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Five numbers, E, I, pi, 1, and 0.

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The base of natural logarithms, the imaginary unit,

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the circle constant, the multiplicative identity, and nothing.

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Five numbers from completely different areas of mathematics.

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One man discovered they're connected by the most beautiful equation ever written.

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E to the I pi plus 1 equals 0.

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And that was just one afternoon's work for Leonhard Euler.

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Last time, we met the Bernoulli family,

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eight mathematicians across three generations

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who mastered Leibniz's calculus and advanced probability and physics.

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Johann Bernoulli's greatest student would outshine them all.

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Welcome back to Men of Mathematics.

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Today we meet Leonhard Euler, the most prolific mathematician in history.

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over 866 papers, contributions to every branch of mathematics that existed,

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and much of the notation we still use today.

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And he did nearly half of it while completely blind.

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Leonhard Euler was born on April 15, 1707, in Basel, Switzerland.

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His father was a pastor who had studied mathematics under Jacob Bernoulli.

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Young Euler received private lessons from Johann Bernoulli,

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who quickly recognized extraordinary talent.

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At 13, Euler enrolled at the University of Basel, the youngest student there.

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By 20, he had written a paper that almost won the Paris Academy Prize.

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The winner? A 19-year-old.

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Euler spent most of his career at the St. Petersburg Academy in Russia

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and the Berlin Academy in Prussia.

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He had 13 children, only five of whom survived childhood.

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He reportedly could do mathematics while bouncing a baby on his knee or while children played at his feet In 1738 Euler lost sight in his right eye By 1771 he was completely blind His response his output increased He developed astonishing mental calculation abilities

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dictating papers from memory. Assistance would transcribe as Euler calculated entirely in his

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head. Euler discovered that exponential and trigonometric functions are intimately connected

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in the complex plane, e to the i theta equals cosine theta plus i sine theta. When theta equals

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pi, this becomes e to the i pi plus 1 equals 0. Five fundamental constants, e, i, pi, 1, and 0,

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united in one equation. Addition, multiplication, and exponentiation. Nothing contrived, just pure

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mathematics revealing deep connections. In 1736, Euler solved a puzzle about the bridges of

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Königsberg. Can you cross all seven bridges exactly once and return to your starting point?

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Euler proved it's impossible, and in doing so invented graph theory. He reduced the problem

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to vertices, landmasses, and edges, bridges, showing that a solution requires all vertices

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to have even degree. The problem isn't about geometry. It's about connectivity.

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This abstraction launched an entirely new branch of mathematics. For nearly a century,

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mathematicians struggled with the sum 1 plus 1 fourth plus 1 ninth plus 1 sixteenth and so on.

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The Bernoullis tried and failed. Then the 28-year-old Euler solved it. The sum of 1 over n

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squared equals pi squared over 6. The appearance of pi in this sum about integers shocked the

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mathematical world Why should circles have anything to do with squares of whole numbers For any convex polyhedron count the vertices v edges e and faces f Euler discovered v minus e plus f equals 2

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A cube, 8 minus 12 plus 6 equals 2.

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A tetrahedron, 4 minus 6 plus 4 equals 2.

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This simple formula launched topology, the study of properties unchanged by continuous deformation.

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Much of mathematical notation comes from Euler.

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E for the base of natural logarithms, 2.71828.

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I for the imaginary unit, the square root of negative 1.

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Pi for the circle constant, popularized by Euler.

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Sigma for summation, and F of X for function notation.

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Before Euler, mathematics looked different.

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After Euler, it looked modern.

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Euler published approximately 866 papers and books.

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more than any mathematician in history. His collected works, the opera Omnia,

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fills over 80 volumes and is still being edited over 200 years after his death.

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Even accounting for his long life, Euler averaged about one paper per month for his entire career,

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including the years he was blind. When asked how to learn mathematics, Laplace answered,

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Read Euler, read Euler, he is the master of us all.

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The phrase captures Euler's unique position.

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He wasn't just prolific, he was foundational.

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Nearly every branch of mathematics traces key developments to Euler.

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Beyond pure mathematics, Euler contributed to physics,

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Euler's equations for rigid body rotation,

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astronomy, lunar theory, and orbital mechanics,

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music theory, mathematical analysis of harmony,

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engineering, ship design, and lens systems,

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and number theory, Euler's totian function,

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and proofs about primes.

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When a problem existed Euler probably worked on it On September 18 1783 Euler spent the morning calculating the orbit of the newly discovered planet Uranus He had lunch with his family

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discussed mathematics with a colleague, and while playing with his grandson,

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suffered a brain hemorrhage. As one contemporary wrote, he ceased to calculate and to live.

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Leonhard Euler was the most prolific mathematician who ever lived, and arguably one of the greatest.

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His formula connects exponentials to trigonometry.

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His graph theory connects problems to their abstract structure.

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His polyhedron formula connects geometry to topology.

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He invented notation, solved problems, founded fields, and did nearly half of it while blind.

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There's something inspiring about Euler's response to blindness.

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Most people would despair.

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Euler calculated faster.

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His mind held entire mathematical structures that his eyes could no longer see.

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When the external world dimmed, the internal world burned brighter.

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Read Euler.

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Read Euler.

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He is the master of us all.

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As E.T. Bell wrote,

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Euler calculated without apparent effort as men breathe or as eagles sustain themselves in the wind.

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Next time on Men of Mathematics, we meet Joseph Louis Lagrange,

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the mathematician who reformulated physics without a single diagram.

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His mécanique analytique replaced Newton's geometric methods with pure algebra.

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His Lagrange multipliers solve constrained optimization.

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His work on polynomial equations would inspire Abel and Galois.

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Thank you for joining us on this journey through mathematical history.

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If you're inspired by the master of us all, subscribe and hit the notification bell.

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New episodes release every week.

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