The Strange Occurrence of Pi Everywhere

[robert Ihnot]

GibZow does it related to the observation of the arc and sin of a smaller and smaller angle? Did you mean the tan of 1? But yes that is correct, involving radians will bring pi into the matter, naturally

I am talking about the derivative of the sine, which is the cosine, we have to find $$lim \frac{sinh}{h}\rightarrow1$$ $$h\rightarrow0$$ This can be found in almost any Calculus book.

It's when we get to the Calculus that we have to use radians. And it's only at the time of the Calculus that the Leibniz found his formula.

The original definition of the sine is the side opposite over the hypotenuse. But in the Calculus we are using, what can be called circular trigonometric functions, where the angle is defined in terms of the unit circle.

 

[Gib Z]

O ok see what you were talking about now. Thank you for your patience and help Robert.

 

[robert Ihnot]

Your welcome!

 

[jwalker1196]

Is it possible, given the infinite possible "base" number systems that can be used (we use base 10 of course) that e and pi have a very simple definition in one "native" base system?

This lends itself to a creator of course. Or an influencer. ZapperZ made a very good analogy.

I myself am a firm agnostic... but that doesn't mean there isn't something out there. We just haven't found it yet, so I'll wait.

 

[Gib Z]

I don't see the problem, pi and e have the same definitions in any base, and the aren't extremely complex anyway.

In any base e is the unique number that makes the integral true: $$\int^e_1 \frac{1}{dx} dx = 1$$ IE e is the unique number where the area under the graph of 1/x from 1 to e is equal to 1.

Also, in any base, pi units is half the length of the circumference of the unit circle, (though only in a Euclidean space), or the smallest positive value of x for which sin (x) is equal to 0.