Euler’s identity

Here’s some stuff about math I never knew until today.
First, a few things I did know, tidbits most of you probably knew as well.

  • e, the base of the natural logarithm, = 2.7182818…
  • i is the “imaginary unit,” such that i2 = -1
  • The value of π, the artio of a circle’s circumference to its diameter, is 3.141592…

Are these numbers related? On the surface, there’s no reason why thy should be. Yet the following relationship has been proven:


e = -1.
Neat, huh? This is Euler’s identity and is commonly regarded as one of the greatest…equations…evah.
Its derivation — and Euler himself probably wasn’t the first to perform it — is straightforward:
eix = cos x + i sin x
This is true for any real number x. If x = π, then
e = cos π + i sin π
Since cos π = -1 and sin π = 0, then
e = -1.
So memorize that one and spring it on your next Match.com or Yahoo! Personals blind date. Won’t he or she be impressed!

7 thoughts on “Euler’s identity”

  1. If you re-express the equation as:
    e + 1 = 0
    then it includes 5 (rather than 4) of the most fundamental mathematical constants. Is that artistically more satisfying?

  2. I learned about Euler’s formula in complex analysis. I learned about it in some earlier reading material I needed to assimilate for 3D graphics/rendering, but it was treated as little more than a tautology. Which is a shame, since many of the results that Euler obtained through it are very interesting.

  3. You can also use Euler’s formula to make sense out of trigonometry.

    And by extension it is the basis for making sense out of Fourier analysis and related transforms.
    Both beautiful and powerful – not much can resist that. :-P

  4. You can also use Euler’s formula to make sense out of trigonometry.

    And by extension it is the basis for making sense out of Fourier analysis and related transforms.
    Both beautiful and powerful – not much can resist that. :-P

  5. Erm, just noticed this, but the first sentence in my above post should read “I learned about Euler’s formula fully in complex analysis.” Doesn’t really make much sense as written…
    That is all.

  6. I don’t use Euler’s formula but I use all of the “parts” you have mentioned in my circuits classes.
    It should be noted that we electrical engineers do not use “i”, we use “j”. This is because we use “i” to stand for current (as in “intensity”, “c” is used for capacitance). Our freshman programming course uses the Python language. Besides the numerous references available (to Monty), it has the unique character of native support for complex numbers using the j operator (instead of i).

Comments are closed.

Euler’s identity

Here’s some stuff about math I never knew until today.
First, a few things I did know, tidbits most of you probably knew as well.

  • e, the base of the natural logarithm, = 2.7182818…
  • i is the “imaginary unit,” such that i2 = -1
  • The value of π, the artio of a circle’s circumference to its diameter, is 3.141592…

Are these numbers related? On the surface, there’s no reason why thy should be. Yet the following relationship has been proven:

(more…)

One thought on “Euler’s identity”

  1. You can also use Euler’s formula to make sense out of trigonometry.

    And by extension it is the basis for making sense out of Fourier analysis and related transforms.
    Both beautiful and powerful – not much can resist that. :-P

Comments are closed.