Euler's identity facts for kids
Euler's identity, sometimes called Euler's equation, is this equation:
It features the following mathematical constants:
It also features three of the basic mathematical operations: addition, multiplication and exponentiation.
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.
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
Now, if we replace x with , we have:
Since we know that and , we have:
which is the statement of Euler's identity.
pl:Wzór Eulera#Tożsamość Eulera
Images for kids
In this animation N takes various increasing values from 1 to 100. The computation of (1 + iπ)N is displayed as the combined effect of N repeated multiplications in the complex plane, with the final point being the actual value of (1 + iπ)N. It can be seen that as N gets larger (1 + iπ)N approaches a limit of −1.
Euler's identity Facts for Kids. Kiddle Encyclopedia.