Am Montag, 10. Februar 2014 11:51:05 UTC+1 schrieb thenewc...@gmail.com:
> But pi and sqrt(17) are not numbers, they are incommensurable magnitudes. There is a big difference between a magnitude and a number.
Of course. Before I wrote my book I considered to call then irrationalities in order to emphasize the difference. But the notation real number has been generally accepted (like the notation complex number has been accepted) and it would require always long and winded explanations to deviate from that agreement.
> In fact, there are infinitely many rational numbers that cannot be represented in decimal.
Yes, if you restrict your notation on the fixed base 10 (deci), that is right, for example for 1/3. But there are other bases such that every rational can be expressed without error. >
> Actually e can be represented exactly! e is exactly represented by f(0) where f(x)= (1+x)^(1/x).
Amusingly, Euler has used that kind of notation: (1 + 1/oo)^oo, often with an "i" for numerus infinitus. But it is easy to confuse the meaning, because 1 + 0 = 1 and 1^whatever is 1.
> There is no need to consider limits at all, which are inherently ill defined.