JAKOB BERNOULLI: FAMILY, STUDIES, CONTRIBUTIONS TO MATHEMATICS - BIOGRAPHIES - 2024

Jakob Bernoulli (January 6, 1655 - August 16, 1705), was a renowned mathematician of Swiss origin. Together with his brother Johann Bernoulli, he postulated and introduced the first principles related to the calculus of variations.

Among other significant contributions is the discovery of the fundamental mathematical constant "e" and the demonstration of the "law of large numbers" within the field of probability.

Portrait of Jakob Bernoulli

Bernoulli

Within his family, Jakob Bernoulli was the first member to embark on the study of mathematics, followed by his brother. From this resulted the two generations of mathematicians that distinguish the Bernoulli family within the history of this science.

Among other studies, Bernoulli obtained academic training in theology by the opinion of his parents, in addition, he studied mathematics and astronomy. He was a proponent of the Leibnizian infinitesimal calculus during the period of great debate between Isaac Newton and Gottfried Leibniz over determining priority over the discovery of the infinitesimal calculus.

One of the most relevant publications in Bernoulli's career was his work in the field of probability, known as "Ars Conjectandi", with which he gave life to what would later be called "Bernoulli numbers" and the aforementioned " Law of the big numbers".

Family and studies

Jakob Bernoulli's parents were part of the spice trade in Basel, Switzerland, although the link between his parents in the drug markets is also discussed.

Originally from Belgium, the Bernoulli family's grandparents became refugees from the persecution of the anti-Protestant tyranny, settling permanently in Basel, the great commercial capital of central Europe at the time. It is here that Jakob Bernoulli's parents as well as himself and his siblings were born.

Jakob Bernoulli begins his academic life in the field of theology fulfilling the wishes of his parents. However, after obtaining the degree in 1676, he dedicated the following years of his life to learning mathematics, opposing his family.

He made a variety of continuous trips to learn about the new discoveries of science from the most important characters of the moment.

He had instances in Geneva, France, the Netherlands and England where he made various links with mathematicians and scientists such as Johannes Hudde, a mathematician who studied maximum and minimum equations; Robert Boyle, founding member of the Royal Society; and Robert Hooke, renowned English scientist. In this way he had a long correspondence with characters linked to his field of interest.

After his return to Switzerland, he began working as a professor of mechanics from 1687 at the University of Basel. He was also the tutor of his brother, Johann Bernoulli, 12 years his junior, who would also be the most prominent member of the family in mathematics.

From this moment on, both brothers began to work in the field of calculus, using Leibniz's studies on calculus as a great reference. They also had as a reference the works of von Tschirnhaus. It should be noted that the Bernoulli brothers were among the first to try to understand the complexity of Leibniz's proposals.

Contribution to mathematics

Discovery of the mathematical constant "

Through the study of compound interest, in 1683, Jakob Bernoulli managed to discover the value of the constant "e", trying to reduce compounding periods to a minimum. It is currently listed as an irrational number of great importance. It is also known as "Euler's number." It is the base of the natural logarithm. Its value is equal to or approximately 2.71828…

The number "e" is part of an exponential function that refers to "growth", for this reason it can be used in various ways. It allows, for example, to describe cell growth or decrease, helps in dating fossils and is used in statistics within the exponential function.

The number "e" is irrational, it cannot be reflected as a fraction and it has a transcendent character, which means that it is not a root or a result of polynomial equations.

Law of the big numbers

It is a theorem applied within probability theory and has a fundamental role. The law states that an experiment repeated many times with a tendency to infinity will result in the frequency of that specific event being carried out almost constant.

This resulting constant becomes the probability of occurrence. Jakob Bernoulli was the one who managed to demonstrate this law in his publication “Ars Conjectandi” (1713), helping to develop the probabilistic theory. It should be noted that the publication was made after Bernoulli died, being his nephew Nicholas who brought it to light.

The law indicates that the frequency with which an event occurs is unstable in the first instance but that the increase in the occurrence of the event can generate a stabilization in the frequency of the phenomenon under study.

For example, rolling a die to roll the number 1 has a probability of ⅙. The law states that the more tosses are made of the die, the closer the frequency of occurrence of that event will be to a constant. The constant has a value equal to the probability that it would be ⅙ or 16.66% of the tosses.

The roll of the dice is an event that can be an example of the Law of Large Numbers

Image by Willi Heidelbach from Pixabay

Each roll of the dice is an independent phenomenon that does not affect or influence the previous or subsequent rolls, so that after 30 numbers there may be a frequency of 6%, but perhaps, after 100 rolls, the probability will go away. approaching 16.66%.

It is likely that after thousands of tosses, the frequency is already close enough to probability to check the Law of Large Numbers.

Calculation of variations

Jakob Bernoulli, together with his brother, developed the first result of the calculus of variations with the brachistochron curve, first suggested by Johann Bernoulli. This was one of the contributions that the Bernoulli family made within the branch of variational calculus. Later, it was the Swiss mathematician Leonhard Euler, who formulated the first theory of the calculus of variations.

Commemorations

- Bernoulli lunar crater, is a crater located northeast of the lunar surface. It commemorates the mathematicians of the Bernoulli family, among whom is Jakob Bernoulli.

- "2034 Bernoulli". It is an asteroid discovered at the Bern-Zimmerwald Observatory in Switzerland. His name also fulfills the function of honoring the brothers Jakob and Johann Bernoulli. This asteroid was discovered in 1973 by Paul Wild, a Swiss astronomer.

- Jakob Bernoulli was a member of the Royal Academy of Paris and Berlin.

Death

Jakob Bernoulli held his post as professor at the University of Basel until the end of his days. He died in 1705 at the age of 50. Among his fascinations were logarithmic spirals, of which he asked to have an engraving on his tombstone. Besides, he included the phrase "Eadem mutata resurgo" (I'll get up the same, even if I have changed). After his death, his brother Johann would take up his position as a teacher.

The Bernoulli family

The entry of science into the Bernoulli family is started by Jakob. He was the first to oppose his parents' wishes and dedicate himself to the study of mathematics. Later, both his brother Johann and his nephew Daniel, continued the academic foray into these areas and gave the surname Bernoulli a place of recognition within the world of science.

Thanks to the diversity of studies, contributions and recognitions, the Bernoulli are remembered as an outstanding family of mathematicians.

Johann bernoulli

Twelve years younger than his brother Jakob, Johann decided to study mathematics after having fought against pressure from his father, first to run the family spice business and second to study medicine.

Due to the age difference with his brother, Johann was even tutored by Jakob, and from that moment they began to work together on the studies of Leibnizian calculus.

Johann was one of the most outstanding members of his family for his various contributions, such as his work on the calculation of variations carried out with his brother. He is also recognized for solving the Catenary problem, through an equation obtained by him together with Gottfried Leibniz and Christian Huygens in the year 1691.

Daniel Bernoulli

He figured as one of the main representatives of the second generation of mathematicians within his family. He was the second son of Johann Bernoulli. Apart from mathematical sciences he also studied medicine, philosophy and logic. He obtained a position within the Russian Academy of Sciences.

In 1738 he published Hydrodynamica, where he studied the properties of fluid flow and established the fundamental connection between pressure, density and speed. He established the “Bernoulli principle”, with which he affirmed that the increase in the speed of a fluid decreases its pressure.

He obtained ten prizes between 1720 and 1750, awarded by the Royal Academy of Sciences in Paris for his diversity of work in fields such as astronomy, gravity, magnetism, ocean currents and more. He also shared the award with his father for his work on planetary orbits.

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