# Leonhard Euler

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### 4 Responses to Leonhard Euler

1. s.A says:

Euler's identity

2. Anonymous says:

the scientists said "we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth."
God might answer "I possess the power of break this ring circle and human' ability only of stretch and extend"
but where is God? Gods are everywhere. you can meet some even at museums, your silent prays their silent respond. they seem to say:
follow your conscience, seek peace you will receive happiness. don't against self, against nature, don't escape/avoid difficult and problems, seek truth…
Life is to learn to live, to face and to deal
this is life

3. shinichi says:

Euler’s identity

Wikipedia

https://en.wikipedia.org/wiki/Euler%27s_identity

Mathematical beauty

Euler’s identity is often cited as an example of deep mathematical beauty. Three of the basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation. The identity also links five fundamental mathematical constants:

• The number 0, the additive identity.
• The number 1, the multiplicative identity.
• The number π, which is ubiquitous in the geometry of Euclidean space and analytical mathematics (π = 3.14159265…)
• The number e, the base of natural logarithms, which occurs widely in mathematical analysis (e = 2.718281828…).
• The number i, the imaginary unit of the complex numbers, a field of numbers that contains the roots of all polynomials (that are not constants), and whose study leads to deeper insights into many areas of algebra and calculus.

(Both π and e are transcendental numbers.)

Furthermore, the equation is given in the form of an expression set equal to zero, which is common practice in several areas of mathematics.

Stanford University mathematics professor Keith Devlin has said, “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”. And Paul Nahin, a professor emeritus at the University of New Hampshire, who has written a book dedicated to Euler’s formula and its applications in Fourier analysis, describes Euler’s identity as being “of exquisite beauty”.

The mathematics writer Constance Reid has opined that Euler’s identity is “the most famous formula in all mathematics”. And Benjamin Peirce, a noted American 19th-century philosopher, mathematician, and professor at Harvard University, after proving Euler’s identity during a lecture, stated that the identity “is absolutely paradoxical; we cannot understand it, and we don’t know what it means, but we have proved it, and therefore we know it must be the truth”.

A poll of readers conducted by The Mathematical Intelligencer in 1990 named Euler’s identity as the “most beautiful theorem in mathematics”. In another poll of readers that was conducted by Physics World in 2004, Euler’s identity tied with Maxwell’s equations (of electromagnetism) as the “greatest equation ever”.

History

It has been claimed that Euler’s identity appears in his monumental work of mathematical analysis published in 1748, Introductio in analysin infinitorum. However, it is questionable whether this particular concept can be attributed to Euler himself, as he may never have expressed it. (Moreover, while Euler did write in the Introductio about what we today call “Euler’s formula”, which relates e with cosine and sine terms in the field of complex numbers, the English mathematician Roger Cotes also knew of this formula and Euler may have acquired the knowledge through his Swiss compatriot Johann Bernoulli.)

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“Euler’s identity” は日本語では「オイラーの等式」という。

オイラーの等式

ウィキペディア

https://ja.wikipedia.org/wiki/オイラーの等式

数学誌のThe Mathematical Intelligencer の読者調査によると、この等式は「数学における最も美しい定理」 (The most beautiful theorem in mathematics) に選出されている。また、2004年に実施された Physics World 誌での読者調査ではマクスウェルの方程式と並び、「史上最も偉大な等式」(Greatest equation ever) に選出されている。

ポール・ネイヒン（ニューハンプシャー大学 (en) 名誉教授）の著書「オイラー博士の偉大な式」(Dr. Euler’s Fabulous Formula) [2006] では、この等式のために400ページも充てている。本著書ではこの等式を「数学的な美の絶対的基準」(The gold standard for mathematical beauty) としている。

コンスタンス・レイド (en) は、オイラーの等式を「全ての数学分野において最も有名な式」(The most famous formula in all mathematics) であると主張した。

カール・フリードリッヒ・ガウスは「この式を見せられた学生がすぐにその意味を理解できなければ、その学生は第一級の数学者には決してなれない」(If this formula was not immediately apparent to a student on being told it, the student would never be a first-class mathematician.) と指摘している。

この等式がベンジャミン・パース (19世紀の数学者、ハーバード大学教授) の講義で紹介されたあと、「全く逆説的なことだ、我々はそれを理解できないし、それがどんな意義を持っているかも分からない。だが我々はそれを証明したし、それゆえにそれが間違いのない真実であると知っている」(It is absolutely paradoxical; we cannot understand it, and we don’t know what it means, but we have proved it, and therefore we know it must be the truth.) と付け加えた。

スタンフォード大学の数学の教授、キース・デブリン (en) は「愛の本質そのものをとらえるシェークスピアのソネットのように、あるいは、単なる表面でなくはるかに深い内面から人間の形の美しさを引き出す絵画のように、オイラーの等式は存在の遥かな深遠にまで到達している」(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.) と記している。

Bob Palaisが2001年に公開したエッセイ “π is wrong!” の中では、円周率πの代わりに、τ、円の周の半径に対する比率を用いれば、この式はe^{i \tau} = 1\,という、よりシンプルな表現になると述べられている。

4. shinichi says:

(sk)

数学を知る人の多くが、オイラーの等式を美しいと言う。キース・デブリン（Keith Devlin）は、オイラーの等式は「存在のとても深いところまで到達している（reaches down into the very depths of existence）」と言い、ポール・ネイヒン（Paul Nahin）は、オイラーの等式を「絶妙な美しさ（of exquisite beauty）」と説明する。

数学を知らない人は、オイラーの等式が美しいとは思わない。知識のあるなしが、これほどくっきりと別れるのもめずらしい。