✎ Exponential Function [The Physics Travel Guide]

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====== Exponential Function ======

<blockquote>Describing e as “a constant approximately 2.71828…” is like calling pi “an irrational number, approximately equal to 3.1415…”. Sure, it’s true, but you completely missed the point.

Pi is the ratio between circumference and diameter shared by all circles. It is a fundamental ratio inherent in all circles and therefore impacts any calculation of circumference, area, volume, and surface area for circles, spheres, cylinders, and so on. Pi is important and shows all circles are related, not to mention the trigonometric functions derived from circles (sin, cos, tan).

e is the base rate of growth shared by all continually growing processes. e lets you take a simple growth rate (where all change happens at the end of the year) and find the impact of compound, continuous growth, where every nanosecond (or faster) you are growing just a little bit.

e shows up whenever systems grow exponentially and continuously: population, radioactive decay, interest calculations, and more. Even jagged systems that don’t grow smoothly can be approximated by e.

Just like every number can be considered a scaled version of 1 (the base unit), every circle can be considered a scaled version of the unit circle (radius 1), and every rate of growth can be considered a scaled version of e (unit growth, perfectly compounded).

So e is not an obscure, seemingly random number. e represents the idea that all continually growing systems are scaled versions of a common rate.
<cite>[[https://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e/|An Intuitive Guide To Exponential Functions & e]] by Kalid Azad</cite></blockquote>

<note tip>
Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.
</note>
  
<tabbox Student>

* The best introduction is [[https://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e/|An Intuitive Guide To Exponential Functions & e]] by Kalid Azad
<tabbox Researcher>

<note tip>
The motto in this section is: //the higher the level of abstraction, the better//.
</note>

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