Euler's identity Facts for Kids

Euler's identity, sometimes called Euler's equation, is this equation:

$e^{i\pi} + 1 = 0$

It features the following mathematical constants:

$\pi$ , pi

$\pi \approx 3.14159$
$e$ , Euler's Number

$e \approx 2.71828$
$i$ , imaginary unit

$i = \surd{-1}$

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
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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 $e ^{x}$ can be written as the Taylor series

$e ^{x} = 1 + x + {x^{2}\over{2!}} + {x^{3}\over{3!}} + {x^{4}\over{4!}} \cdots = \sum_{k=0}^\infty {x^{n}\over n!}$

As well, the sine function can be written as

$\sin{x} = x - {x^{3} \over 3!} + {x^5 \over 5!} - {x^{7} \over 7!} \cdots = \sum_{k=0}^\infty {(-1)^{n}\over (2n+1)!} {x^{2n+1}}$

and cosine as

$\cos{x} = 1 - {x^{2} \over 2!} + {x^4 \over 4!} - {x^{6} \over 6!} \cdots = \sum_{k=0}^\infty {(-1)^{n}\over (2n)!} {x^{2n}}$

Here, we see a pattern take form. $e^{x}$ 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 $e^{ix} = \cos(x) + i \sin(x)$ .

So, on the left side is $e^{ix}$ , whose Taylor series is $1 + ix - {x^{2} \over 2!} - {ix^{3} \over 3!} + {x^{4} \over 4!} + {ix^{5} \over 5!} \cdots$

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 $\cos(x) + i \sin(x)$ , whose Taylor series is the Taylor series of cosine, plus i times the Taylor series of sine, which can be shown as:

$( 1 - {x^{2} \over 2!} + {x^{4} \over 4!} \cdots) + (ix - {ix^{3} \over 3!} + {ix^{5} \over 5!}\cdots)$

if we add these together, we have

$1 + ix - {x^{2} \over 2!} - {ix^{3} \over 3!} + {x^{4} \over 4!} + {ix^{5} \over 5!} \cdots$

Therefore,

$e^{ix} = \cos(x) + i \sin(x)$

Now, if we replace x with $\pi$ , we have:

$e^{i\pi} = \cos(\pi) + i \sin(\pi)$

Since we know that $\cos(\pi) = -1$ and $\sin(\pi) = 0$ , we have:

$e^{i\pi} = -1$
$e^{i\pi} + 1 = 0$

which is the statement of Euler's identity.

pl:Wzór Eulera#Tożsamość Eulera

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In this animation $N$ takes various increasing values from 1 to 100. The computation of $(1 + iπ / N) N$ is displayed as the combined effect of $N$ repeated multiplications in the complex plane, with the final point being the actual value of $(1 + iπ / N) N$ . It can be seen that as $N$ gets larger $(1 + iπ / N) N$ approaches a limit of −1.

Euler's identity facts for kids

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Mathematical proof of Euler's Identity using Taylor Series

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