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Euler's Identity as seen above is often described as the most beautiful equation in the world, where the most fundamental constants of mathematics come together to produce a simple equation.
The term e refers to the base of the natural logarithm, ln, of value 2.71828....
π, 3.14159… is the ratio of the circumference of the circle to its diameter.
The term i refers to the imaginary number of the square root of -1.
The proof for the equation is easily presented through the use of an argand diagram http://en.wikipedia.org/wiki/Complex_plane
The algebraic proof of the equation can also be achieved fairly easily with some prior knowledge on complex numbers, differentiation, Maclaurin's series and a little bit of algebraic manipulation.
Firstly, obtain the Maclaurin's series for ex. A Maclaurin's series is an expression of a function as a power series. It can be obtained by substituting the variable 'x' of each term of the function with 0, with the first term of the function as the function itself, and each subsequent term as a differentiated form of the previous term.
The Maclaurin's series for sinx and cosx should then be obtained. (Obtaining the first few terms for each series should suffice.)
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| List of several Maclaurin's expansion for various common functions. |
Next, replace the 'x' term of the ex series with 'ix' and we will soon realise that the real part of the solution corresponds to the Maclaurin's series for cosx while the imaginary part of the solution corresponds to that of sinx. Thus, replace the series for the real and imaginary parts with cosx and sinx respectively.
Thereafter, we would obtain the equation e^ix = cosx + isinx
e^ix = cosx + isinx
By replacing x with π, we will obtain e^iπ = -1. Hence e^iπ+1=0
Sources:
Proof: The pleasures of Pi,e and other interesting numbers (By Yeo Adrian)
Photos are obtained from:
1. http://en.wikipedia.org/wiki/Euler's_identity (Equations)
2. http://www.seab.gov.sg/aLevel/2014Syllabus/ListMF15.pdf (Standard series)
