E

A page for e, i, π, and φ

From R.Knott's Golden Mean page tan x = cos x and, since tan x = sin x / cos x, we have: sin x =(cos x)2 =1-(sin x)2 because (sin x)2+(cos x)2=1. or (sin x)2 + sin x = 1 and solving this as a quadratic in sin x, we find sin x = (-1+sqrt5)/2 or sin x = (-1-sqrt5)/2 It's the geometric way of creating the quadratic structure yielding phi... The real component of 2eiπ/5 is Φ because Φ = 2cos(π/5); And also, the imaginary component of 2eiπ/10 is φ since φ = 2sin(π/10);

The role that the exponential function (e^x) plays is extremely significant to an understanding of the laws of nature and harmonic organisation.

[math]e^x = \sum_{n = 0}^{\infty} {x^n \over n!} = 1 + x + {x^2 \over 2!} + {x^3 \over 3!} + {x^4 \over 4!} + \cdots[/math]

As an infinite sum it is Σ(xⁿ/n!)   =   1   +   x / 1   +   x.x / 1.2   +   x.x.x / 1.2.3   +   x.x.x.x / 1.2.3.4 + ...  

We can look at the construction of this relation more easily by describing it by an algorithm. We can remove the power and factorial operations by basing the current numerator and denominator values on the previous values:

+e.pl

But a better conceptual representation is to restructure it such that the division occurs outside the main loop since its also a higher-level operation. So each iteration is now maintaining the value of a numerator and denominator which represent the entire preceeding series as a single fraction which can be reduced afterwards.

+alt e.pl

Here's a snipit in C which uses the taylor series of cosine to make &Phi from π

+phi.c

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See also

• Exponential function
• E (mathematical constant)
• Euler's identity