# The Hidden Gems of Mathematics: Gauss (3 of 4)

*This is part 3 of a 4-part series where I share how math has changed my career and my life. I’m sharing three heroes of mine that have impacted actuarial science, mathematics, and many parts of our lives. I discussed my first hero, Leonhard Euler, in my last blog. *

My second hero is Carl Friedrich Gauss, born in Germany on April 30, 1777, and died on February 23, 1855. To give you context, Gauss was born 70 years after Euler was born and 6 years before Euler died. Like Euler, Gauss had his challenges to overcome. Gauss was born to a poor family. Pursuing a career in math and science was not a sure way to pay the bills. But being poor didn’t slow Gauss’s pursuit of excellence.

Actuaries benefit directly from Gauss’s development of the normal distribution and method of least squares. But, let’s start at his beginnings.

Gauss became well known when he calculated where a newly discovered dwarf planet Ceres would reappear. A true data scientist, he predicted where the planet would appear within 0.5 of a degree though he was only given data for less than 3 degrees. Gauss is a giant in number theory as he is credited for making modular arithmetic real math and proving the law of quadratic reciprocity.

A key part of number theory are the prime numbers. The order of prime numbers was an unsolved mystery for mathematicians before Gauss. Gauss found a key clue to this mystery by identifying the probability that a random number between 1 and N is prime equals 1/ln(N). That means the expected number of primes between 1 and N is N/ln(N). This was a major breakthrough in understanding primes.

I mentioned in the previous blog that Euler defined i to represent √-1. √-1 was defined as the imaginary number by Rene Descartes centuries before Euler used the letter i. √-1 was not considered an important concept until Euler and Gauss uncovered its value, hence the term “imaginary.” Ironically, almost all of our modern conveniences can be traced back to this number. Why did the industrial revolution and modern inventions (light bulb, telephone, etc.) occur in the late 18th century and most of the 19th century? This progress was possible because of the advances in mathematics, with the imaginary number and complex analysis at the center of this progress. The concept of the imaginary number started around 1572. That means it took approximately 200 years before mathematicians identified the potential for this discovery. One thing that makes mathematicians great, such as Gauss and Euler, is they find “diamonds in the rough of math.” Gauss and Euler found the imaginary number as a diamond by developing the new math that expanded real numbers to complex numbers. After mathematicians developed the math, scientists used it to change the world.

Some consider Gauss as the greatest mathematician of all time, or at least the “prince of mathematicians.” This is not because of the number of papers he published. Gauss was the opposite of Euler who published nonstop. Even though Gauss was a great mathematician, he was not confident about publishing his work until he was absolutely sure it was perfect. So much of Gauss’s work was found unpublished after his death. I learned two important lessons from Gauss here.

One is not to be afraid to share my work. I often doubt my work even though I pursue professional quality. Maybe it’s the actuary in me that sees worst-case scenarios ( someone finding major mistakes somewhere). But, if I don’t publish, others don’t benefit. So I’m working on moving past those doubts and fears. I had doubts and fears about publishing this 4-part series on mathematicians. The history and importance of math is outside the normal conversation for actuaries, so many may struggle to see the relevance. However, these concepts have made a major impact on my life, so perhaps they can help others as well. So I publish.

A second lesson for me is to encourage others to publish as well. That means creating a culture where they feel comfortable taking the chance to publish. This not only includes actuaries but also many other professionals such as programmers who must move their code to production.

His discovery of a non-Euclidean geometry is a specific example of work that Gauss chose not to publish. Euclid’s geometry was the only valid geometry for 2,300 years. Imagine the pressure Gauss felt when he uncovered an equally elegant way to define geometry where the sum of angles for a triangle can be less than or greater than 180 degrees. It’s not likely he saw all the potential for alternative geometries, but other mathematicians and scientists took this idea and made it into one of the most important discoveries of all time.

We’ll continue this story in our next blog where we’ll meet my third hero, Bernhard Riemann. Now that we’ve digested the published and unpublished work of Gauss, we admire his mathematical genius. However, perhaps his greatest contribution was the impact he had on Riemann. Gauss mentored Riemann and convinced him to pursue a career in mathematics. Just like Gauss took Euler’s work to new heights, Riemann took Gauss’s work to a whole new world. So one final lesson from Gauss is the importance of mentoring others. Society reaps exponential gains when we invest our time mentoring others. Or, in Gauss’ own words, “Mathematicians stand on each others’ shoulders.”