Euler's Formula

Pronunciation: /ˈɔɪlərz ˈfɔrmyələ/ Explain

Click on the blue point and drag it to change the figure

Manipulative 1 - Euler's Formula Created with GeoGebra.

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. Paul J. Nahin. Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills. Princeton University Press. April 10, 2006. Buy the book
  2. 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. Last Accessed 8/6/2018. Buy the book

Cite this article as:

McAdams, David E. Euler's Formula. 7/10/2018. All Math Words Encyclopedia. Life is a Story Problem LLC.

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Revision History

7/5/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (McAdams, David E.)
1/29/2010: Added "References". (McAdams, David E.)
12/23/2008: Initial version. (McAdams, David E.)

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