Zeta 137: 18. Complex formulas

The following equation relates the well-known mathematical constants e, pi, i, to derive another famous constant:

the golden number φ = 1.61803398874989...

As with Euler's identity, also in this case some of the most important mathematical constants appear simultaneously in the same formula.

To demonstrate the relationship, we must start with the Golden Ratio.

It involves dividing a segment AB into 2 parts (which we will call AC and CB) in such a way that the continuous proportion AB : AC = AC : CB holds.

Euclid used this formula when working on pentagons.

We set the smallest segment CB = 1 and AC = x, from which AB = 1 + x

The required condition is therefore: (1 + x) / x = x / 1

So, we have x² – x – 1 = 0

The solutions of this quadratic equation have as solutions:

The Golden Section was the starting point for the Greek study of regular pentagons, and everything associated with them, such as the decagon, the dodecahedron and the icosahedron.

As we will see later, if we draw a regular pentagon with side 1, then the diagonals have the golden number as their length.

The term Golden Ratio is relatively recent and appears to have been used for the first time by Martin Ohm (brother of the more famous Georg Simon Ohm who gave his name to the law) in his 1835 book.

Before seeing why, let's review some trigonometry notions.

Trigonometry studies right triangles starting from their angles. Its main task consists in calculating the measurements that characterize the elements of the triangle (sides, angles, etc.) by means of special functions starting from already known measurements.

Trigonometric functions (the most important are sine and cosine) are also used independently of geometry, for example in connection with the exponential function.

1) The sine of an angle is the ratio between the length of the opposite side and the length of the hypotenuse.

2) The cosine of an angle is the ratio between the length of the adjacent side and the length of the hypotenuse.

Euler's formula is a formula in the field of complex analysis that shows a profound relationship between trigonometric functions and the complex exponential function. Euler's identity is a special case of Euler's formula.

Euler's formula, named after the mathematician Leonhard Euler, was first proven by Roger Cotes in 1714 and then rediscovered and made famous by Euler in 1748. Neither saw the geometric interpretation of the formula: the vision of complexes numbers like points in the plane arrived only about 50 years later, thanks to Wessel, Argand and Gauss.

The most widespread proof is based on the Taylor series expansion of the exponential function.

Euler's formula also allows the sine and cosine functions to be interpreted as simple variants of the exponential function:

Euler's formula: e^ix = cos x + i sin x

Euler's formula also allows the sine and cosine functions to be interpreted as simple variants of the exponential function:

sin x = ( e^ix - e^-ix ) / 2i cos x = ( e^ix + e^-ix ) / 2

As is known, angles can be expressed in different ways, the most used are sexagesimal degrees and radians. Below, depending on the purpose, both will be considered.

Let's now show some notable angles: 30, 36, 45, 60 and 90 degrees.

In particular, the sine (30°) = ½ (whose square is equal to ¼); similarly, the sine squares of 45, 60 and 90 degrees are respectively: 2/4, 3/4 and 4/4.