Euler's identity Facts for Kids

A visual representation of Euler's identity.

Euler's identity, also known as Euler's equation, is a very famous and beautiful equation in mathematics. It looks like this:

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

This equation brings together five of the most important numbers in mathematics:

$\pi$ (pi): This number is about circles. It's the ratio of a circle's distance around (circumference) to its distance across (diameter). Pi is approximately 3.14159.
$e$ (Euler's Number): This special number is important in areas like growth and decay, and it's used in many science and engineering calculations. It's approximately 2.71828.
$i$ (imaginary unit): This is a special number that helps us work with square roots of negative numbers. By definition, $i = \surd{-1}$ .

The identity also uses three basic math actions: adding, multiplying, and raising to a power (exponentiation).

Euler's identity is named after Leonard Euler, a brilliant mathematician from Switzerland. It's not fully clear if he was the first to discover it.

Many mathematicians and scientists think this identity is incredibly important. In a poll by Physics World magazine, people called it "the most profound mathematical statement ever written" and "filled with cosmic beauty."

How to Understand Euler's Identity
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How to Understand Euler's Identity

You can show that Euler's identity is true using something called a Taylor series. Imagine breaking down a complicated math function into a long list of simpler additions. This is what a Taylor series does.

For example, the exponential function $e ^{x}$ can be written as an endless sum:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): e ^{x} = 1 + x + {x^{2}\over{2!}} + {x^{3}\over{3!}} + {x^{4}\over{4!}} \cdots

The sine function Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sin{x} can also be written as a series:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sin{x} = x - {x^{3} \over 3!} + {x^5 \over 5!} - {x^{7} \over 7!} \cdots

And the cosine function Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \cos{x} can be written as:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \cos{x} = 1 - {x^{2} \over 2!} + {x^4 \over 4!} - {x^{6} \over 6!} \cdots

Now, let's look at a related formula called Euler's formula: $e^{ix} = \cos(x) + i \sin(x)$ . This formula connects the exponential function with sine and cosine.

If we put ix into the series for $e^x$ , we get:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): e^{ix} = 1 + ix + {(ix)^{2}\over{2!}} + {(ix)^{3}\over{3!}} + {(ix)^{4}\over{4!}} \cdots

Since $i^2 = -1$ , $i^3 = -i$ , $i^4 = 1$ , and so on, this series becomes:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): e^{ix} = 1 + ix - {x^{2} \over 2!} - {ix^{3} \over 3!} + {x^{4} \over 4!} + {ix^{5} \over 5!} \cdots

Now, let's look at the series for $\cos(x) + i \sin(x)$ :

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \cos(x) + i \sin(x) = ( 1 - {x^{2} \over 2!} + {x^{4} \over 4!} \cdots) + i(x - {x^{3} \over 3!} + {x^{5} \over 5!}\cdots)

If we combine these terms, we get:

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

You can see that the series for $e^{ix}$ is exactly the same as the series for $\cos(x) + i \sin(x)$ ! This proves that:

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

To get Euler's identity, we just replace x with $\pi$ (pi) in this formula:

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

We know that Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \cos(\pi) (cosine of 180 degrees) is -1, and Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sin(\pi) (sine of 180 degrees) is 0. So, the equation becomes:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): e^{i\pi} = -1 + i(0)
$e^{i\pi} = -1$

Finally, if we add 1 to both sides, we get Euler's identity:

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

Images for kids

This animation shows how the value of Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): (1 + {{sfrac|''iπ''|''N''}})''N'' gets closer and closer to -1 as N gets larger.

Euler's identity facts for kids

Contents

How to Understand Euler's Identity

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