Leonhard Euler

⚡ Key Takeaways

Leonhard Euler (1707–1783) was a Swiss mathematician and physicist, the most prolific mathematician in history and arguably the most influential of the 18th century. One-line statement: Euler founded or transformed virtually every branch of modern mathematics — analysis, number theory, graph theory, mechanics, and the calculus of variations — while introducing notation ( $e$ , $i$ , $f(x)$ , $\Sigma$ ) still universal today. Key result in words: he produced over 850 works (886 in the standard catalogue), including the identity $e^{i\pi}+1=0$ , the solution of the Basel problem, the gamma function, and the Königsberg bridges theorem that birthed topology. Where it applies: all of pure and applied mathematics, theoretical physics, engineering, and astronomy. Who/when: Basel, Saint Petersburg, and Berlin, across the High Enlightenment.

💡 The Core Insight

Euler's defining method was the bold, fearless manipulation of infinite processes — series, products, and limits — guided by analogy and confirmed by computation. He treated transcendental functions as algebraic objects that could be expanded, multiplied, and rearranged like polynomials of infinite degree. This "analytic optimism," though not always rigorous by modern standards, repeatedly produced correct and profound results. His single deepest organizing principle was the primacy of the function as the central object of analysis, replacing the geometric curves that dominated 17th-century mathematics.

DefinitionEuler's Number and the Exponential

The constant $e$ is defined as

1

\quad e = \lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n} = \sum_{k=0}^{\infty}\frac{1}{k!} \approx 2.718281828.

Derivation hint: expand $(1+1/n)^n$ by the binomial theorem and let $n\to\infty$ term-by-term; each binomial coefficient term $\binom{n}{k}n^{-k}\to 1/k!$ . Here $k! = 1\cdot 2\cdots k$ (with $0!=1$ ) is the factorial, and the series converges absolutely for all real arguments when generalized to $e^x=\sum_k x^k/k!$ .

TheoremEuler's Formula

For every real $\theta$ ,

2

\quad e^{i\theta} = \cos\theta + i\sin\theta,

where $i=\sqrt{-1}$ . The special case $\theta=\pi$ gives Euler's identity

3

\quad e^{i\pi} + 1 = 0,

linking the five fundamental constants $0,1,e,i,\pi$ . Derivation hint: substitute $x=i\theta$ into the power series for $e^x$ , separate real and imaginary parts, and recognize the Taylor series of $\cos\theta$ and $\sin\theta$ .

TheoremSolution of the Basel Problem

The sum of reciprocal squares is

4

\quad \sum_{n=1}^{\infty}\frac{1}{n^{2}} = \frac{\pi^{2}}{6}.

Derivation hint: treat $\sin x / x$ as an infinite polynomial with roots at $\pm n\pi$ , form the Weierstrass-type product $\prod(1-x^2/n^2\pi^2)$ , and match the coefficient of $x^2$ against its Taylor expansion.

TheoremThe Euler Product and the Zeta Function

For $\operatorname{Re}(s)>1$ ,

5

\quad \zeta(s) = \sum_{n=1}^{\infty}\frac{1}{n^{s}} = \prod_{p\ \text{prime}}\frac{1}{1-p^{-s}}.

Derivation hint: expand each factor as a geometric series $\sum_k p^{-ks}$ and invoke the unique factorization of integers into primes (the fundamental theorem of arithmetic).

TheoremEuler's Polyhedron Formula

For any convex polyhedron with $V$ vertices, $E$ edges, and $F$ faces,

6

\quad V - E + F = 2.

Derivation hint: project onto a plane to form a planar graph; remove edges/faces one at a time, showing each removal preserves $V-E+F$ , until a single triangle (or vertex) remains.

DefinitionThe Gamma Function

Euler extended the factorial to non-integers by the integral

7

\quad \Gamma(z) = \int_{0}^{\infty} t^{z-1}e^{-t}\,dt, \qquad \operatorname{Re}(z)>0,

satisfying $\Gamma(n+1)=n!$ for non-negative integers $n$ . Derivation hint: integrate by parts to obtain the recurrence $\Gamma(z+1)=z\,\Gamma(z)$ , and evaluate $\Gamma(1)=\int_0^\infty e^{-t}dt=1$ .

ProofEuler's Formula via Power Series

Step 1. Begin with the absolutely convergent series defining the exponential of a complex argument:

8

\quad e^{z} = \sum_{k=0}^{\infty}\frac{z^{k}}{k!}.

Derivation hint: this is the analytic continuation of the real series in [1]; absolute convergence permits rearrangement.

Step 2. Set $z=i\theta$ and split the sum by the parity of $k$ , using $i^0=1,\ i^1=i,\ i^2=-1,\ i^3=-i$ (period 4):

9

\quad e^{i\theta} = \sum_{m=0}^{\infty}\frac{(-1)^{m}\theta^{2m}}{(2m)!} \;+\; i\sum_{m=0}^{\infty}\frac{(-1)^{m}\theta^{2m+1}}{(2m+1)!}.

Derivation hint: even powers $k=2m$ give real terms with sign $(-1)^m$ ; odd powers $k=2m+1$ give imaginary terms.

Step 3. Recognize the two series as the Taylor expansions of cosine and sine:

10

\quad \cos\theta = \sum_{m=0}^\infty \frac{(-1)^m\theta^{2m}}{(2m)!},\qquad \sin\theta = \sum_{m=0}^\infty \frac{(-1)^m\theta^{2m+1}}{(2m+1)!}.

Derivation hint: these follow from repeated differentiation of $\cos$ and $\sin$ evaluated at $0$ .

Step 4. Substitute to conclude $e^{i\theta}=\cos\theta+i\sin\theta$ , which is [2]. Setting $\theta=\pi$ gives $e^{i\pi}=\cos\pi+i\sin\pi=-1$ , hence [3]. $\blacksquare$

∎

mathematician · Leonhard Euler | 1707–1783

Born in Basel on 15 April 1707 to a Calvinist pastor, Euler studied under Johann Bernoulli, the leading mathematician of the age, who recognized the boy's genius and gave him private lessons. He completed his master's degree in philosophy at thirteen and finished theology training before turning fully to mathematics. Euler possessed a phenomenal memory (he could recite the Aeneid and recall the first six powers of every integer up to 100) and an unparalleled capacity for mental calculation. He fathered thirteen children, of whom five survived; he worked easily with a child on his lap. After losing sight in his right eye in 1738 and the left around 1766, he remained the most productive mathematician alive, declaring that the loss freed him from distraction. He died of a brain hemorrhage in Saint Petersburg on 18 September 1783, having calculated the orbit of the newly discovered Uranus that very day — "he ceased to calculate and to live," wrote the eulogist Condorcet.

🧠 Intuition

Picture Euler's worldview as treating the infinite as just another arena for ordinary algebra. Where a modern student sees an infinite series and worries about convergence, Euler saw a "polynomial of infinite degree" that he could factor, multiply, and rearrange. The Basel problem is the cleanest example: $\sin x$ vanishes at $0,\pm\pi,\pm 2\pi,\dots$ , so Euler reasoned that $\frac{\sin x}{x}$ — which is $1$ at the origin — must factor like a polynomial built from its roots, $\prod_n(1-x^2/n^2\pi^2)$ . Comparing the $x^2$ coefficient of this product with the known Taylor coefficient $-1/6$ instantly delivers $\sum 1/n^2=\pi^2/6$ . The audacity is breathtaking and the answer is exactly right; the rigorous justification (Weierstrass factorization) came a century later. Euler's instinct was that analogy with the finite, carried boldly into the infinite, tells the truth more often than not — and his computational checking kept the analogies honest.

Figure

Euler's formula on the complex unit circle:

e^{i\theta}

traces the unit circle, with real part

\cos\theta

and imaginary part

\sin\theta

.

ExampleComputing the Basel-Type Sum Numerically

Given: verify [4] by partial summation. Compute $S_N=\sum_{n=1}^{N}1/n^2$ for $N=1000$ and compare to $\pi^2/6$ .

Steps: 1. The target is $\pi^2/6 = 9.8696044.../6$ . Using $\pi=3.14159265$ , $\pi^2=9.8696044$ , so $\pi^2/6 = 1.6449341$ . 2. The tail estimate: $\sum_{n>N}1/n^2 \approx \int_N^\infty x^{-2}\,dx = 1/N$ . For $N=1000$ , the tail $\approx 0.001$ . 3. Therefore $S_{1000}\approx 1.6449341 - 0.001 = 1.6439\ldots$ (direct summation gives $1.6439346$ ). 4. The discrepancy $1.6449341-1.6439346 = 0.0009995 \approx 1/N$ , confirming the tail estimate.

Answer: $\boxed{\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}\approx 1.6449341}$ (dimensionless).

ExampleApplying the Polyhedron Formula to a Soccer Ball

Given: a truncated icosahedron (the classic soccer-ball pattern) has $F=32$ faces (12 pentagons + 20 hexagons). Find the number of vertices $V$ .

Steps: 1. Count edges by faces: each pentagon contributes 5 edges, each hexagon 6, but every edge is shared by 2 faces:

11

\quad E = \frac{12\cdot 5 + 20\cdot 6}{2} = \frac{60+120}{2} = 90.

Derivation hint: sum the edge counts of all faces and divide by 2 because each edge borders exactly two faces. 2. Apply Euler's formula [6]: $V - E + F = 2 \Rightarrow V = 2 + E - F = 2 + 90 - 32$ . 3. Compute: $V = 60$ .

Answer: $\boxed{V = 60 \text{ vertices}}$ — matching the 60 carbon atoms of the buckminsterfullerene molecule $C_{60}$ .

⚠️ Common mistake

"Euler discovered the number $e$ / invented $\pi$ ." Neither is true. The constant $e$ appears implicitly in earlier work (Napier's logarithms, Jacob Bernoulli's compound-interest limit); Euler named it and developed its analytic theory. Likewise $\pi$ is ancient; Euler standardized its notation. Crediting him with discovery rather than systematization and deep development misrepresents the history.

"Euler's manipulations were always rigorous." Many were not by modern standards. He summed divergent series, freely interchanged limits and sums, and treated infinite products informally. His genius lay in knowing when such moves were safe — a judgment later formalized by Cauchy, Abel, and Weierstrass.

CounterexampleWhy naïve series manipulation can fail

Euler famously assigned the divergent series the value

12

\quad 1 - 1 + 1 - 1 + \cdots \stackrel{?}{=} \tfrac{1}{2}

Derivation hint: substitute $x=1$ into the geometric series $\sum (-x)^n = 1/(1+x)$ , ignoring that the series diverges at $x=1$ . By ordinary partial sums this series oscillates between $1$ and $0$ and has no limit. The "value" $1/2$ is the Abel/Cesàro summation, a meaningful but distinct notion. The lesson: Euler's bold extrapolations capture genuine structure (Abel summation is rigorously consistent), but treating [12] as an ordinary convergent sum is an error. The unrestricted rearrangement of conditionally convergent series can produce any limit whatsoever — a warning Euler's successors had to make precise.

connection

Euler's work is the hub from which much of modern mathematics radiates. His Euler's Formula is the bridge between Complex Analysis and Trigonometry, rendering all trigonometric identities as algebraic consequences of exponential rules. The Euler Product in [5] launched Analytic Number Theory and is the historical seed of the Riemann Zeta Function and the still-open Riemann Hypothesis. The Königsberg bridges problem and Euler's Polyhedron Formula founded Graph Theory and Topology, the formula being the first topological invariant (the Euler characteristic). His Calculus of Variations (the Euler–Lagrange equation) underlies all of Lagrangian Mechanics and modern physics, while the Gamma Function connects to Probability Theory, Special Functions, and quantum field theory.

🔬 Where this appears

⏳ History

- 1707 — Born 15 April in Basel, Switzerland, son of pastor Paul Euler.

- 1720–1726 — Enters the University of Basel at 13; studies under Johann Bernoulli; earns a master's in philosophy.

- 1727 — Joins the Imperial Russian Academy of Sciences in Saint Petersburg, on the recommendation of Daniel and Nicolaus Bernoulli.

- 1734 — Solves the Basel problem (

\sum 1/n^2=\pi^2/6

), [4], catapulting him to European fame at age 28.

- 1736 — Publishes the Seven Bridges of Königsberg solution, founding graph theory; also publishes Mechanica, recasting Newtonian mechanics in analytic form.

- 1738 — Loses sight in his right eye, possibly from overwork or a fever.

- 1741 — Moves to Berlin at the invitation of Frederick the Great, joining the Prussian Academy; produces ~380 works during 25 years there.

- 1748 — Publishes Introductio in analysin infinitorum, the foundational text on functions and the source of Euler's formula [2].

- 1755 — Publishes Institutiones calculi differentialis, systematizing differential calculus.

- 1766 — Returns to Saint Petersburg under Catherine the Great; soon becomes almost totally blind.

- 1768–1770 — Produces Institutiones calculi integralis and Letters to a German Princess, a hugely popular science exposition, all while blind.

- 1783 — Dies 18 September in Saint Petersburg, computing to the end.

open-problem

Several questions Euler opened remain unresolved or were only settled long after him. The Riemann Hypothesis, the deepest descendant of his zeta-function work [5], concerning the location of the nontrivial zeros of $\zeta(s)$ , remains the most famous open problem in mathematics. Euler conjectured a generalization of Fermat's Last Theorem (Euler's sum-of-powers conjecture) that was disproved in 1966 by computer search — a reminder that even his intuition could fail. The rigorous foundation of his divergent-series manipulations awaited the 19th-century theories of Cauchy, Abel, and the 20th-century summability methods of Cesàro and Borel; understanding precisely which of his formal computations are valid, and under what summation, remains a subtle subject. Finally, the regularity of solutions to the 3D Euler equations of fluid flow — whether smooth initial data can develop singularities in finite time — is a major open problem in analysis.

Formal Statement of Euler's Central Results

Why It Holds — Derivation of Euler's Formula

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also