The Duality of Time Theory

Since we are reviewing the main important concepts in the history that shaped the modern theories of physics and cosmology, which we are going review in chapter III, and because the Duality of Time is deeply related to complex numbers, we want to quickly mention how they were conceived, since this happened at around this same period when all these important observations and theories in physics and astronomy were being established.

Complex numbers first arose in the 16th century when algebraic solutions for the roots of cubic and quartic polynomials were discovered by mathematicians. It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers.

This looked first like nonsense, but formal calculations with complex numbers can simplify the solutions of cubic equations with real roots. Furthermore, later mathematicians showed .that the use of complex numbers is unavoidable.

The Italian mathematician Gerolamo Cardano (1501-1576 AD) is the first person known to have introduced complex numbers. He called them “fictitious”, but Rafael Bombelli (1526-1572 AD) was the first to explicitly address the seemingly paradoxical solutions of cubic equations, and he developed the rules for complex arithmetic trying to resolve these issues.

Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex numbercan be then identified with the pointin the complex plane.

A complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the complex numbers are a field extension of the ordinary real numbers, in order to solve problems that cannot be solved with real numbers alone. The term “imaginary” was coined by Descartes in 1637, although he was at pains to stress their imaginary nature.

In the 18th century, complex numbers gained wider use, as it was noticed that formal manipulation of complex expressions could be used to simplify calculations involving trigonometric functions. They now have practical applications in many fields, including physics, chemistry, biology, economics, electrical engineering, and statistics.