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let think x is real number, then \begin{align} x&=x\\ x&=(x/2)+(x/2), \\ x&=(x/3)+(x/3)+(x/3), \\ x&=(x/4)+(x/4)+(x/4)+(x/4), \\ &\vdots \\ x&=(x/n)+(x/n)+(x/n)+ \cdots +(x/n), \\ \end{align} If $n$ is too large and equal to infinity, then

$x=(x/\infty)+(x/\infty)+(x/\infty)+(x/\infty)+\cdots+(x/\infty)$

then

$(x/\infty)= \epsilon$ and this $\epsilon$ is equal ant to zero. Because in goes below planck scale and we can't observe it. my question is that who described it?

Brian
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  • Welcome to Math.SE! Please format your question using MathJax. [This page](https://math.meta.stackexchange.com/q/5020) should give you a start at learning how to typeset mathematics here so that your posts say what you want them to, and also look good. – Brian Mar 26 '19 at 12:32
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    Planck scale is a physical properties of the space - time. You refere to a mathematical assumption about infinity – Riccardo.Alestra Mar 26 '19 at 12:36
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    That seems weird. What does Planck length have to do with real numbers? We don't need to observe real numbers, in order to calculate with them. – Matti P. Mar 26 '19 at 12:36
  • What your describing looks a lot like non-standard analysis, for which I once wrote a good introduction: https://www.universiteitleiden.nl/binaries/content/assets/science/mi/scripties/masterclaassens.pdf. Let me know if anything's unclear. – Floris Claassens Mar 26 '19 at 12:57

1 Answers1

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I do not believe a reliable reference can be found since the method you describe is inherently flawed. The value $n$ is never "equal to" $\infty$. Likewise, there is no such number as "$(x/\infty)$," and it is most certainly not equal to 0.

Rather, it would appear that you are attempting to evaluate the limit of the series $$ \sum_{i=1}^{n}\frac{1}{n} $$ as $n$ approaches $\infty$ by treating the series as

$$ \lim_{n\to\infty}\sum_{i=1}^{n}\frac{1}{n} = \sum_{i=1}^{n}\lim_{n\to\infty}\frac{1}{n} = \underbrace{\lim_{n\to\infty}\frac{1}{n} +\lim_{n\to\infty}\frac{1}{n} + \cdots + \lim_{n\to\infty}\frac{1}{n}}_{n\ \text{times}} $$ which is nonsensical, since the number of elements in the series depend the value of $n$, which no longer "exists" outside of the sum.

You may find the answers to this semi-related question helpful, which concludes that $1=2$ by computing $$\frac{\mathrm d}{\mathrm d x}x^2 = \frac{\mathrm d}{\mathrm d x} \underbrace{x+x+\cdots+x}_{x \ \text{times}}$$

As noted in the comments, this is mathematics, not physics, and so we do not have to worry about physical limitations such as "going below the plank scale." Real numbers are not quantized, and we are free to talk about finite values that are as small as we like.

Brian
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