Circle in the Euclidean space using Euler's Number

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Discussion Overview

The discussion revolves around the concept of Euler's Number and its relationship to various mathematical ideas, including limits, circles in Euclidean space, and complex numbers. Participants explore the implications and interpretations of these concepts without reaching a consensus.

Discussion Character

  • Exploratory
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants mention the expression (1 + 1/n)^n and its connection to Euler's Number, noting the limit as n approaches infinity.
  • There is confusion regarding the phrase "0 to 1 in Euclidean space," with questions about what is changing in this context.
  • Participants inquire about the meaning of "1 to 0 with the circle" and its relevance to the discussion.
  • One participant suggests a connection to the complex plane, specifically mentioning e^{iθ} as a representation of a circle with radius 1.
  • Another participant references the series expansion for (1 + 1/n)^n, indicating a more technical perspective on Euler's Number.
  • There is a general appreciation for the significance of Euler's Number in various mathematical contexts, though the specific connections remain unclear.

Areas of Agreement / Disagreement

Participants express confusion and seek clarification on the initial statements, indicating that multiple interpretations exist without a clear agreement on the intended meaning.

Contextual Notes

Limitations in understanding arise from ambiguous phrasing and the need for clearer definitions of terms used in the discussion.

OrthoJacobian
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0 to 1 in Euclidean space.

(1 + 1/n)^n using Euler's Number.

1 to 0 with the circle.

How amazing is Euler's Number?!
 
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OrthoJacobian said:
0 to 1 in Euclidean space.

(1 + 1/n)^n using Euler's Number.

1 to 0 with the circle.

How amazing is Euler's Number?!

What...?

But welcome to PF!
 
What do you mean by "0 to 1 in Euclidean space"? What is changing from 0 to 1?

What do you mean by "(1+ 1/n)^n using Euler's number"? Yes, the limit, as n goes to infinity is Euler's number but I would not say "with" Euler's number.

And, finally, what do you mean by "1 to 0 with the circle"? What is changing from 1 to 0 and what does that have to do with the circle?
 
I'm so confused by this post. Are you talking about how ##e^{i\theta}## is a circle in the complex plane with radius ##1##, or how the series expansion for ##(1+\frac{1}{n})^n## is ##e-\frac{e}{2n}+O(\frac{1}{n^2})##, or something else?

Regardless, e certainly is an amazing number and pops up in tons of (un)expected places.
 

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