Approach of the value of a rent when non-central moments of the capitalization factor are known: an R application with interest rates following normal and beta distributions


Ratio Mathematica                                             Volume 36, 2019, pp.89-95  

 

              
 

89 

 

 

 

 

The theorem of the complex exponentials 
 

 

 

Alberto Daunisi* 

 
 

 

 

Abstract  

This paper describes a new theorem that relates the lengths of the legs of a right 

triangle with the ratio of three complex exponentials. The big novelty of the 

theorem consists in transforming two real measures of legs derived from 

Euclidean geometry into a combination of imaginary elements obtained from 

the complex analysis. 

 

Keywords: complex analysis, complex geometry 

 

2010 subject classification: 37K20.† 
 

 

                                                      
* Researcher in Mathematics, Bologna (Italy). alberto.daunisi@gmail.com 
† Received on December 10th, 2018. Accepted on April 24th, 2019. Published on June 30th, 

2019. doi: 10.23755/rm.v36i1.472. ISSN: 1592-7415. eISSN: 2282-8214. ©Alberto Daunisi 

This paper is published under the CC-BY licence agreement. 



Alberto Daunisi 

90 

 

1. Introduction  

It seems difficult, apparently, to imagine that the difference between the 

lengths of the legs of a right triangle may have some connection with the ratio 

of complex numbers, or vice-versa, that a ratio of complex numbers may be 

obtained from the difference of the legs of a right triangle, but this theorem, that 

we will call, precisely, Theorem of the complex exponentials, shows that it is 

possible. 

Let’s start with some historical and mathematical considerations, from which 

our research is inspired. In ancient Greece, the right triangles were basically 

solved by the first and second theorem of Euclide (IV-III century B.C.) and by 

the theorem of Pythagoras (about 575-495 B.C.). This happened because the 

Greek trigonometry, that was only applied to the study of astronomy, was based 

on the measurement of the ropes of a circle (subtended by a certain angle), rather 

than on that of sines and cosines. The functions sine and cosine, developed by 

the Indians in the IV-V century A.C., have been imported in the Arab world 

around the VIII century A.C., and then, to the West world, a few centuries later. 

From this moment, the triangles started to be solved by the relations that bind 

the lengths of the sides of the triangle with the values of the trigonometric 

functions of its angles. In particular, two fundamental trigonometric theorems 

were introduced, through which it has been possible to solve any problem 

related to the elements of a triangle: the theorem of sines and the theorem of 

Carnot. The first one states that in any triangle the ratio between one side and 

the sine of the opposite angle is always constant and equal to the diameter of 

the circle circumscribed to the given triangle; the second one states that in any 

triangle the square of one side is equal to the sum of the squares of the other 

two, plus their product to the cosine of the angle included. From the theorem of 

sines, applied to the right triangles, it descends the theorem according to which 

in a right triangle a leg is equal to the product of the hypotenuse for the sine of 

the angle opposite to the leg. Finally, we arrive at the XVIII century A.C., 

where, in another branch of mathematics completely different from the above 

one, is developed, in all its entirety, the theory of complex numbers of the form 

x+iy, with x and y real numbers and i = √-1 the imaginary unit. In particular, 
the studies of Abraham de Moivre (1667-1754) and Leonhard Euler (1707-

1783) provided to the complex numbers a definitive and systematic structure 

from which descended the complex trigonometric functions and the complex 

exponential functions. De Moivre left us the famous formula (1739) that 

calculates the power of a complex number expressed in the form trigonometric 

(cos α + sin α)ᵐ= cos(mα) + i sin(mα), while Euler left us the equally famous 

formula (1748) that binds the trigonometric functions sine and cosine to the 

complex exponential function  = cos w + i sin w.  



The theorem of the complex exponentials 

 

91 

 

Our theorem is inspired by a very specific motivation: considering that the 

sides of a right triangle may be expressed by a trigonometric function, and this 

one by a complex variable, we wanted to discover if the same sides may have a 

relationship with the elements of the complex analysis and, in case of positive 

response, in which way and form. With this purpose, we discovered two 

important results: the first one is that the ratio between the difference of the legs 

of a right triangle and the difference of their projections on the hypotenuse, 

multiplied by the cosine of half-difference of two angles opposite to the legs, is 

always constant; the second one is that this constant is given by the ratio between 

complex exponentials,  or their powers, where the most important constants of 

the all mathematics are appearing: the constant of Napier e (or of Euler), 

introduced by John Napier on 1618 and used systematically by Euler (1736) for 

its exponentials; the imaginary unit i, officially introduced by Friedrich Gauss 

(1777-1855) in an essay of 1832; the constant of Archimedes (287-212 B.C.) π, 

calculated with approximation by the greek mathematician in the III century 

B.C. and definitively calculated, with 35 decimal digits, by Ludolf van Ceulen 

on 1610. Both the above important results are set forth and proved in the 

following theorem. 

 

2. The theorem of the complex exponentials 

                                                                       

Statement: In a right triangle CAB (rectangle in A), where a is the 

hypotenuse, c and b the legs, m and n the projections of the respective legs c 

and b on the hypotenuse, γ and β the angles respectively opposed to c and b, it 

results: 

 

where e=2,71….is the Napier’s constant, π=3,14…..is the pi and i=  is 

the imaginary unit. 

 

 

Proof. Let us consider the right triangle CAB of Figure 1, rectangle in A 

(α=90°), having hypotenuse a, height h,  minor leg b and major leg c, n and m 

the respective projections of b and c on the hypotenuse a, γ the angle opposed 

to c and β the angle opposed to b. 

 

 

 

 

 



Alberto Daunisi 

92 

 

 

 

 

 

 

  

 

 

                                                                            
                                                   

Figure 1 

 

 

With reference to the right triangle of Figure 1, we know that the first Euclid’s 

theorem asserts: 

 

                                                             

  

                                                                                                                    

 

from which, subtracting member to member, it derives: 

 

 

                                          
                                              

 

namely:                                

  

 

 

 

 

Taking into account that it is: c=a sinγ and b=a sinβ, from (3) it derives: 

 

 
 

Simplifying and applying the formulas Prosthaphaeresis to the denominator 

of second member (4), from (4) it’s obtained:  

(1) 

(2) 

(4) 

(3) 



The theorem of the complex exponentials 

 

93 

 

          

         

 
 

 

But we know that  γ + β = 90°, so in the denominator of (5) it’s   2 sin 

 , therefore from (5) it derives: 

 

 
                                                                                                                                                              

namely:                                                                                                                                     

 
                                                                                            

        

We know, from complex number’s theory, that the trigonometric form of the 

complex number 1+i is: 

 

 1+i =                                                

 

                                                                                                                                                                   

For the formula of Euler it’s: 

 

cos  

                                                                                                                                                               

Replacing (9) in (8), we obtain: 

 

1+i=  

 

namely:                                                                                                                                                                   

 

(5) 

(6) 

(7) 

(8) 

(9) 

(10) 



Alberto Daunisi 

94 

 

 
 

Let us remind now that it is: 

 

1=     and    i=  

 

Replacing (12) in (11), we obtain: 

 

 
 

Finally, replacing the second member of (7) with the second member of (13), 

we obtain: 

 

 
 

And the theorem is thus proven. 

 

 

Conclusions 

 We have shown a theorem born from the motivation to investigate and solve 

a problem: to link a geometric result of III century B.C., although it reworked 

by the trigonometric functions of XVI century, to the last theories of complex 

numbers of XVIII century, apparently irreconcilables with the Euclidean 

geometry. We think to have got two relevant teachings: on the one hand we have 

bound the elements of a right triangle (legs and angles) to a constant  of complex 

analysis, given by the combination of three most important constants of 

mathematics; on the other hand we have notably pointed out a precise 

methodological procedure of the proof, based strictly on the deductive method, 

where, starting from a general axiom alleging geometric structure of the right 

triangles, we reached, through a series of rigorous logical concatenations, a 

particular result alleging  new structure of complex analysis. 

We finally think that from this article we also can draw another useful 

teaching: to discover this theorem allowed us to investigate on three completely 

different (among their) branches of mathematics (Euclidean geometry, 

trigonometric functions, complex analysis), born and developed in different  

(11) 

(12) 

(13) 



The theorem of the complex exponentials 

 

95 

 

ages, transmitted by several men separated by time and by different languages, 

cultures and religions, who, although not knowing themselves with each other, 

have always improved the ideas of their predecessors and transmitted it to the 

future generations. They have been united only by their love for mathematics, 

in addition to the desire to contribute to its development. We think that we all 

must pick up an example from this act of faith, that only mathematics, between 

all sciences, is able to provide. 

 

 

 

References 

[1] Alberto Daunisi (2014). L’Ultimo Teorema di Fermat,  Storia 

Matematica, BookSprint Edizioni, Salerno-Italy. 

[2] Howard Levi (1960). Foundations of geometry and trigonometry, 

Prentice-Hall, Inc. Englewood Cliffs, New Jersey. 

[3] Joseph Bak and Donald Newman (1997). Complex Analysis, Springer-

Verlag New York Inc. 

[4] Morris Kline (1998). Calculus, Dover Publications,  Inc. Mineola,  New 

York. 

[5] Barry Mazur (2004). Imaging numbers, Picador (Farrar,  Straus and 

Giroux), New York.