Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

What is Euler's identity really saying?

  1. May 4, 2012 #1
    So it is true that ei∏+1=0. But what does this mean? Why are all these numbers linked?
  2. jcsd
  3. May 4, 2012 #2

    They are linked precisely by that equation, and since the equality [itex]e^{i\theta}:=\cos\theta+i\sin\theta\,\,,\,\,\theta\in\mathbb{R}\,\,[/itex] follows at once say from the definition

    of the complex exponential function as power series (or as limit of a sequence), the above identity is really trivial.

  4. May 4, 2012 #3


    User Avatar
    Science Advisor

    Look at the MacLaurin series for those functions:
    [tex]e^x= 1+ x+ x^2/2!+ x^3/3!+ \cdot\cdot\cdot+ x^n/n![/tex]
    [tex]cos(x)= 1- x^2/2!+ x^4/4!- x^6/6!+ \cdot\cdot\cdot+ (-1)^nx^{2n}/(2n)![/tex]
    [tex]sin(x)= x- x^3/3!+ x^5/5!- x^7/7!+ \cdot\cdot\cdot+ (-1)^nx^{2n+1}/(2n)![/tex]

    If you replace x with the imaginary number ix (x is still real) that becomes
    [tex]e^{ix}= 1+ ix+ (ix)^2/2!+ (ix)^3/3!+ \cdot\cdot\cdot+ (ix)^n/n![/tex]
    [tex]e^{ix}= 1+ ix+ i^2x^2/2!+ i^3x^3/3!+ \cdot\cdot\cdot+ i^nx^n/n![/tex]

    But it is easy to see that, since [itex]i^2= -1[/itex], [itex](i)^3= (i)^2(i)= -i[/itex], [itex](i)^4= (i^3)(i)= -i(i)= -(-1)= 1[/itex] so then it starts all over: [itex]i^5= (i^5)i= i[/itex], etc. That is, all even powers of i are 1 if the power is 0 mod 4 and -1 if it is 2 mod 4. All odd powers are i if the power is 1 mod 4 and -i if it is 3 mod 4.

    [tex]e^{ix}= 1+ ix- x^2/2!- ix^3/3!+ \cdot\cdot\cdot[/tex]

    Separating into real and imaginary parts,
    [tex]e^{ix}= (1- x^2/2!+ x^4/4!- x^6/6!+ \cdot\cdot\cdot)+ i(x- x^3/3!+ x^5/5!+ \cdot\cdot\cdot)[/tex]
    [tex]e^{ix}= cos(x)+ i sin(x)[/tex]

    Now, take [itex]= \pi[/itex] so that [itex]cos(x)= cos(\pi)= -1[/itex] and [itex]sin(x)= sin(\pi)= 0[/itex] and that becomes
    [tex]e^{i\pi}= -1[/tex]
    [tex]e^{i\pi}+ 1= 0[/tex]

    I hope that is what you are looking for. Otherwise, what you are asking is uncomfortably close to "number mysticism".
    Last edited by a moderator: May 4, 2012
  5. May 5, 2012 #4
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - Euler's identity really Date
I Intuitive understanding of Euler's identity? Thursday at 4:52 AM
I Intuition for Euler's identity Nov 1, 2016
I Can Euler's Identity be used? Oct 18, 2016
B Implications of e^i*pi = -1 Aug 17, 2016
Problem in apparent contradiction in Euler's Identity? Dec 25, 2014