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I was reading about real number system briefly as a part of my pre-calculus study and read about Archimedean property of real numbers which says that

If $ x \gt 0 $ and if * y * is an arbitrary real number, there exists a positive integer * n * such that $ nx \gt y $.

They stated it as a consequence of least upper bound axiom for real numbers. Geometrically they said it means that * any line segment, no matter how long, may be covered by a finite number of line segments of a given positive length, no matter how small. *

I could well understand the statement but could not relate it with the Archimedean property . I thought that may be in this geometric intuition n may mean that finite number of small line segments and x and y be length of the small and large line segments respectively. But I'm not sure about this and need help in this matter. Any suggestions or help is appreciated.

Thanks .

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It basically tells you that there is no such a thing called largest number when you take the Real number line. What they mean by any line segment can be covered by a finite number of line segments is you since you are using some natural number n you can cover that segment in finite number of steps though it can be very large still you can find some n. I read this somewhere, it's like you can measure a very large length using a meter stick of a known length in a finite number of attempts. For example let y be 675.47778, you can cover this by taking x as 5.677 and your n as 120. That is if your measuring tape is approximately 5.677 unit long you can measure that distance of 665.47778 by measuring 120 times.

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    $\begingroup$ I'm not sure this is the same thing as "no largest real number". There are ordered fields (e.g. ultrapowers of $\mathbb R$) that don't satisfy the Archimedean property, but still don't have a largest number. $\endgroup$ Apr 6 '17 at 12:19
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Let $y$ be the length of your line segment. Let $x$ be the length of another line segment. Then we can cover the original line segment with $n$ segments of length $x$ since $nx > y$.

Another form that will be probably be more useful to you in the future is the case $y=1$. Now it says, no matter how small $x$ is, there exists a natural number $n$ such that $$x > \frac{1}{n}. $$

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In ordered fields like the real numbers, there is a notion of an element being infinitesimal in comparison to another. Given non-zero elements $x$ and $y$ of such a field, we say that $x$ is infinitesimal compared to $y$ if, for all positive integers $n,$ we have $nx<y.$ We could also say that $y$ is infinite compared to $x.$ The Archimedean property then states that there is no such thing as real numbers which are infinite(simal) in comparison to others.

As for your suspicion about the geometric interpretation, you are exactly right.

See here for much more.

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