Euler's identity
Euler's identity, sometimes called Euler's equation, is this equation:
It features the following mathematical constants:
, pi
, Euler's Number
, imaginary unit
It also features three of the basic mathematical operations: addition, multiplication and exponentiation.
Euler's identity is named after the Swiss mathematician Leonard Euler. It is not clear that he invented it himself.
Respondents to a Physics World poll called the identity "the most profound mathematical statement ever written", "uncanny and sublime", "filled with cosmic beauty" and "mind-blowing".
Mathematical proof of Euler's Identity using Taylor Series
Many equations can be written as a series of terms added together. This is called a Taylor series.
The exponential function can be written as the Taylor series
As well, the sine function can be written as
and cosine as
Here, we see a pattern take form. seems to be a sum of sine and cosine's Taylor series, except with all of the signs changed to positive. The identity we are actually proving is .
So, on the left side is , whose Taylor series is
We can see a pattern here, that every second term is i times sine's terms, and that the other terms are cosine's terms.
On the right side is , whose Taylor series is the Taylor series of cosine, plus i times the Taylor series of sine, which can be shown as:
if we add these together, we have
Therefore,
Now, if we replace x with , we have:
Since we know that and , we have:
which is the statement of Euler's identity.
Related pages
-1 (number)
Euler's formula
References
Complex analysis
pl:Wzor Eulera#Tozsamosc Eulera