The Hidden Gems of Mathematics: Euler (2 of 4)

This is Blog 2 of a 4-part series where I share how math has changed my career and my life. My hope is that it will inspire you to see math in a new light and, perhaps, as a new friend. I’m going to share three heroes of mine that have impacted actuarial science, mathematics, and many parts of our lives. My first hero is Leonhard Euler, born in Switzerland on April 15, 1707, died in Russia on September 18, 1783.

Euler had an interesting and full life. Euler was fortunate to be tutored at a young age by the great mathematician Johann Bernoulli. Euler took advantage of the opportunity, which is a good reminder for us to seek mentors and learn from them. Nothing stopped Euler from his work, neither hardship nor fame. He served King Frederick the Great of Prussia but was persecuted by the king who didn’t appreciate Euler’s mathematical genius. Euler struggled with eyesight most of his life and was essentially totally blind his last 16 years. For him, losing his sight removed a distraction from math, as demonstrated by an increase in his publications after he became blind. Euler and his wife were blessed with 13 children but suffered the loss of eight who died in childhood. It was common for Euler to do his work while holding a baby.

Euler is the ultimate math machine. He may have published more than any other mathematician. He covered most of the major topics. Euler made major contributions to basic topics such as algebra and geometry. Actuarial topics include combinatorics, logarithms, and the infinite series. A small sampling of other advanced topics include number theory, analytic number theory, and complex variables.

One of Euler’s most famous formula, named Euler’s formula, is also one of the most important formulas (the first equation above). If we set ??=??, then the equation simplifies to either of the two bottom formulas in the image above. Notice this simple formula includes three of the most important mathematical constants (e, i, and ??) and two of the most important numbers (1, the multiplicative constant, and 0, the additive constant). Euler was instrumental in defining the notation for . This formula is the foundation for wave theory which is the foundation for sight, sound (including music), gravity, magnetism, and electricity. Isn’t it amazing that these numbers are connected? Not just connected, but woven into many parts of our lives. Euler even devised Latin Squares, which is the math foundation for Sudoku.

Euler solved the Basel equation which is an infinite sum (see above). It is fascinating that is part of the solution. Here is a link to an inspiring proof of this formula. In generalizing this infinite sum by replacing 2 with any real number s>1, Euler found an incredible connection between this infinite sum and an infinite product. Not just any infinite product, but a product over all the primes.

This equation converts prime numbers as a “random” set of numbers to a very orderly set in relation to natural numbers.

When I first saw the Euler’s Method during my preparation for the LTAM exam (formerly, MLC exam), it was intimidating. The method is a way to approximate a continuous change in reserves using discrete results. To be honest, at first I only learned the bare minimum. That basically required using the formula to solve sample problems. But, after I studied Euler and felt connected to him, I wanted to understand the formula, not just memorize it and know how to use it. Learning about Euler motivated me to dig deeper and understand.

Euler worked until the very end. On Euler’s last day at age 76, he gave his grandchild a math lesson, discussed a recent discovery of planet Uranus with colleagues, performed calculations on balloons, then suddenly had a brain hemorrhage and died. Nicolas de Condorcet eulogized Euler by stating he “ceased to live and calculate.”

If you’re looking for a role model that made a huge impact by solving important and difficult problems while displaying outstanding character during hardship, Euler is your guy! If Euler considered losing his eyesight a blessing, it begs the question: Are there distractions in our lives we can remove so we can more clearly focus on what is important? Laplace famously suggested “Read Euler, read Euler, he is the master of us all.” If interested, you can learn more about Euler in William Dunham’s The Master of Us All. In my next blog I’ll share my second hero, Carl Gauss, who took Euler’s work and took it to new heights.