Avraham de Moivare - the father of probability theory

De Moivre's contributions to the world of probability were so important that the mathematician and historian Todhunter wrote in 1865 that "probability theory owes most of it to De Moivre, with one exception and that is Laplace."

My series of biographies continues, and this time I will bring the story of a special and influential mathematician from France who, like his predecessors, did not, to put it mildly, receive a renewal in his life. This sounds puzzling since France has given the world of mathematics more talent and research than any other nation engaged in this field. It is even more puzzling that the saying - "Tell me who your friends are and I will tell you who you are" is not credited to the man since his closest friends are among the greatest minds of the human race, and yet he remained small and depressed throughout his life.

The next French mathematician would today fall under the category of "brain drain" because at one point he fled to England in order to seek a life that did not include persecution and harassment. And yet, although he was a mathematician of superior grace, this was not to his credit when the average Englishman standing before him heard his mother tongue assimilated into a jumble of words of the local language.
We are all different but we are all equal? Not at our school!

Abraham de Moivre was born in 1667 in the Champagne region near Paris to a middle-lower class family, where he was educated at the knees of the current of Protestant Christianity which was a source of oppression and intolerance mainly from the Catholic Church of that time. This situation led de Moaver, who studied at the Protestant Academy until the age of 15, to immigrate to the city of Somore where he began to study, among other things, the basics of logic at a local institution until he reached the age of 17. Although this field did not formally clash with the world of mathematics, its conceptual proximity is what brought de -Movar read and study by himself in his free time many mathematical books; The topic that fascinated him more than all the others came from the articles of the great mathematician Huygens and was related to predicting the results of games that include "chance", that is, gambling or games with probabilities. In the same year, de Moivre's parents returned to Paris and he enrolled in university where he received a formal education in mathematics - this is the beginning of his mathematical life.

However, this beginning was accompanied by many difficulties; The oppression we mentioned at the beginning was manifested in the most radical way during the reign of Louis XIV, which resulted in the imprisonment of de Moivre in a monastery for a long period of time because of his religious beliefs. Immediately after he was released, de Moaver fled to London, England where he worked as a private math teacher, when he himself comes to the homes of his students and even teaches some of them in local cafes. His mathematical knowledge at that time was relatively large and he was well versed in the standard writings, but during a short visit to a city located in the southwest of the country he encountered for the first time the most important book in the world of mathematics (and in science in general) - Isaac Newton's Principia (called in full - Mathematical Principles of Natural Philosophy ) and he immediately understood that it was a real work of art that was several levels above all the mathematical writings he had studied so far; The infinite beauty in Newton's theorems of immortality created in him an uncontrollable urge to study and deepen the principles presented in the book until they were complete; De Moaver acquired a copy of the book and his great eagerness to speak led him to tear the pages at their ends so that he could carry them in his hands at all times, the reason he wanted to do this was to learn as he went from student to student while he was teaching privately. Just to explain the point, it is worth emphasizing once again that this book can be defined as dividing the world of science into two parts, before it and after it, and the ideas presented in it are revolutionary and at the same time extremely difficult to understand, but despite the real difficulties, de Moiver studied everything written to its depth of interest in a very short time while doing so only in the segments where he walked on foot in the streets of London.

De Moaver decided to upgrade his academic life and tried to be accepted as a full member of the mathematics faculties at one of the local universities, but Protestant foreigners, especially the French, were not welcome in England and his path was not successful in this direction. De Moaver realized that he needed an entrance ticket to the world of English science and thus began to form friendships with local scientists, the two main ones being Edmund Halley and Isaac Newton, who was his good friend for many years. His first mathematical paper was on the concept of "flexions" that Newton mentioned in his great book, this paper attached to another important book of Newton's led to de Moaver's election to the "Royal Society", which was a kind of club of friends of famous scientists.

Before I continue, I will explain that flexion is the term that Newton gave to what is now known as the differential method. This method as well as the one called the integral method (which was also invented by Newton) are considered the two most important and useful achievements in the world of mathematics in particular and science in general. It is hard to think of a scientific field that is not based on these two, usually on the first term. This term, the differential, describes a method in which it is possible to define a change that occurred at a certain moment in something (such as the speed of a vehicle in a certain section of the road) and also other works caused the method to find ways in which a maximum and a minimum can be obtained from some action. It was said for example that I own a restaurant and I have fixed payments every month such as rent, electricity, etc. On the other hand, I have income that should cover these costs. In order to attract diners, I want to give a discount to every group of X people that enters the restaurant. I will build a sort of equation of income versus expenses, where income from regular meals will be described as some variable, and meals with a discount will be described as the same variable but less the amount that I deduct. Using the differential method, you can easily find two important things - one, assuming and I know how many diners will come to the restaurant in a certain month, I can know what the minimum price for a regular dish (which also applies to a meal with a discount) that I would like to demand in order to cover the monthly expenses; The second thing I can find is, for example, knowing how many diners I need to dine at my place (or what size group will receive a discount) in order to cover the expenses again. When it comes to everyday life, using this math is extremely simple and can help us optimally (because that's how it's been proven mathematically).

Let's get back to our topic - in 1710 De Moaver was chosen as one of the representatives in the team on behalf of the Royal Society that examined the claims regarding the great fight between Newton and Leibniz over the copyrights in the discovery of the two methods I mentioned above, this appointment was due to the fact that he was Newton's close friend and obviously that the result was "tailored" in advance. Even stranger was the fact that, although de Moaver did not hold a position at any university in England, he was nevertheless appointed to be a judge on a team that examined one of the most fascinating and controversial issues in the world of mathematics.

During the time de Moaver began to publish articles independently and was actually one of the founders of the mathematical fields known today as analytic geometry and probability theory. The first concept basically describes a method in which all kinds of engineering shapes, such as a circle, are examined using algebraic tools, i.e. equations with numbers and vanishings. The second concept is clear and familiar, and in this regard de Moiver was asked by one of the dukes in England to expand his research since the previous works in this field done by Montmort (de Montmort) and Huygens (whom he read as a child) were insufficient. De Moivre did expand and change some of Montmore's words, which made the latter very angry with him, but unlike the war between Leibniz and Newton, this matter ended amicably. I will only add that in this book by de Moaver is mentioned for the first time, together with many probability problems in dice games, the concept called - statistical independence, which is fundamental and extremely important.

De Moivre's contributions to the world of probability were so important that the mathematician and historian Todhunter wrote in 1865 that "the theory of probability owes most of it to De Moivre, with one exception and that is Laplace." Another field that de Moaver studied was the probability of living and the theories of allowances (that is, an allowance or grants given to a certain party by another party), he based these studies on the articles of Huygens and especially on tables that examined the probability of a person living according to data from the city of Wrocław in Poland. The main use of these tables was, as expected, by insurance companies of that time.

In 1730, de Moivre wrote a very important book that dealt with analysis and, among other things, provided a proof of the equation with which he is most identified - "De Moivre's Equation", and it is -

cos x + i sin x)^n = cos nx + i sin nx)

I went out of my way and this time I brought a mathematical formula because it is a critical matter for the following things.
This formula deals with the combination of trigonometry and complex numbers and is an important basis for many studies in the world of mathematics. Trigonometry deals with numerical relationships within engineering forms; Let's say that I am an architect who wants to build a structure similar to the triangular Azrieli Tower and for this I am required to plan the size of the base (which is of course triangular) according to the lot I purchased. Trigonometry helps us to easily find how the building should be constructed solely by means of fixed relationships that exist in any triangle; The words that we use in this field are what appear in the formula as COS and SIN.

On the other hand, there is also a variable called i in the formula and it represents a very mysterious and fascinating world of its own, in which imaginary numbers are hidden, a distorted alchemy of our reality. I will not be able to explain in this context even a little bit what it is about, but I will only summarize and say that the first to discover these numbers was the Italian mathematician, one of our acquaintances, Girolamo Cardano, who used them to solve the same quadratic equation that we discussed quite a bit in the article on Tartalia (Tartaglia); Until not long ago, these numbers were indeed called imaginary because they were completely against the entire world of mathematical thought (however, they were used in many essential ways without any problem), but new discoveries in the world of physics show that these strange numbers exist within us as a reality for everything (which is not always possible understand it).

A combination of these two fields creates a gift whose value is priceless in the mathematical world, any such branch that is combined with another branch brings fantastic results, a reader who wants to understand how important this combination is will find it in the historical process of solving Fermat's last theorem (de Fermat).

From all these things it seems that de Moiver lived a happy life, but the truth is quite different. His only livelihood was always from private lessons and his financial situation was very bad and he suffered from severe poverty. De Moaver begged the great mathematician Jacob Bernoulli to contact Leibniz so that the latter would recommend him for some position at the University of Cambridge", but this attempt and another attempt by Leibniz to approve him for a professorship in Germany did not go well. So serious is the thing that even Newton and Halley failed to provide him with any position in the entire English academy. Indeed, this is a mistake, since it was a mathematical exaggeration for everything, and Newton would even say to the mathematicians who questioned him about his results in his book Principia: "If you want to know something from what I wrote down, ask de Moaver, he understands it better than I do."

De Moaver never married and died in severe poverty at the age of 87 in London. A fascinating anecdote about this was that de Moaverd, together with Cardano, are famous because they both knew how to accurately predict the day of their death. According to him, he slept 15 minutes more each night than he should have and calculated the total amount of time that he supposedly "wasted" in vain. His conclusion was that he would die on the day he slept exactly 24 hours, and that's what happened!