A close approximation for square root of 2.

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Discussion Overview

The discussion centers around the exploration of mathematical coincidences, particularly focusing on approximations related to the square root of 2 and other constants like pi and e. Participants examine whether such coincidences are meaningful or merely happenstance, and they consider the implications of these observations in mathematical contexts.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant presents an approximation involving a series and questions whether such coincidences are significant or just random occurrences.
  • Another participant expresses skepticism about the likelihood of these coincidences being meaningful and suggests that proving pi can be expressed as a ratio would be a major breakthrough.
  • A different participant notes that the series mentioned is divergent and that the closeness of the partial sum to another value may not be as surprising as it seems, implying that such occurrences could happen frequently.
  • Another participant shares a personal observation about the square root of 79 and suggests that there may be more to learn about these coincidences, indicating a belief in underlying patterns.
  • One participant reiterates skepticism about the likelihood of finding a method to calculate pi to any decimal place or proving it as a ratio, emphasizing the complexity of such a task.

Areas of Agreement / Disagreement

Participants express differing views on the significance of mathematical coincidences, with some finding them intriguing while others remain skeptical about their relevance. There is no consensus on whether these coincidences hold deeper meaning or are simply random.

Contextual Notes

Participants reference divergent series and the nature of mathematical coincidences without resolving the implications of these observations. The discussion reflects a range of perspectives on the relationship between mathematical expressions and their approximations.

Boorglar
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By chance I stumbled on this "almost" equality:

[itex]\frac{1}{5}(1/2+2/3+3/4+4/5+5/6+6/7+7/8+8/9+9/10) ≈ √2 - 7.2×10^{-6}[/itex]

I'm just wondering, are these funny coincidences simply, well, coincidences :biggrin: or is there some kind of explanation?
I've see a ton of other funny stuff like [itex]e^{\pi} - {\pi}≈19.99909998[/itex].
How likely is it for a relatively simple expression involving unrelated constants to work out almost nicely? Is it actually easy to come up with these meaningless things?
 
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I don't think it's very likely. Now, if someone could by chance stumble upon a way to calculate pi to any decimal place (i,e, whoever runs the equation can calculate pi to as many decimal places as he pleases) or prove us all wrong by showing that pi can be expressed as a ratio, that would be great.
 
moonman239 said:
I don't think it's very likely. Now, if someone could by chance stumble upon a way to calculate pi to any decimal place (i,e, whoever runs the equation can calculate pi to as many decimal places as he pleases) or prove us all wrong by showing that pi can be expressed as a ratio, that would be great.

There are many known series for pi.

http://mathworld.wolfram.com/PiFormulas.html

As far as the OP's question, note that the series is divergent. It just happens to be the case that a particular partial sum is close to something else. Take one more term of the series and the coincidence disappears. It seems that would happen fairly often ... it's not as big a coincidence as some series or expression actually converging close to something else, like the ones here ...

http://en.wikipedia.org/wiki/Mathematical_coincidence
 
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nice I like that with [itex]e[/itex][itex]\pi[/itex]-[itex]\pi[/itex]

I accidentally found [itex]\sqrt{79}=8.888194417...[/itex] which I like to say/present as [itex]\sqrt{69+10}[/itex] but I don't think these things are coincidental , probably we have a lot more to learn.
 
moonman239 said:
I don't think it's very likely. Now, if someone could by chance stumble upon a way to calculate pi to any decimal place (i,e, whoever runs the equation can calculate pi to as many decimal places as he pleases) or prove us all wrong by showing that pi can be expressed as a ratio, that would be great.

∏=[itex]\frac{c}{d}[/itex] :cool:
 

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