In the history of mathematics, one person was able to begin matching the real to the imaginary, the rationals to the irrationals. He was Leonard Euler (1707-1783). In the 1740s while doing researches at the St. Petersburg Academy of Sciences he discovered the mathematical fact that the irrational base of natural logarithms raise to the power of the product of the transcendental irrational ratio of any arbitrary circle’s circumference over its diameter with the imaginary square root of negative unity is equal to the rational integer of negative unity. However, this is only half the true story of mathematical discoveries.

The other half is the fact that the negative same power of the irrational base of natural logarithms is also equal to negative unity such that their product gives the zero power of the irrational base of natural logarithms which is truly positive unity as square of negative unity or the 4th power of the imaginary unity, thus matching the set of rational numbers to the set of imaginary numbers creating the set of complex numbers. Nonetheless, more analytical complications could not provide the intimate disjointed directional connection at the rational value of zero which prompted further discoveries of sets of hypercomplex numbers in order to differentiate between scalar and vector quantities, and awaiting the eventual discovery of a principle of directional invariance.