Leonhard Euler (1707-1783)

Read Euler, read Euler. He is the master of us all. Laplace

Life Story

Leonard Euler was born near Basel, Switzerland. He was a precocious youth, blessed with a gift for languages and an extraordinary memory. Euler eventually carried in his head an assortment of curious information, including orations, poems, and lists of prime powers. He also was a fabulous mental calculator, able to perform intricate arithmetical computations without benefit of pencil and paper. These uncommon talents would serve him well later in life.

After entering the University of Basel at age 14, Leonard encountered his most famous professor, Johann Bernoulli. Not Euler's teacher in a modern sense of the term, Bernoulli instead became a guide for the young scholar, suggesting mathematical readings and making him available to discuss those points that seemed especially difficult.

At university, Euler's education was not limited to mathematics. He spoke on the subject of temperance, wrote on the history of law, and eventually completed a master's degree in philosophy.

Ultimately, the call of mathematics was stronger than anything. His progress was rapid. He had many scientific outputs and proofs related to number theory, logarithms, infinite series, analytic number theory, complex variables, algebra, geometry, combinatorics and graph theory. Some mathematicians argue that his mathematics was not as precise as of today's and he had sometimes proceeded heuristically. However, one can question whether modern mathematics would exist without him.

His grave is in St Petersburg.

Dunham, W. Euler: The Master of Us All, The Mathematical Association of America, Washington (1999). The Dolciani Mathematical Expositions, 22.


Works

Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's equation reaches down into the very depths of existence. Keith Devlin, as quoted in Dr. Euler's Fabulous Formula : Cures Many Mathematical Ills (2006)

Theorem: For any real x, e ix =cosx+isinx. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiaacY cacaWGLbWaaWbaaSqabeaacaWGPbGaamiEaaaakiabg2da9iGacoga caGGVbGaai4CaiaadIhacqGHRaWkcaWGPbGaci4CaiaacMgacaGGUb GaamiEaiaac6caaaa@45DD@

Proof:

e x =1+x+ x 2 2! + x 3 3! + x 4 4! +... e ix =1+ix+ (ix) 2 2! + (ix) 3 3! + (ix) 4 4! +... e ix =1+ix x 2 2! i x 3 3! + x 4 4! + i x 5 5! x 6 6! ... e ix =(1 x 2 2! + x 4 4! x 6 6! + x 8 8! ...)+i(x x 3 3! + x 5 5! x 7 7! +...) e ix =(cosx)+i(sinx) e iπ =(cosπ)+i(sinπ) e iπ +1=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGLb WaaWbaaSqabeaacaWG4baaaOGaeyypa0JaaGymaiabgUcaRiaadIha cqGHRaWkdaWcaaqaaiaadIhadaahaaWcbeqaaiaaikdaaaaakeaaca aIYaGaaiyiaaaacqGHRaWkdaWcaaqaaiaadIhadaahaaWcbeqaaiaa iodaaaaakeaacaaIZaGaaiyiaaaacqGHRaWkdaWcaaqaaiaadIhada ahaaWcbeqaaiaaisdaaaaakeaacaaI0aGaaiyiaaaacqGHRaWkcaGG UaGaaiOlaiaac6caaeaacaWGLbWaaWbaaSqabeaacaWGPbGaamiEaa aakiabg2da9iaaigdacqGHRaWkcaWGPbGaamiEaiabgUcaRmaalaaa baGaaiikaiaadMgacaWG4bGaaiykamaaCaaaleqabaGaaGOmaaaaaO qaaiaaikdacaGGHaaaaiabgUcaRmaalaaabaGaaiikaiaadMgacaWG 4bGaaiykamaaCaaaleqabaGaaG4maaaaaOqaaiaaiodacaGGHaaaai abgUcaRmaalaaabaGaaiikaiaadMgacaWG4bGaaiykamaaCaaaleqa baGaaGinaaaaaOqaaiaaisdacaGGHaaaaiabgUcaRiaac6cacaGGUa GaaiOlaaqaaiaadwgadaahaaWcbeqaaiaadMgacaWG4baaaOGaeyyp a0JaaGymaiabgUcaRiaadMgacaWG4bGaeyOeI0YaaSaaaeaacaWG4b WaaWbaaSqabeaacaaIYaaaaaGcbaGaaGOmaiaacgcaaaGaeyOeI0Ya aSaaaeaacaWGPbGaamiEamaaCaaaleqabaGaaG4maaaaaOqaaiaaio dacaGGHaaaaiabgUcaRmaalaaabaGaamiEamaaCaaaleqabaGaaGin aaaaaOqaaiaaisdacaGGHaaaaiabgUcaRmaalaaabaGaamyAaiaadI hadaahaaWcbeqaaiaaiwdaaaaakeaacaaI1aGaaiyiaaaacqGHsisl daWcaaqaaiaadIhadaahaaWcbeqaaiaaiAdaaaaakeaacaaI2aGaai yiaaaacqGHsislcaGGUaGaaiOlaiaac6caaeaacaWGLbWaaWbaaSqa beaacaWGPbGaamiEaaaakiabg2da9iaacIcacaaIXaGaeyOeI0YaaS aaaeaacaWG4bWaaWbaaSqabeaacaaIYaaaaaGcbaGaaGOmaiaacgca aaGaey4kaSYaaSaaaeaacaWG4bWaaWbaaSqabeaacaaI0aaaaaGcba GaaGinaiaacgcaaaGaeyOeI0YaaSaaaeaacaWG4bWaaWbaaSqabeaa caaI2aaaaaGcbaGaaGOnaiaacgcaaaGaey4kaSYaaSaaaeaacaWG4b WaaWbaaSqabeaacaaI4aaaaaGcbaGaaGioaiaacgcaaaGaeyOeI0Ia aiOlaiaac6cacaGGUaGaaiykaiabgUcaRiaadMgacaGGOaGaamiEai abgkHiTmaalaaabaGaamiEamaaCaaaleqabaGaaG4maaaaaOqaaiaa iodacaGGHaaaaiabgUcaRmaalaaabaGaamiEamaaCaaaleqabaGaaG ynaaaaaOqaaiaaiwdacaGGHaaaaiabgkHiTmaalaaabaGaamiEamaa CaaaleqabaGaaG4naaaaaOqaaiaaiEdacaGGHaaaaiabgUcaRiaac6 cacaGGUaGaaiOlaiaacMcaaeaacaWGLbWaaWbaaSqabeaacaWGPbGa amiEaaaakiabg2da9iaacIcaciGGJbGaai4BaiaacohacaWG4bGaai ykaiabgUcaRiaadMgacaGGOaGaci4CaiaacMgacaGGUbGaamiEaiaa cMcaaeaacaWGLbWaaWbaaSqabeaacaWGPbGaeqiWdahaaOGaeyypa0 JaaiikaiGacogacaGGVbGaai4Caiabec8aWjaacMcacqGHRaWkcaWG PbGaaiikaiGacohacaGGPbGaaiOBaiabec8aWjaacMcaaeaacaWGLb WaaWbaaSqabeaacaWGPbGaeqiWdahaaOGaey4kaSIaaGymaiabg2da 9iaaicdaaaaa@E408@

"Euler's Identity" video from Youtube


Euler as a Teacher