Often described as the most beautiful equation in mathematics, Euler’s identity is an identity usually taught in high school mathematics and is as shown above. Having an interest in mathematics, I also came across this identity some time in high school and was amazing by the way in which it relates some of the most fundamental constants in mathematics. However, what is often not talked about especially at school is where this identity comes from, which is why I wish to outline a simple derivation in this article.
We begin begin by considering the function eˣ and it’s Maclaurin series. If you are not quite familiar with the Maclaurin series, this video by 3Blue1Brown has a great explanation for the Taylor series which is essentially a generalized version of the Maclaurin series. A Maclaurin series is just a Taylor series centered at x = 0.
With this in mind, we can now express eˣ as the following infinite series:
Now, what we can do with this is let x = iθ where i is the imaginary number and θ is just a variable. Substituting this in and using the fact that i² = -1, we can simplify as shown below:
Notice how in the last step I separated the real and and imaginary parts of the infinite series to create two separate infinite series. At this point you may recognize that these two infinite series are the Maclaurin series for cos(θ) and sin(θ) respectively which can now be substituted to produce the following equation known as Euler’s formula:
This formula is one that is used quite often in high school mathematics (at least in IB Math AA HL), and I find it interesting to see how it can be derived using the Maclaurin series. Finally, to reach Euler’s identity all that remains is to let theta equal to π as shown below:
