical role in allowing a broad, unified vision of number. Moreover, this insight teaches an important lesson about the developments we want to consider in this book, because the logic of numbers and the logic of history did not work in parallel here. Indeed, it was not the case that first a clear idea developed of fractions and integers being entities of a same generic type (numbers) and thereafter a notational system developed for conveniently expressing this idea. Rather, it was exactly the other way round: since it was realized that one and the same approach can be followed in writing fractions and integers, the idea was gradually (and not easily) developed that it may make sense to begin seeing fractions and numbers as essentially representing a common, general, idea. This important process will be discussed in some detail in Chapter 7. WRITING NUMBERS NOWADAYS
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