Ann Varela: Johann Carl Fredrich Gauss

Feb 21, 2017 by

An Interview with Ann Varela: Johann Carl Fredrich Gauss

Michael F. Shaughnessy –

1) In my opinion, there is almost no other name, so synonymous with statistics than Gauss.  When and where was he born, and what was his early childhood and early educational experiences like?

In 1777, Johann Carl Friederich Gauss was born in Braunschweig, Germany.  He came from a line of bricklayers and gardeners, and was expected to follow in the family tradition.  Though his family was quite poor and not formally educated, his mother and uncle wanted a future with more profitable opportunities for him.

From the time he was a young boy, Gauss had a natural talent for mathematics and apparently corrected his father’s payroll ledger at the young age of three.  By the age of 9, Gauss’ teachers were unable to keep up with his mathematical abilities.  It was evident that Gauss was a child prodigy.

At the age of 11, Gauss attended the Gymnasium, a senior secondary school in Germany, and then attended Brunswick Collegium Carolinum, as a result of receiving a stipend from the Duke of Brunswick.  By age 18, Gauss began his studies at Gӧttingen University where he independently rediscovered several significant mathematical theorems.

2) Apparently, he was a multi-talented mathematician in that he worked in algebra, geometry, and number theory. Can you describe some of his contributions in each of these fields?

As far as algebra is concerned, Gauss was the first to discover some kind of specific arrangement in the occurrence of prime numbers.  Gauss dealt with the problem from a different perspective by graphing the frequency of primes as the numbers became larger. He noticed a trend:  as the numbers increased by 10, the likelihood of prime numbers existing decreased by a factor of approximately 2.

In 1801, Gauss discovered a procedure for finding the best-fitting curve to a given set of points, known as least squares fitting.  He found that the difference between vertical and perpendicular fits is relatively insignificant.  See Figure 1.

Figure 1

Another noteworthy discovery of Gauss’ was his proof of the Fundamental Theorem of Algebra, which states: for every polynomial equation containing complex coefficients and degree larger or equal to 1, there exists at least one complex root of the form a + bi.

In the field of geometry, Gauss showed that a 17-sided polygon, the regular heptadecagon, can be constructed with a straightedge and compass.  See Figure 2.  This discovery was deemed the greatest development in geometry since the time of Greek mathematics.  It was noted that he wanted this geometric figure engraved on his headstone, but the stonemason refused to etch a regular 17-sided polygon because he feared that the subsequent figure would be impossible to differentiate from a circle.

Figure 2

Number theory is primarily dedicated to the study of the integers.   It was reportedly Gauss’ favorite field of mathematics.  Gauss made significant contributions to number theory with his 1801 book Disquisitiones Arithemticae, in which he utilized the symbol ≡ to denote similarity.

3) Now, I would say that just about anyone who has taken any kind of statistics course- be it parametric or non-parametric- has been exposed to the Gaussian or Bell shaped curve. How did he come about with this conception?

At first, the normal distribution began as a hypothetical distribution for errors in fields in which variabilities in nature were believed to act unsystematically.  It seems that in his development of the least squares method, Gauss was able to minimize the effect of such measurement errors, or variabilities, which resulted in the discovery of the normal distribution.  In other words, when constructing a curve that has the best fit to a set of data points, a smooth curve results.   See Figure 3.

Figure 3

4) I would go so far as to say that the Gaussian, or normal distribution or Bell shaped curve is as pervasive a symbol of math as the sigma sign or addition sign. In the big scheme of things- how important is it?

The notion or development of the normal distribution is related to the Central Limit Theorem, which says that if you take a sample from a population with a random distribution, the sample means will likely be normally distributed with the same mean as the population.  Therefore, if numerous sample means are calculated from the given population and plotted using a histogram, one will see the formation of a normal curve.

The occurrence and application of normal distribution in practical problems are numerous.  Here are some examples:  a) Thermal radiation measurements have a normal distribution over longer timescales.  b) Binomial distributions are used to model the number of successes in a sample of size n obtained with replacement from a population of size N.  c) In biology, living tissue measurements and blood pressure are normally distributed. d) Standardized testing, such as the SAT, is based on a normal distribution. e) Quite a few scores are derivative of the normal distribution, including percentile statuses, normal curve equivalents, stanines, T-scores, and z-scores.

5) Any interesting stories about him? And where and when did he work, and how was his work received during this lifetime?

After he earned his doctorate degree from the University of Helmstadt in 1799 for his work on the Fundamental Theorem of Algebra, Gauss began working on number theory and received a stipend from the Duke of Brunswick.   Upon the death of the duke, Gauss needed steady employment to support his family.  Fortunately, Gauss was considered and ultimately chosen as director of the Observatory at Gӧttingen.  He held this position for the remainder of his live.  This employment gave Gauss the opportunity to contemplate the real-world or practical applications of mathematics.

6) What have I neglected to ask?

Who wouldn’t find the autopsy of Gauss’ brain fascinating?  According to medical examiner Rudolf Wagner, the mass of Gauss’ brain was slightly larger than average and had highly advanced complexities.

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