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30 May 2019

Gauss, the Prodigious Mathematical Mind

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At the age of just three, he is said to have corrected his father’s calculation error, and by the time he was 19 he had enunciated his first theorem. Gauss was above all a child prodigy who, as he grew up, knew how to keep his curious and extraordinary mind active. “It is not knowledge, but the act of learning, not possession, but the act of getting there, which grants the greatest enjoyment,” he wrote. His mathematical works revolutionised arithmetic, astronomy and statistics, a field in which he is known for the famous Gaussian bell curve. During his 77 years of life, he also had time to treasure a personal library of around 6,000 books. His scientific advancements earned him the posthumous title of Princeps mathematicorum (Latin for “the foremost of mathematicians”).

Carl Friedrich Gauss monument in Brunswick, Germany. Source: Wikimedia

Although he was born into a poor and illiterate family in Brunswick, a small town west of Berlin, Carl Friedrich Gauss (April 30, 1777 – February 23, 1855) was soon noticed thanks to his prodigious mind. When he was only seven years old, he surprised his teacher and classmates by calculating, effortlessly and rapidly, the sum of all the natural numbers from 1 to 100 —the task with which his entire class had been assigned as a punishment. Gauss’s intellectual abilities caught the attention of the Duke of Brunswick, who decided to subsidise his secondary and university studies.

A problem inherited from Ancient Greece

In 1796 he published his first great achievement, the demonstration that a heptadecagon, a regular polygon with 17 sides, could be constructed with ruler and compass, a problem inherited from Ancient Greece that had remained unanswered for two thousand years. Shortly thereafter he enunciated the Prime Number Theorem, which consists of a description of how primes are distributed in the set of natural numbers. It is one of the most important theorems in the history of mathematics, which allowed the further development of the investigation of prime numbers. Starting in that prolific year, Gauss began to keep a diary in which he wrote down all the mathematical discoveries he made from 1796 to 1814, with a total of 146 entries.

Regular heptadecagon with annotation. Credit: László Németh

Gauss became particularly famous in 1801, when he was 24. At the beginning of that year, astronomers observed what they thought was a new planet, Ceres, that soon vanished from sight. Gauss described its orbit with mathematical precision and determined that it was actually an asteroid (nowadays Ceres is considered a dwarf planet). To everyone’s admiration it reappeared at the end of that year, precisely where Gauss had predicted it would.

A short time later, he accepted the position of Professor of Astronomy at the Göttingen Observatory, about 100 kilometres from his native Brunswick, where he became the director and remained so for the rest of his life. There, in 1809, he determined how to calculate the orbit of a planet with unprecedented precision. During those years, his personal life was no less fruitful. His first wife died giving birth to his third child and the boy also died shortly afterwards, which led to a deep depression in Gauss. Even so, he remarried and had three more children.

A pillar of statistics

Around 1820, while working on the mathematical determination of the shape and size of the terrestrial globe, Gauss developed different tools for data processing. The most important was the Gaussian function or bell curve, which is one of the pillars of statistics. This is the visual representation of the frequency of a given group of data, those that are generated by random causes. An example is the temperature of a city: if we represent the data of the temperature of our city as a function of the days on which it is reached, we will discover that the most extreme temperatures are repeated a few times while the more moderate temperatures have a much higher frequency. The resulting graph is bell-shaped and symmetrical, with the most moderate temperatures in the centre and the most extreme at the edges of the bell. We say then that this variable follows a normal distribution, and the ease with which it can be used to model very diverse situations makes it a fundamental tool of many studies.

Gaussian bell curve. Credit: Inductiveload

Gauss died in his sleep on February 23, 1855 and was buried in the Göttingen cemetery. He was so proud of his youthful achievement of the heptadecagon that he asked it to be carved on his tombstone, just as Archimedes had a sphere inside a cylinder inscribed on his. His desire, however, could not be fulfilled because the stonecutter given the task found it impossible to sculpt a heptadecagon without it looking like a circle. Gauss’s brilliant mind would surely have known how to do it, but he was no longer around to explain it.

Bibiana García


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