The principle of least action in analytical mechanics (in complex domain of quantum mechanics this is equivalent to Lagrangian formalism) raises the fact that nature is frugal not lazy. Nature does not waste any unnecessary expenditure of its reserved energy sources. Moreover, in order to conserve and optimize these energy expenses, nature is wise; it uses minimum principles describable by mathematics. One of these found in the real domain is given by the following: There exist two real numbers, a and b such that their products are equal to their differences, ab = a - b if and only if a = b / (1 - b) and b = a / (1 + a).
If a and b are dimensions of force then equality of forces implies that a = b = 0. If a = ∞ then b = 1 and if a = 1 then b = ∞.
If a and b are dimensions of time then the interval Dt = t(a) – t(b) is meaningful only if t(a) ≠ t(b) otherwise Dt = t(a) Ä t(b) = 0. This suggest that the physical definition of acceleration as distance per unit time per unit time implies t(a) Ä t(b) = t(a) – t(b). This can be used to remove time parameterization in quantum mechanics and quantum field theories. Hence it proved the independence of the phase factor exp(iq) in all quantum mechanical wavefunctions.
If a and b are dimensions of length then if a metric (distance) is defined as m = a - bthen m = aÄbcan be used to prove the differential form of Stoke’s theorem in vector analysis.
If a and b are dimensions of energy then if a= 1is the kinetic energy and b= ˝is the potential energy then ab = a - bsatisfies the virial theorem. Furthermore, square of energy as ab implies that ab = (a - b)(a + b) where the factor (a + b) is normalized into a unit circle a + b = 1 implying that a + b = a + b = 1 which is true if and only if a= 1and b= 1/∞ or if a= 1/∞and b= 1.
If a and b are dimensions of speed then the g-factor of special relativity becomes the real part of an imaginary number.


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with polar form
and normalized duals
with polar form
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