Euler's Pen and the Language of Math: How One Swiss Scholar Gave Us f(x), e, i and π

Leonhard Euler (1707–1783) matters in mathematics because he shaped both what mathematicians know and how they write it down: he founded graph theory, made major discoveries in number theory, calculus and complex analysis, and introduced much of the notation every student still uses, such as f(x) for a function, e for the base of the natural logarithm and i for the square root of minus one. He is regarded as arguably the most prolific contributor in the history of mathematics and science, and his 866 publications and letters were later gathered into a giant collected edition. In short, if you have ever written an equation in a school notebook, you have probably written it partly in Euler's language.

A pastor's son from Basel

Euler was born in Basel, Switzerland, on 15 April 1707, the son of Paul Euler, a pastor of the Reformed Church, and he spent most of his childhood in the nearby village of Riehen. His father had studied with the famous mathematician Jacob Bernoulli and gave the boy his first lessons in mathematics. The family expected Leonhard to follow his father into the church, and for a while he even studied theology.

At thirteen he entered the University of Basel, which was not unusual at the time, and there he met Johann Bernoulli, one of the leading mathematicians in Europe. Bernoulli was too busy for private lessons, so he gave the young student a different kind of help: he told him to work through difficult books on his own and to bring his questions every Saturday afternoon. Euler later wrote that whenever Bernoulli solved one of his difficulties, ten others disappeared at once. With Bernoulli's support, Euler persuaded his father to let him become a mathematician instead of a pastor. It is a lovely example of a teacher who did not hand over answers but taught a student how to find them.

Saint Petersburg, Berlin and back again

Switzerland had few jobs for a young mathematician, so Euler looked abroad. Johann Bernoulli's sons Daniel and Nicolaus were working at the new Imperial Russian Academy of Sciences in Saint Petersburg, and after Nicolaus died, Daniel recommended his friend for a post there. Euler arrived in May 1727, learned Russian, moved quickly from the medical department to mathematics and even worked for a time as a medic in the Russian Navy. When Daniel Bernoulli returned home, Euler took over as head of the mathematics department, and in 1734 he married Katharina Gsell, the daughter of a painter at the academy.

Russia in those years was politically unstable, and in 1741 Euler accepted an invitation from Frederick the Great of Prussia to join the Berlin Academy. He stayed twenty-five years and wrote several hundred articles there, including his famous textbook on functions, the Introductio in analysin infinitorum, in 1748. The relationship with the king was not warm. Frederick preferred witty conversation, especially from Voltaire, and he found the quiet, religious mathematician unsophisticated outside the world of numbers. The king even complained about a fountain at his palace that did not work, although historians now think Euler's calculations were probably correct and the problem lay elsewhere.

After Catherine the Great came to the Russian throne, Euler returned to Saint Petersburg in 1766 on generous terms, and he spent the rest of his life there.

Working in the dark

One of the most striking parts of Euler's story is his eyesight. In 1738, after a serious fever, he became almost blind in his right eye, and in 1766 a cataract was found in his left eye; an operation helped for a short time, but complications left him almost totally blind. Surprisingly, his output did not fall. With the help of scribes he dictated his work, and in 1775 he produced on average one mathematical paper per week. His memory was famous: he could recite Virgil's Aeneid and knew the first hundred prime numbers along with their powers up to the sixth. When he lost the sight of one eye, he is reported to have said that now he would have fewer distractions.

On 18 September 1783, after lunch with his family, Euler was discussing the orbit of the newly discovered planet Uranus when he collapsed and died of a brain hemorrhage. The French thinker Condorcet ended his eulogy with a famous line: he ceased to calculate and to live.

The notation we all write

For most people, Euler's biggest gift is invisible because it feels so ordinary. He was the first to write f(x) for the value of a function f at x, he introduced the modern way of writing trigonometric functions, and he used e for the base of the natural logarithm, Σ for sums and i for the imaginary unit. The symbol π for the ratio of a circle's circumference to its diameter came from the Welsh mathematician William Jones, but it was Euler who made it popular. Because his textbooks were read so widely across Europe, his habits became everyone's habits.

Think of notation as the grammar of mathematics. A good symbol is like a good word: it packs a long idea into something short that people can share. Before such symbols became standard, writers often described mathematical relationships in long sentences. Euler's clear symbols made it easier to read, check and build on other people's work, which is one reason his influence lasted so long.

e^(iπ) + 1 = 0, explained gently

The formula most connected with his name is Euler's identity, e^(iπ) + 1 = 0. It brings together five of the most important numbers in mathematics: 0, 1, π (about 3.14159), e (about 2.71828) and i, the number whose square is minus one. It also uses addition, multiplication and raising to a power exactly once each.

You do not need advanced math to feel why people find it surprising. The number π comes from circles, e comes from growth and interest rates, and i comes from trying to take the square root of a negative number. These ideas grew up in different corners of mathematics, yet one short line ties them together and the answer is simply zero. The identity is a special case of Euler's formula, e^(ix) = cos x + i sin x, which links the exponential function with the geometry of circles.

Readers of The Mathematical Intelligencer voted it the most beautiful theorem in mathematics in 1990. There is an honest twist, though: Euler published the general formula in 1748, but it is unclear whether he himself ever wrote down this exact five-constant version, and it seems to be unknown who first stated it explicitly. The name honours the man whose work made it possible.

Seven bridges and the birth of graph theory

In 1735 Euler solved a puzzle from the Prussian city of Königsberg, which sat on a river with two islands joined to the rest of the city by seven bridges. The question was whether a person could take a walk that crossed each bridge exactly once. Euler proved it was impossible, and his solution is considered the first theorem of graph theory. His trick was to ignore distances and shapes and look only at what is connected to what. That same way of thinking now sits behind route planning, computer networks and social media maps.

He also discovered that for a solid shape without holes, the number of corners minus the number of edges plus the number of faces always equals 2, a result now called the Euler characteristic. You can test it with a cube: 8 corners, 12 edges and 6 faces give 8 minus 12 plus 6, which is 2.

Beyond pure mathematics

Euler did not stay inside pure theory. He solved the Basel problem, finding the exact sum of the reciprocals of the square numbers, a question that had defeated leading mathematicians for decades. He rewrote Newton's laws of motion to describe rigid bodies, wrote equations for the flow of fluids, and studied ships, optics, the bending of beams and the loads that columns can carry. He was also a gifted teacher for ordinary readers: his more than two hundred letters to a German princess, explaining physics and mathematics in plain language, became more widely read than any of his technical works.

Why he still matters

Later mathematicians spoke of him with unusual respect. Pierre-Simon Laplace told students to read Euler because he is the master of us all, and Carl Friedrich Gauss said that studying Euler's works was the best school for mathematics. He appeared on Swiss banknotes and many stamps, and an asteroid and a crater on the Moon carry his name.

For language learners there is a quiet lesson in his life. Euler moved between Swiss German, Latin, Russian and French-speaking courts, and he spent his career turning complicated ideas into clear, shared symbols. Learning a language is a similar job: you collect small, reliable pieces and use them again and again until they feel natural. Euler's steady habit of working every day, even without his sight, is a good model for any long project.

Learn with Euler: English phrases

Phrase	Meaning	Example
to crunch the numbers	to do the calculations	Euler could crunch the numbers in his head.
a household name	someone everyone knows	In math, Euler is a household name.
against the odds	despite great difficulty	He kept working against the odds after losing his sight.
to leave a mark on	to have a lasting effect on	His notation left a mark on every textbook.
it all adds up	it makes sense	Once you see the bridges as dots and lines, it all adds up.

Try this: Write three English sentences about a skill you are learning, using "against the odds" and "it all adds up".