Meet Leonhard Euler—a mathematician who solved the impossible and shaped how we see the world. In this blog, Coach K shares how Euler’s resilience and genius transformed everything from prime numbers to actuarial science.
This is the second blog in a four-part series where I share how math has shaped my career and my life. My hope is to inspire you to see math in a new light and, perhaps, as a new friend. I’m going to introduce you to three of my heroes who have had a profound impact on actuarial science, mathematics, and many aspects of our lives.
My first hero is Leonhard Euler, born in Switzerland on April 15, 1707, and passed away in Russia on September 18, 1783.
Euler’s Life and Legacy
Euler led an extraordinary and eventful life. He was fortunate to be tutored at a young age by the renowned mathematician Johann Bernoulli, and he made the most of this opportunity—a reminder to us all of the value of seeking mentors and learning from them.
Despite facing significant challenges, Euler’s commitment to his work never wavered. He served King Frederick the Great of Prussia but endured persecution from the king, who failed to appreciate Euler’s mathematical genius.
He also struggled with poor eyesight for most of his life and was almost completely blind during his final 16 years. Euler considered his blindness a blessing, as it removed distractions and allowed him to focus entirely on his research—leading to an increase in his mathematical publications.
Euler and his wife had 13 children but experienced the heartbreak of losing eight of them in childhood. Nevertheless, it wasn’t uncommon for Euler to do his mathematical work with a baby in his arms.
The Ultimate Math Machine
Euler is often regarded as one of the most prolific mathematicians in history. He published more papers and made contributions to more topics than perhaps any other mathematician. His work spanned foundational areas like algebra and geometry, as well as actuarial-related fields such as combinatorics, logarithms, and infinite series. He also made significant contributions to advanced areas like number theory and complex variables.
One of Euler’s most famous contributions is Euler's formula:
The equation simplifies to:
The result is surprisingly simple and elegant—one of those rare equations that mathematicians describe as "beautiful." Euler was also instrumental in establishing the notations we still use for these mathematical constants today.
Euler’s discoveries underpin our understanding of waves, which affects light, sound, music, gravity, magnetism, and electricity. Isn’t it amazing that these ideas are connected? Euler even devised Latin Squares, a concept that later became the foundation for the modern puzzle game Sudoku.
Solving the Basel Problem
Euler solved the Basel problem, which involves finding the value of the infinite sum shown below:
Surprisingly, the solution includes the value of pi squared, revealing a deep connection between an infinite sum of fractions and a fundamental mathematical constant. For a more detailed explanation, there are various proofs showcasing how this formula emerges.
Building on this, Euler generalized the infinite sum by replacing the exponent of two with any real number greater than one, resulting in the following expression, now known as the Riemann zeta function:
But Euler’s insights didn’t stop there. He discovered an incredible link between this sum and an infinite product that involves all prime numbers:
This equation transforms prime numbers, often considered random and unpredictable, into an ordered structure that relates directly to natural numbers.
Euler’s discovery demonstrated that prime numbers, though seemingly scattered, play a crucial role in defining the behavior of infinite series.
Euler’s Influence on My Actuarial Journey
I first encountered Euler’s Method while preparing for the LTAM exam (formerly known as the MLC exam). It’s a numerical method for approximating continuous changes in reserves using discrete calculations. Initially, I found it intimidating and approached it by simply memorizing the formula and practicing sample problems.
But after learning about Euler’s life and contributions, I felt a deeper connection to him—and that connection inspired me to truly understand the method, not just memorize it. Learning about Euler encouraged me to dig deeper and embrace the “why” behind the formula.
Euler worked tirelessly until the very end of his life. On the day he died at age 76, he gave his grandchild a math lesson, discussed the recent discovery of the planet Uranus with colleagues, performed calculations involving balloons—and then, without warning, suffered a brain hemorrhage and passed away.
The mathematician Nicolas de Condorcet eulogized him by saying that Euler “ceased to live and calculate.”
A Role Model for Perseverance
If you’re looking for a role model who made groundbreaking contributions while displaying extraordinary character in the face of hardship, Euler is your guy. His ability to turn adversity into strength raises an important question: Are there distractions in our lives that we can remove to focus more clearly on what truly matters?
The mathematician Pierre-Simon Laplace famously said, “Read Euler, read Euler; he is the master of us all.” If you’re curious to learn more about Euler, I highly recommend The Master of Us All by William Dunham.
In my next blog, I’ll introduce you to my second hero, Carl Gauss—who took Euler’s work and elevated it to new heights.