Why is epsilon not a rational number?

Question

I was wondering why epsilon, the smallest positive number, isn't a rational number. I was watching a video a few days ago about surreal numbers, and I've learned that, in the field of surreal numbers, o.(9) is not equal to 1, in contrast to the field of the real numbers, where they represent the same number. In the field of surreal numbers, you would get epsilon by subtracting 0,(9) from 1. If you were to do this in the rationals, you would just get 0. But I think there is a method do get epsilon even in the rationals, you would just take the following limit:

$$\lim\limits_{n \to\infty} \sum_{i=0}^n {1\over 10^i}$$ Am I making a wrong mathematical assumption or...? Is there a reason for which epsilon couldn't be a rational number?

Answer 1

See Surreal number :

Consider the smallest positive number in $S_ω$:

$\varepsilon =\{S_{-}\cup S_{0}|S_{+}\}=\{0|1,{\tfrac {1}{2}},{\tfrac {1}{4}},{\tfrac {1}{8}},...\}=\{0|y\in S_{*}:y>0\}$.

This number is larger than zero but less than all positive dyadic fractions. It is therefore an infinitesimal number, often labeled $ε$.

Thus epsilon is, "by definition" less than (and so different from) all rational in the $(0,1)$ interval.

Answer 2

Okay. I do not know much about the surreal numbers. So I looked on wikipedia and these were the VERY FIRST two sentences.

"In mathematics, the surreal number system is an arithmetic continuum containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. The surreals share many properties with the reals, including a total order ≤ and the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an ordered field."

This answers why epsilon, the smallest positive (surreal) number, can not be rational. The rational numbers and the real numbers have the archimedian property that between any two rationals and between any two reals there exist a third. So it is impossible to have a smallest positive real or rational number. So if a smallest positive number is possible (as apparently it is) it can not be real or rational otherwise it would be the smallest real or rational number which is impossible.

So, I interpret the OP question to be why doesn't his construction of $\lim_{n \rightarrow \infty} \prod_{i=1}^n\frac 1{10^i} $ produce a least rational number?

Because the lim = 0. (It also needn't be rational {although it is}. But it does need to be real-- and there is no least positive real either.)

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Okay, I wasn't aware that epsilon exists in the field of surreal numbers and does have a definition as the smallest positive surreal number (I guess).

But here, fundamentally, the surreals are not the reals. The rationals are the rationals and as $0 < 1/(n+1) < 1/n$ there can be no smallest positive rationals. Thus any system that allows a smallest positive number (which the reals do not), such a number can not be rational.

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1) epsilon is not the smallest positive number. epsilon is not a set number at all. It is a variable to represent possible numbers. It is no more or less a number than, x, n, or i are numbers for exactly the same reasons and in exactly the same way.

2) There is no smallest positive |real| number. This is fundamental to mathematics real analysis. [ed: and so any smallest positive number can not be real.]

3) The limit of a sequence (or sum) of rational numbers need not be rational at all. Indeed it is a fundamental principal of real [ed: emphasis added later] numbers that all real [ed: but not surreal] numbers (rational and irrationals) are limits of sequences of rational numbers.

4) The limit of the infinite product you gave is 0.

Answer 3

It seems you are assuming that "multiplying infinitely many rational numbers yields a rational number," and this is not necessarily true. Doing things "infinitely many times" (really, taking a limit of a process) is a pretty good way to leave the rationals, even when we stay within the realm of real numbers.

Every (nonnegative) decimal number you can write down is a sum of rational numbers of the form $\frac{d}{10^n}$, where $d$ is an integer between $0$ and $9$, and $n$ is some integer. So, for example,

$$\pi = \frac{3}{10^0} + \frac{1}{10^1} + \frac{4}{10^2} + \frac{1}{10^3} + \ldots,$$

but $\pi$ certainly isn't rational, despite being the sum of infinitely many rational numbers.

For products, we can use Euler's product formula for the Riemann zeta function to write

$$\frac{\pi^2}{6} = \left( \frac{1}{1 - 2^{-2}} \right)\left( \frac{1}{1 - 3^{-2}} \right) \left( \frac{1}{1 - 5^{-2}} \right) \left( \frac{1}{1 - 7^{-2}} \right) \cdot \ldots$$

a clearly irrational number as a product of infinitely many rational numbers, although the value of $\pi^2/6$ is highly nontrivial.