Euler Identity Explained in 5 Minutes

Euler Identity Explained in 5 Minutes

Imagine calculating this..

(1+(1/n))^n

where n is an integer n=1,2,3,..

You would notice something interesting. As n gets bigger and bigger the value seems to approach a specific number. It does, as n approaches infinity the value of this expression approaches the number e. It's a famous mathematical constant..

e=2.718281828459045..

So mathematician say that..

e=limit as n goes to infinity of (1+(1/n))^n

The number e shows up in many areas of math and physics and is probably as famous as pi. Like pi it is also an irrational number and the decimal places go on forever.

e can also be written in other ways. For example, here's e as an infinite series..

e=1+1/1+1/(1*2)+1/(1*2*3)+1/(1*2*3*4)+...

e is sometimes called Euler's number after the Swiss mathematician Leonhard Euler and the letter e was chosen in his honor.

e can be used to define a very useful function, the "exponential function"..

f(x)=e^x

and this function can be written as an infinite series which is a generalization of the series for e..

e^x=1+x/1+(x^2)/(1*2)+(x^3)/(1*2*3)+(x^4)/(1*2*3*4)...

This series can also be used to discover many interesting things about f(x), for example, f(x) has the unique property that its derivative is equal to itself, meaning..

df(x)/dx=f(x)

Another property is the famous Euler identity..

If you had to pick the three most famous numbers in math you might choose pi, e and i, where i is the imaginary number i.

You would surely believe these fundamental numbers are independent and not related in any way. You would be wrong, because the Euler identity gives this elegant result..

e^(pi*i)=-1

Then, if you write this identity as e^(pi*i)+1=0 you have a mysterious relationship between the 5 most famous numbers in mathematics!

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Content written and posted by Ken Abbott abbottsystems@gmail.com