Euler’s Identity

21.06.2015 |



Episode #10 of the course “Most important numbers in the world”

Described as a kind of mathematical poem or art, Euler’s identity, which is also known as Euler’s equation, is named for the 18th-century mathematician Leonhard Euler. However, it is not known if Euler actually discovered the simple equation; although he published it, the equation may have been the work of someone else. Regardless, Euler’s identity is known for its beauty because each of the basic mathematical operations—addition, multiplication, and exponentiation —each occur exactly once. Because two of the numbers are transcendental, they are anomalies in math and exhibit properties not commonly found or explained. Without repeating their decimal places, both e and π (pi) explain patterns of geometric shapes—measurements of changes in growth and curved points in space.

Euler’s equation combines a number of known and commonly used mathematical symbols and concepts in a theoretical puzzle that describes no understood practical concept. A 19th-century Harvard University professor, Benjamin Peirce, once stated that the identity is “paradoxical”—because mathematicians have proven it, it must be true, yet no one knows exactly what it means. The identity has remained popular throughout time, being voted on as one of the most beautiful or perfect mathematical equations. It has become something referenced in popular culture as well.

The concepts used in Euler’s identity are common for other mathematical functions:

e= -1

e is the base for the natural logarithm, Euler’s Constant, or approximately 2.71828

i is the “imaginary” number, √-1

π (pi) is the ratio of a circle’s circumference to its diameter, or approximately 3.14159

Although the numbers seem to have little in common, Euler’s equation shows that if imaginary numbers could be graphed in space along with real numbers, the relationship between these transcendental numbers would bring together several formulas and mathematical concepts in a unique way.

Euler’s identity has been extrapolated to be utilized generally to quaternion exponentiation, octonions, calculus, and other fields of advanced mathematics.


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