Archive for November, 2013

Complex Numbers and Quaternions

November 29, 2013

complex_numbersTwo conceptualizations of four directions in mathematics are Complex numbers and Quaternions. For complex numbers, two of the directions are the opposite or negative of the other two. Complex numbers are like Cartesian coordinates in that they combine an (x,y) coordinate pair of real numbers into one complex number x + yi. Plus, complex numbers can by manipulated by extensions of arithmetic operations, like addition, subtraction, multiplication, and division.
quaternionsQuaternions are an extension of the complex numbers where the directions are all different, and each direction is perpendicular to the other three. Like complex numbers are a notion of numbers that cover a plane, quaternions are a notion of number that fill four dimensional space. Thus they combine a 4-tuple (w,x,y,z) into one quaternion number w + xi + yj + zk. Like complex numbers, quaternions can be manipulated by further extensions of arithmetic operations.

http://en.wikipedia.org/wiki/Complex_number

http://en.wikipedia.org/wiki/Quaternion

[*7.189]

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Ohm’s Law

November 8, 2013

ohms_lawCharts for the equations of Ohm’s Law usually consist of the fourfold relation between Voltage (Volts), Current (Amps), Resistance (Ohms), and Power (Watts). For each electrical quantity, there are three equations that represent it in terms of two others, making twelve equations total.

Ohm’s Law is really only between voltage, current, and resistance. Apparently, power is more correctly introduced by the formula for “Joule heating”.

Both voltage and current have been in two fourfolds previously (Four Basic Electrical Components and System Dynamics) but not power and resistance. Well, resistance did make an appearance as a relation between current and voltage.

The twelve equations can be generated by the following identities (where E = voltage, I = current, R = resistance, and P = power):

I*R/E = P/(I*E) = P*R/E^2 = R*I^2/P = 1

http://en.wikipedia.org/wiki/Ohm%27s_law

http://en.wikipedia.org/wiki/Joule%27s_first_law

Google search for images for Ohm’s Law.

[*8.4, *8.5, *8.8, *8.9, *8.10]

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The Four Temperaments

November 8, 2013

four_temperamentsWhat would a list of fourfolds be without mentioning the Four Temperaments: Sanguine, Choleric, Melancholic, and Phlegmatic?

Indeed, these personality types from ancient medicine and philosophy are still a source of classification and inspiration for modern psychology, even though the reasons for the differences (the four humors) has been discredited. The four humors were blood, yellow bile, black bile (they were big on bile), and phlegm, and were also related to the Four Seasons and the Four Elements.

Below is psychologist David Keirsey’s modernization of the four temperaments, consisting of Idealist, Artisan, Guardian, and Rationalist.

keirsey_temperamentsReferences:

http://en.wikipedia.org/wiki/Four_temperaments

http://en.wikipedia.org/wiki/Humorism

http://en.wikipedia.org/wiki/Myers-Briggs_Type_Indicator

http://en.wikipedia.org/wiki/David_Keirsey

http://en.wikipedia.org/wiki/Keirsey_Temperament_Sorter

Images of the Four Temperaments.

[*6.122, *6.123, *7.194, *8.14, *8.94, *8.139]

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