Near Equalities in Mathematics

  • Context: High School 
  • Thread starter Thread starter AlbertEinstein
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Discussion Overview

The discussion revolves around the concept of near equalities in mathematics, exploring various examples that illustrate how certain mathematical expressions can yield results that are very close to well-known constants or integers. The scope includes theoretical musings and numerical curiosities without reaching definitive conclusions.

Discussion Character

  • Exploratory
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant introduces near equalities related to pi, specifically mentioning the fractions 22/7 and 355/113, and presents a method to derive a number close to pi using the fourth root of a division involving 2143 and 22.
  • Another participant references the expression e^{\pi\sqrt{163}}, noting that it is very nearly an integer, providing a specific numerical approximation that highlights the limitations of calculator precision.
  • A different participant humorously mentions a near equality involving the expression (9876543210 + .0123456789) / 9876543210, suggesting it is close to 1 but not exactly.
  • One participant challenges the notion of what constitutes "thrilling and exciting" in mathematics, indicating a difference in perspective regarding the examples presented.

Areas of Agreement / Disagreement

Participants express differing views on the excitement and significance of the examples provided, with some finding them intriguing while others question their value. The discussion remains unresolved regarding the overall importance of these near equalities.

Contextual Notes

Some examples rely on specific numerical approximations and may depend on the limitations of computational tools, which could affect perceptions of their accuracy.

AlbertEinstein
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Near equalities...

Mathematics usually contains a few coincidences which themselves give an account on how thrilling and exciting mathematics is.
Here I have started a new thread whisch enlists some near(or very near equalities. Hope you guys would extend the thread!

1. We all know the near equalities of pi such as 22/7 ,355/113 etc (which are a result of continued functions).But do you know :Write 1234 as 2143 and then divide 2143 by 22. NOW take the 4th root of the result(i.e. take the sqrt and again take the sqrt).Isn't the number which you get now tantalizing close to pi !
pi^4 ~ 2143/22where ~ denotes near equality.

2. 3^2 + 4^2 = 5^2 from the old pythagoras theorem , but do you know 3^3 + 4^3 + 5^3 = 6^3. Exciting!
 
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You have a very strange idea of what "thrilling and exciting" are!
 
The one that pops to mind for me is the expression:

[tex]e^{\pi\sqrt{163}}[/tex]

This value is very nearly an integer. But, not quite. The value is:

262537412640768743.99999999999925007259...

If you type the above expression into a non-graphing calculator, the result will come out as an integer because the number of 9's exceeds the calculator's floating point abilities.
 
Yeah my favorite is (9876543210 + .0123456789) / 9876543210

It comes out amazingly close to 1 but not quite. [j/k]
 

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