NUMBERS: FROM REAL TO IMAGINARY
The humankind firstly developed the numeric system of natural numbers; however, this system was not enough to solve crucial and also daily problems:

How to represent the financial state of our family in order to notice that we have debts to pay for?

How can we estimate the remaining amount of water in the jar after having used some proportion of it?

How can we measure the length of the diagonal of a square?

The very ancient question: how can we measure the amount of space inside a sphere or a circle? Is it possible to know exactly the length of a round shaped circuit with given radius?

How can we describe the movement of a photon?
The answers to these problems came after the creation of systems of Integers, Rational, and Irrational numbers: all of these called Real Numbers, and on the other hand, the imaginary numbers came as the answer of the last problem concerning the quantum mechanics. Let us have a short travel through this very long history of problems and solutions.
Each time that the human invented a new numeric system it happened because of the necessity to give a solution to an equation. For the first problem, it was necessary to solve the equation
Where n is a natural number. The answer is then denoted by
From here we arrive to the set of Integer Numbers
ordered from less to greater, left to right, and the general intuitive new rule is that negative numbers represent debts, or also temperature under the freezing point of water, or some time before something interesting happened.
In order to solve the second problem, it resulted natural at some point, to think in a unity to compare with, then divide this unity in an integer (natural) number of identical parts and then to say explicitly how many of this little parts were already used. Formally we have invented the solution to the equation with integer coefficients p and q, given by
This solution is written as
and should be understood as: “you divide a unity in q parts and take p pieces of that size”. We have then the set of fractions or Rational Numbers
whose operations are based on the idea of just fractions of the same size can be really operated in the way:
For a long period of time, it was satisfactory to solve many problems in daily life using rational numbers, however what happened when appeared the third problem?
The mathematician Pythagoras gave a solution to this problem with his theorem:
Figure taken from internet.
The Pythagoras’s Theorem is one of the most celebrated theorems in the history of the humankind, and for sure you remember to have seen it before somewhere. From this theorem, we can deduce that for a square whose sides are of length 1, the corresponding diagonal should have a length satisfying.
We invite the reader to think on why no rational number can be a solution for this equation, and in consequence a new symbol needs to be included. The solution is denoted by
Since this quantity is not rational, but it is for sure real since it is a fact that the diagonal of the square exists and have a finite length, it appeared natural to call
Maybe the most famous irrational number is denoted by
This is the symbol of a Greek letter, which curiously represented also the number 80 for the Greek culture. Even complete books have been written about this number! For instance, we invite you to read [1].
But,
The definition of this quantity comes from the problem of measuring the length of the perimeter of a circle. By the definition.
and does not matter the radius of the circle, this quotient is always the same! This number appears in innumerable formulas about volume, area, number theory and the whole trigonometry and geometry subjects in general. To prove that is not a rational number is not an easy homework because it probably could take you a millennium, indeed Aristotle (384322 B.C.E.) suspected this was true, but we can say that at the day, thanks to the French mathematician AdrienMarie Legendre (17521833) that is an irrational number. We can also say that the most powerful computers have been able to compute more than a trillion of decimal digits in the decimal representation of.
and it appears that not periodic sequences are present.
There exist another important and interesting irrational number, the Euler’s number, denoted by
This number appears as the basis of the natural logarithm and is used for example to estimate the number of prime numbers less or equal than a given natural number. It is the center of attention in some formulas to estimate the growth rate of a given population, and several beautiful formulas, for example:
Again, it is not easy, even more, it is hard to prove that is irrational, but you can be confident that it is!
Figure taken from [2]. Isaac Newton’s work is in front of us!!!!
Isaac Newton took us in front of the number when measuring the area under the hyperbola
he computed some values for the natural logarithm.
The set of Real Numbers contains all the rational and Irrational numbers, so we can say that we already gave a glance to the real quantities or numbers that are in front of our eyes in the world. One could think that no other numeric system would be necessary to complete a description of the world around us, until one gets to the differential equation, the timedependent Schrödinger Equation.
Where
and h is the Planck’s constant. This equation governs the quantum mechanics as it is known nowadays, see [3].
The number
is called the imaginary unity, and we are in presence of the system of the complex numbers
This numeric system is very useful and in particular for us, in quantum computing we will use the following formula
Which is known as Euler´s identity and is the most beautiful formula in the history of humankind, at least it is in the very top three, see [2]. The beauty of this equation for many of us is in the fact that it contains the unities of each numeric system, the real, and the imaginary, and also the irrational already mentioned in this delivery. Finally, the zero is present here to show that “nothing” can become “everything”, at least for us, the romantic researchers.
See you in the upcoming communication on the Spin Quantum Tech Blog!!
Bibliografía
[1] Posamentier A., and Lehmann I., A Biography of the World´s Most Mysterious Number, Prometheus Books, 2004.
[2] Wilson R., Euler´s Pioneering Equation: The Most Beautiful Theorem in Mathematics, Oxford University Press, 2018.
[3] Bowers P., Lectures on Quantum Mechanics: A Primer for Mathematicians, Cambridge University Press, 2020.
About the authors