1) His Biography
Abraham de Moivre was born on May 26, 1667, in Vitry-le-François, a small town in the Champagne region of France. His early years coincided with a period of religious turmoil in France, especially due to the revocation of the Edict of Nantes in 1685, which caused widespread persecution of Huguenots, French Protestants, of which de Moivre’s family was a part. This act marked a pivotal turning point in de Moivre’s life, as the intolerance forced his family to flee the country. It was this event that set the stage for his eventual settlement in England, where he would achieve renown as a mathematician.
De Moivre’s education began early, and he displayed a precocious interest in mathematics. His formal schooling took place in Protestant academies, but with the increasing threat of persecution, he was sent to Paris to study under the tutelage of various renowned scholars, including Jacques Ozanam, a respected mathematician. De Moivre attended the Collège de Harcourt, where he delved into philosophy, but his passion for mathematics remained constant. Despite his evident talents, opportunities in France remained limited due to his Protestant faith, leading to his decision to leave the country.
In 1688, de Moivre arrived in England, initially seeking refuge from religious oppression, but his stay soon became permanent. He quickly integrated into London’s intellectual circles, where his mathematical talents earned him recognition. At the time, England was home to a flourishing scientific community, and de Moivre’s association with the Royal Society, which he was elected to in 1697, was a testament to his growing reputation. Here, he encountered giants of mathematics such as Isaac Newton, with whom he would later form a close professional relationship.
One of de Moivre’s notable contributions was his work on probability theory, which he began exploring in the late 1690s. His insights into the subject became foundational, and his approach marked a departure from the more casual treatment of probability seen before his time. However, despite his intellectual successes, de Moivre’s personal life was fraught with financial difficulty. As a refugee and an outsider, he struggled to secure a stable academic post, a frustration that would persist throughout his career.
Throughout his life in England, de Moivre supported himself through tutoring, often for modest sums. His lack of a university appointment or patronage meant that he was never financially secure, despite his deep involvement with the Royal Society and his prolific contributions to mathematics. This financial insecurity coloured much of his later life, where, despite his genius, de Moivre remained on the fringes of institutional academia. His income came primarily from tutoring wealthy students and solving mathematical puzzles for gambling patrons, a common occupation for mathematicians of the time.
In his later years, de Moivre’s contributions to mathematics were recognised more widely, especially his work on the normal distribution, which became foundational to probability theory. His relationship with Newton deepened, and it is even said that Newton, in his later years, referred challenging mathematical questions to de Moivre. Despite his brilliance, de Moivre’s later years were marked by a decline in both health and fortune. His most famous work, The Doctrine of Chances, published in 1718, solidified his legacy but did not lead to financial security.
Abraham de Moivre died in London on November 27, 1754, at the age of 87. His death, like his life, was humble, but his intellectual legacy was far-reaching. De Moivre’s work on probability and the normal distribution influenced a range of scientific fields, from statistics to economics, and his name endures in mathematical circles to this day.
2) Main Works
The Doctrine of Chances (1718):
The Doctrine of Chances is arguably de Moivre’s most significant contribution to mathematics and the field of probability theory. First published in 1718, this work laid the foundation for the formal study of probability and cemented de Moivre’s reputation as one of the pioneers in this area. The book provides a mathematical framework for understanding probabilities, particularly in games of chance, which were a popular subject of study during his time. De Moivre’s work delved into complex calculations related to card games, dice rolls, and other gambling scenarios, offering rigorous mathematical solutions to problems that had previously been approached more casually. His methods, such as those for calculating expected values and understanding the outcomes of repeated independent trials, were groundbreaking.
The significance of The Doctrine of Chances extends far beyond gambling, as it also provided tools that would eventually influence the development of modern statistics and actuarial science. De Moivre’s treatment of probability, particularly his application of the binomial distribution, set the stage for later developments, including the discovery of the normal distribution. His derivation of the famous approximation to the binomial distribution, now referred to as the Central Limit Theorem, remains a cornerstone of statistical theory. This work, although initially designed to solve practical problems of the day, became a foundational text that helped transform probability into a rigorous mathematical discipline.
Miscellanea Analytica (1730):
De Moivre’s Miscellanea Analytica is another major work that focused on higher-level mathematics, particularly in the realm of series and infinite products. Published in 1730, this text explored analytic methods, further advancing the work of his contemporaries like Newton and Leibniz. The book is best known for introducing the famous formula (cosθ+isinθ)n=cos(nθ)+isin(nθ), which is now called de Moivre’s Theorem. This theorem provides a crucial link between trigonometry and complex numbers, offering a way to raise complex numbers to powers or extract roots.
De Moivre’s Miscellanea Analytica marked a significant contribution to the development of complex analysis, a field that would only come into full bloom in the 19th century. The theorem named after him is essential in both pure and applied mathematics, particularly in fields such as electrical engineering, quantum mechanics, and signal processing. Despite the profound impact of this work, it was not widely appreciated during his lifetime, largely because de Moivre did not hold an official academic position. Nevertheless, Miscellanea Analytica remains a cornerstone in the history of mathematics, and its influence can be seen in the development of modern complex number theory.
Annuities on Lives (1725):
De Moivre’s Annuities on Lives was one of the earliest works to rigorously apply probability theory to life insurance and annuities. Published in 1725, the book provided a mathematical approach to calculating the value of life annuities, making it a pioneering text in the field of actuarial science. De Moivre developed methods for predicting life expectancy based on statistical data, a critical advancement for insurance companies seeking to assess risk and price life annuities accurately. His formulas allowed for more precise calculations of the premiums required to ensure profitability for insurers while offering fair terms to policyholders.
The ideas presented in Annuities on Lives had a profound impact on the insurance industry and on the development of life tables, which are still used today to assess life expectancy and risk in various financial and health-related fields. De Moivre’s ability to apply abstract mathematical principles to practical problems is exemplified in this work. It also demonstrated his foresight in seeing how probability could be used beyond games of chance, influencing both economics and the emerging fields of demography and financial risk assessment.
De Mensura Sortis (1711):
Before The Doctrine of Chances, de Moivre published De Mensura Sortis (The Measurement of Chance) in 1711. This paper, presented to the Royal Society, laid the groundwork for his later developments in probability theory. In this early work, de Moivre tackled the problem of calculating probabilities related to expected outcomes in games of chance. He introduced the concept of expected value, which allowed mathematicians to predict the average result of random events over time. This notion of expected value became a key concept in probability, economics, and decision theory, influencing later thinkers such as Daniel Bernoulli and Blaise Pascal.
De Mensura Sortis also included discussions on the application of probability to mortality tables, paving the way for his later work on annuities. The paper was crucial in expanding the scope of probability beyond the context of gambling and into more practical applications such as life insurance and actuarial calculations. While shorter and less comprehensive than The Doctrine of Chances, this early work remains a key text in the history of probability, showing de Moivre’s early insights into a field that was still in its infancy.
Philosophical Transactions of the Royal Society (Various Papers):
Throughout his career, de Moivre contributed numerous papers to the Philosophical Transactions of the Royal Society, England’s foremost scientific journal of the time. These papers covered a wide range of mathematical topics, including probability, geometry, and complex numbers. One of his most famous contributions was his work on the Central Limit Theorem, published as part of his ongoing research into the binomial distribution. This theorem states that the distribution of the sum of a large number of independent and identically distributed random variables approximates a normal distribution, a concept now fundamental to statistics.
De Moivre’s papers in the Philosophical Transactions showcased his remarkable breadth as a mathematician and his deep involvement in the intellectual life of the period. Through these publications, he kept a strong connection with the scientific community, ensuring that his ideas were widely disseminated, even though he lacked a formal academic position. His contributions to the journal reflected his ongoing quest to solve practical problems using mathematical theory, leaving a lasting imprint on the development of mathematics during the 18th century
3) His Contribution to Mathematics
Abraham de Moivre made significant contributions to mathematics, particularly in the fields of probability theory and complex analysis. His pioneering work laid foundational principles that would influence a wide array of disciplines, including statistics, actuarial science, and even modern computational techniques. One of his most notable contributions is his establishment of a formal framework for probability, which he explored in depth in his seminal work, The Doctrine of Chances. This text systematically laid out the rules and calculations for determining probabilities associated with various games of chance, offering rigorous mathematical methodologies where previously only intuitive approaches existed.
De Moivre’s innovative work on probability extended beyond the realm of gambling. He introduced the concept of expected value, a fundamental idea that assesses the average outcome of random events over time. By applying his probabilistic methods to practical problems such as insurance and finance, he significantly advanced the understanding of risk assessment. His work in this area helped to shape the foundations of modern statistical theory, allowing for more sophisticated analysis and decision-making based on probabilistic outcomes. This evolution in thinking established probability as a mathematical discipline, moving it from anecdotal reasoning to a rigorous analytical framework.
In addition to his contributions to probability, de Moivre is best known for his work in complex analysis. His famous formula, known as de Moivre’s Theorem, states that (cosθ+isinθ)n=cos(nθ)+isin(nθ), effectively linking trigonometric functions with complex exponentials. This theorem provides a powerful tool for raising complex numbers to powers and for calculating roots of complex numbers, thereby establishing a bridge between algebra and trigonometry. The significance of this theorem cannot be overstated, as it is integral to the field of complex analysis and has far-reaching applications in engineering, physics, and applied mathematics.
De Moivre’s insights into the properties of the normal distribution further enhanced his mathematical legacy. His approximation of the binomial distribution, which led to the formulation of what is now known as the Central Limit Theorem, showed how the sum of a large number of independent random variables tends toward a normal distribution, regardless of the shape of the original distribution. This finding is fundamental in statistics, underpinning much of inferential statistics and hypothesis testing. De Moivre’s work set the stage for future mathematicians, including Carl Friedrich Gauss and Pierre-Simon Laplace, who would further develop these ideas into more comprehensive theories.
Throughout his career, de Moivre demonstrated a unique ability to apply theoretical mathematics to real-world problems. His work on annuities and life tables in Annuities on Lives and De Mensura Sortis illustrated how probability theory could be employed to assess financial risks and calculate insurance premiums. These contributions not only advanced the field of actuarial science but also highlighted the practical utility of mathematics in society. By developing methods for predicting life expectancy based on statistical data, de Moivre helped shape the burgeoning field of demography and contributed to the economic understanding of risk and uncertainty.
De Moivre’s influence extended beyond his immediate mathematical contributions; he was also instrumental in fostering the mathematical community of his time. His involvement with the Royal Society and his numerous publications in the Philosophical Transactions established him as a key figure in the scientific discourse of the 18th century. He mentored and inspired future mathematicians, and his works became part of the essential reading for those studying mathematics and probability. Despite facing financial hardships and being an outsider in the academic world due to his refugee status, de Moivre’s intellectual legacy endured and continued to impact future generations of mathematicians and scientists.
4) His Contribution to Probability Theory
Abraham de Moivre’s contributions to probability theory are foundational and transformative, significantly shaping the field’s development in the 18th century and influencing the work of future mathematicians. His systematic approach to probability established it as a rigorous mathematical discipline, moving away from anecdotal and intuitive reasoning towards formal methodologies. De Moivre’s most notable work in this area is his seminal book, The Doctrine of Chances, first published in 1718. In this comprehensive treatise, he laid out the principles of probability and provided mathematical formulations for calculating the likelihood of various outcomes in games of chance, such as dice rolls, card games, and lotteries.
One of de Moivre’s key contributions in The Doctrine of Chances was the introduction of the concept of expected value. This critical idea assesses the average outcome of a probabilistic scenario, allowing for a clearer understanding of risk and reward in uncertain situations. De Moivre’s work provided the mathematical tools necessary to compute expected values for different games, enabling a more precise approach to decision-making in gambling and financial contexts. His formulation of expected value would later become a cornerstone of modern probability theory and economics, providing a foundational basis for assessing the risk and value of uncertain outcomes.
De Moivre’s exploration of the binomial distribution further solidified his place in the history of probability theory. He developed methods to calculate probabilities associated with binomial experiments, which involve a fixed number of independent trials, each with two possible outcomes (success or failure). His work not only detailed how to determine the probabilities of different outcomes but also introduced what is now known as de Moivre’s Theorem. This theorem describes the relationship between the binomial distribution and the normal distribution, showcasing how the sum of many independent random variables tends toward a normal distribution, regardless of the original distribution’s shape. This idea would later evolve into the Central Limit Theorem, a fundamental concept in statistics that describes the behavior of averages of large samples.
In his earlier work, De Mensura Sortis, published in 1711, de Moivre addressed the measurement of chance in a more conceptual manner, introducing probabilistic reasoning in a way that connected it with practical applications. This publication, presented to the Royal Society, marked one of the first instances of applying mathematical reasoning to assess the likelihood of events, paving the way for future developments in probability theory. De Moivre’s emphasis on quantitative analysis of probability problems set the stage for a more rigorous mathematical approach to the field, influencing the works of subsequent mathematicians such as Pierre-Simon Laplace and Carl Friedrich Gauss.
De Moivre’s influence extended beyond theoretical developments; he was also one of the first to apply probabilistic methods to real-world problems. His studies on life expectancy and mortality rates, as demonstrated in his work on annuities, illustrated how probability could be applied to assess risk in financial contexts. By developing life tables and utilizing statistical data to predict life spans, he helped to establish a framework for actuarial science, which relies heavily on probabilistic principles. This pioneering work transformed the insurance industry, allowing for the accurate pricing of policies and a better understanding of the risks associated with insuring lives.
Moreover, de Moivre’s collaboration and correspondence with other mathematicians of his time facilitated the exchange of ideas, further enriching the field of probability theory. His connections with figures such as Edmond Halley and John Arbuthnot allowed for the cross-pollination of ideas and methods, contributing to the collective advancement of mathematical understanding during this period. De Moivre’s role in the mathematical community underscored the importance of collaboration and dialogue in fostering intellectual growth, highlighting his influence not only as a mathematician but also as a mentor and colleague to others in the field.
Despite his contributions, de Moivre faced challenges during his lifetime, including financial difficulties and a lack of formal recognition in the academic world. Nevertheless, his work laid the groundwork for modern probability theory, and his methods and insights continued to be built upon by later generations of mathematicians. His ability to blend theoretical rigor with practical applications ensured that his ideas remained relevant, and his legacy endures in contemporary discussions of probability and statistics.
5) His Legacy
Abraham de Moivre’s legacy is profound and multifaceted, impacting not only mathematics but also fields like statistics, finance, and actuarial science. His pioneering work in probability theory laid the groundwork for future developments, influencing subsequent generations of mathematicians and researchers. One of the most significant aspects of his legacy is his role in formalising probability as a mathematical discipline. By introducing rigorous methods and calculations in his seminal work, The Doctrine of Chances, de Moivre transformed the study of chance from an intuitive, anecdotal practice into a structured, analytical field. This shift allowed for the development of more sophisticated theories and methodologies, fundamentally changing how probability is understood and applied.
The introduction of concepts such as expected value, developed by de Moivre, has become a cornerstone in both mathematics and economics. Expected value allows for a systematic approach to decision-making in uncertain situations, serving as a crucial tool in various domains, including finance, insurance, and risk assessment. The principles he established continue to inform modern statistical methods, especially in areas such as behavioural economics and game theory, where understanding risk and reward is essential. De Moivre’s work thus not only laid a foundation for probability theory but also created a framework that would be built upon by future mathematicians like Pierre-Simon Laplace and Carl Friedrich Gauss.
De Moivre’s contributions to the field of complex analysis are also a vital part of his legacy. His famous theorem, which connects trigonometric functions with complex exponentials, serves as a key mathematical tool for various applications in engineering, physics, and applied mathematics. The theorem provides insights into the behaviour of complex numbers and is fundamental in fields like electrical engineering and quantum mechanics. This enduring relevance illustrates how de Moivre’s work transcended its original context, influencing a wide range of scientific disciplines.
Moreover, de Moivre’s practical applications of probability and statistics in finance and insurance established a framework for actuarial science. By developing methods to assess risk and calculate life expectancy, he helped to create a systematic approach to pricing insurance policies and annuities. His work in this area laid the groundwork for the modern insurance industry, enabling actuaries to use mathematical principles to evaluate risk and inform financial decisions. This practical aspect of his legacy demonstrates how theoretical mathematics can be applied to real-world problems, making it accessible and relevant to society.
De Moivre’s impact on the mathematical community extends beyond his own writings. He was an active member of the Royal Society and engaged in correspondence with other prominent mathematicians of his time, fostering a collaborative environment that encouraged the exchange of ideas. His connections with figures such as Edmond Halley and John Arbuthnot allowed for the cross-pollination of mathematical concepts, enriching the field as a whole. De Moivre’s legacy is thus not only a testament to his individual contributions but also to the importance of collaboration and dialogue in advancing mathematical thought.
In recognition of his achievements, de Moivre has been commemorated in various ways. Statues, names of theorems, and mathematical concepts bear his name, reflecting the respect and admiration he garnered for his work. His contributions to probability theory and complex analysis are widely taught in mathematics curricula around the world, ensuring that future generations of mathematicians are familiar with his ideas and methods. This enduring presence in educational contexts serves as a reminder of the lasting impact of his work.
Despite facing personal and professional challenges during his lifetime, including financial struggles and limited recognition, de Moivre’s intellectual contributions have endured. His ability to navigate the complexities of his time, while simultaneously advancing mathematical understanding, speaks to his resilience and commitment to the field. His legacy serves as an inspiration for mathematicians and researchers who continue to explore the intricacies of probability, statistics, and analysis.