Euler's Formula

Pronunciation: /ˈɔɪlərz ˈfɔrmyələ/ ?

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Euler's formula is e^i*theta}=cos(theta)+i*sin(theta). [1] It is remarkable in that it relates trigonometric function functions with complex numbers. Euler's formula maps the natural constant e raised to the complex exponent i^theta to the complex plane using the trigonometric expression cos(theta)+i*sin(theta). This equation is true only if the angle θ is measured in radians.

Click on the blue point in manipulative 1 and drag it to change the figure. The axes in manipulative 1 are the real axis and imaginary axis of the complex plane.

Euler's formula is named after Leonhard Euler who published it in its present form in 1748.

Leonard Euler was a very prolific mathematician. Note that the phrase Euler's Formula is used to indicate different formulas in different disciplines.


  1. Euler's formula. Encyclopedia Britannica. (Accessed: 2009-03-12).\'s formula.
  2. Paul J. Nahin. Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills. Princeton University Press, April 10, 2006.
  3. Fine, Henry B., Ph. D.. Number-System of Algebra Treated Theoretically and Historically, 2nd edition, pg 30. D. C. Heath & Co., Boston, U.S.A., 1907. (Accessed: 2009-12-19).

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Cite this article as:

Euler's Formula. 2010-01-29. All Math Words Encyclopedia. Life is a Story Problem LLC.


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Revision History

2010-01-29: Added "References" (McAdams, David.)
2008-12-23: Initial version (McAdams, David.)

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