User:MathMan64/CasusIrreducibilis

Casus
Some cubic equations with real solutions have solutions that can not be written exactly, without the use of imaginary expressions.

I plan to find out how to tell which a cubic equations have solutions that are totally real expressions, and to relate this topic to the exact value of trigonometric functions of angles whose degree measure is not a multiple of three.

Criteria for irreducibile polynomials are given by Eisenstein's criterion

Internal links
I want to read the Quotes on my user page.

Living with Logs
From “Death by Black Hole and other cosmic quandaries” by Neil De Grasse Tyson, Norton, 2007, page 26.

We register the world’s stimuli in logarithmic rather than linear increments. For example, if you increase the energy of a sounds volume by a factor of ten, your ears will judge this change to be rather small, (rather than tenfold.) Increase it by a factor of two and you will barely take notice. The same holds for our capacity to measure light. If you have ever viewed a total solar eclipse you may have noticed that the sun’s disk must be at least ninety percent covered by the moon before anybody comments that the sky has darkened. The stellar magnitude scale of brightness, the well-known acoustic decibel scale, and the seismic scale for earthquake severity are each logarithmic, in part because of our biological propensity to see, hear, and feel the world that way.

(Consider in addition, the pH measurements for acid and base, the musical scale, and the way we discuss the size of the universe in powers of ten, from the galaxies near the large end and the quarks near the small end.)

Piano keys
To find the piano key that corresponds to a given frequency, use the following formula.


 * $$n= b \cdot ln \left ( a \cdot p \right )$$

where p is the pitch or frequency and n is the number of the key on the keyboard starting from the left and counting both white and black keys.


 * $$a = 0.038525930704 \ $$


 * $$b = \sqrt[12] 2 \ = 0.943874312682$$

Least Squares
The common computational procedure to find a first-degree polynomial function approximation in a situation like this is a follows.

Use $$n \ $$ for the number of data points.

Find the four sums: $$\sum x$$, $$\sum x^2$$, $$\sum y$$, and $$\sum xy$$.

The calculations for the slope, m, and the y-intercept, b, are as follows.


 * $$m= \frac{(\sum y)(\sum x)-n(\sum xy)}{( \sum x)^2 - n(\sum x^2)}$$

and


 * $$b=\frac {(\sum x)(\sum xy)-(\sum y)(\sum x^2)}{( \sum x)^2 - n(\sum x^2)}$$

Cosine of pi over powers of 2

 * $$\cos \frac{\pi}1 = -1$$


 * $$\cos \frac{\pi}2 = 0$$


 * $$\cos \frac{\pi}4 = \frac 1 2 \sqrt 2 $$


 * $$\cos \frac{\pi}8 = \frac 1 2 \sqrt{\sqrt{2}+2} $$


 * $$\cos \frac{\pi}{16} = \frac 1 2 \sqrt{\sqrt{\sqrt{2}+2}+2} $$


 * $$\cos \frac{\pi}{32} = \frac 1 2 \sqrt{\sqrt{\sqrt{\sqrt{2}+2}+2}+2} $$

Add and Divide and copy
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