متطابقة أويلر
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In mathematics, Euler's identity[n 1] (also known as Euler's equation) is the equality
حيث
- e is Euler's number, the base of natural logarithms,
- i is the imaginary unit, which satisfies i2 = −1, and
- π is pi, the ratio of the circumference of a circle to its diameter.
Euler's identity is named after the Swiss mathematician Leonhard Euler. It is considered to be an example of mathematical beauty, perhaps a supreme example as it shows a profound connection between the most fundamental numbers in mathematics.
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شرح
Euler's identity is a special case of Euler's formula from complex analysis, which states that for any real number x,
where the inputs of the trigonometric functions sine and cosine are given in radians.
In particular, when x = π, or one half-turn (180°) around a circle:
حيث
و
مما يستتبع أن
التي تنتج متطابقة أويلر:
الجمال الرياضي
تشتهر متطابقة أويلر بشكل ملحوظ لجمالها الرياضي. [3] Three of the basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation. كما تجمع هذه المتطابقة بين خمس من أهم الثوابت الرياضية:
- الصفر.
- الواحد.
- ط أو π.
- هـ أو e.
- العدد التخيلي i, the imaginary unit of the complex numbers.
انظر أيضاً
ملاحظات
- ^ The term "Euler's identity" (or "Euler identity") is also used elsewhere to refer to other concepts, including the related general formula eix = cos x + i sin x,[1] and the Euler product formula.[2]
المراجع
- ^ Dunham, 1999, p. xxiv.
- ^ Stepanov, S. A. (7 February 2011). "Euler identity". Encyclopedia of Mathematics. Retrieved 18 February 2014.
- ^ Gallagher, James (13 February 2014). "Mathematics: Why the brain sees maths as beauty". BBC News Online. Retrieved 26 December 2017.
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المصادر
- Conway, John H., and Guy, Richard K. (1996). The Book of Numbers (Springer) ISBN 978-0-387-97993-9
- Crease, Robert P., "The greatest equations ever", Physics World, 10 May 2004 (registration required)
- Dunham, William (1999), Euler: The Master of Us All. Mathematical Association of America ISBN 978-0-88385-328-3
- Euler, Leonhard. Leonhardi Euleri opera omnia. 1, Opera mathematica. Volumen VIII, Leonhardi Euleri introductio in analysin infinitorum. Tomus primus (Leipzig: B. G. Teubneri, 1922)
- Kasner, E., and Newman, J., Mathematics and the Imagination (Simon & Schuster, 1940)
- Maor, Eli, e: The Story of a number (Princeton University Press, 1998) ISBN 0-691-05854-7
- Nahin, Paul J., Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills (Princeton University Press, 2006) ISBN 978-0-691-11822-2
- Paulos, John Allen, Beyond Numeracy: An Uncommon Dictionary of Mathematics (Penguin Books, 1992) ISBN 0-14-014574-5
- Reid, Constance, From Zero to Infinity (Mathematical Association of America, various editions)
- Sandifer, C. Edward. Euler's Greatest Hits (Mathematical Association of America, 2007) ISBN 978-0-88385-563-8
- Zeki, S.; Romaya, J. P.; Benincasa, D. M. T.; Atiyah, M. F. (2014), "The experience of mathematical beauty and its neural correlates", Frontiers in Human Neuroscience 8, doi: