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Euler's Identity - For the math/geometry lovers

IJesusChrist

Holofractale de l'hypervérité
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22 Juil 2008
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Hey guys - I've recently brought up Euler's identity back into my life. It is an equation that is fairly simple, yet extremely complex. If you haven't had calculus it may be hard to visualize how amazing this equation really is. But lets take a look, eh?

9e9a547076c6820b95e439dd1a5d6a32.png


So there it is. In all it's glory.

I'm making this post for those of us who don't have a background in college-level mathematics. So I'll go on to explain the equation, and why your jaw needs to drop when you see it:

First. None of these numbers are directly connected in how they were discovered - this is why it shocks so many people the first time they see it. Each of the numbers in the equation are some of the most important numbers in mathematics, yet seemingly unrelated, lets look at e first.

When bankers first came around they wanted a way to make interest on loans. One of the very first equations they used was:
3c61f664e4b9ae0ea85f89dff6b52548.png

Where,

* A = final amount
* P = principal amount (initial investment)
* r = annual nominal interest rate (as a decimal)

(it should not be in percentage)

* n = number of times the interest is compounded per year
* t = number of years

So if we look at the above equation again, we can simplify it down to the very basic essentials of the equation - the parts that are most important in finding out interest, in a more... philosophical means. The equation can be simplified down to

A = (1+1/n)^n

Where we have made P = 1, r = 1, t = 1. Keep your mind focused on this equation right here A = (1+1/n)^n. It is important to understand e.
The standard use of this equation is in finding how many times per year to add interest, and by what amount. The way bankers use it is they will take a year, and add interest to the debt n times per year. So lets say most bankers add interest to the account or debt once a month. That means n = 12, for 12 months in a year. But what does this do to the equation? Lets take a look at what happens when we change the number n, and the total number A, which would represent the total amount owed.

First we should take the equation apart, for those of us unfamiliar with this type of math;
1/n -> as n gets larger, this quantity gets smaller. As n approaches to infinity, the entire quantity, 1/n approaches zero. We can easily see this with 1/2, 1/3, 1/4, 1/8. Anyone who buys weed knows that 1/4 is bigger than 1/8. So, in this quantity it is safe to say, as n -> infinity, 1/n -> 0.

Fair enough, lets look at the exponential. For those of us unfamiliar with exponential numbers, lets look at 2^n.
2^1 = 2.
2^2 = 2 * 2 = 4
2^3 = 2 * 2 * 2 = 8
2^4 = 2 * 2 * 2 * 2 = 16
We can see that as n gets larger, the quantity of 2^n gets very large, very quick! So what do we see, as n -> infinity, 2^n -> infinity very quickly.

So, we have something confusing going on.
In the equation A = (1+1/n)^n,
as n -> infinity, we have;
1/n -> 0
(x)^n -> infinity.


These types of equations are very interesting to mathematicians. What actually happens as n approaches infinity in this equation, gives rise to a asymptote, which means that as n gets larger and larger it gets closer and closer to a certain number. Can you guess what that number is? it is e, or 2.71828... It is very intriguing that by plugging in two types of infinities into the equation, we come across a (somewhat) quantifiable number. But the exact number, much like pi, has an infinite number of non-repeating digits after the decimal. You can plug this equation into your calculator, if you have a decent one. Give n the quantity 99999 or so to approximate infinity.

What is the "big" picture of e, then? Well - lets see where it came from. Remember bankers used the equation
A = (1+1/n)^n to add compound interest (growth) to a certain amount by figuring out how many times to add interest to an account. To get the MOST money, and add the most interest, you would want to make n large, i.e. you would want to continually add interest. However, for ever increase in n, the gains obtained by interest get smaller. Meaning eventually it just wouldn't make sense, say, past n = 50. It would take too much time and accounting and the gains wouldn't be enough to keep adding more (now with computers it is very easy though). But, what if something was to continually grow? To not grow in increments, but rather... organically, naturally - like a tree? Well. Those things grow in proportion to "e". Nature doesn't grow in increments - it is ALWAYS growing. Hence, often "e" is called 'natural growth rate'.

PHEW! We finished e (that is the hardest one). Lets look at i.

"i" is very easy to explain, yet very hard to imagine. Mathematicians call it the imaginary number, because it has no place in our reality, yet is paramount to quantum mechanics and physics. The simplest way to imagine i is to look at it's occurence in equations.

7b7d0fac292ad47f7a7b75b6db518bfd.png


If you've never seen i before, this won't make much sense. But if you remember this rule:
2^2 = 4
(-2)^2 = 4
You see something important. You cannot get a negative number by squaring any number. Meaning there is no number that we can deal with in reality that gives a negative number. We deal with positive numbers all day, and often negative numbers - debt. We can square negative numbers or positive numbers, our brains do it all the time! But the numbers we end up with are always positive. So how then, can we get negative numbers from a square? Only if we use i.
i is equal to
1d1553468ab9a8e79faaa5e937bf7d05.png
There is not much else to explain it. It simply "is there". Some speculate it is the quantity to express a 4th spatial dimension...

Now, pi. Pi is simple. Pi is the ratio of the circumference to the radius of any circle, any sphere. It is a universal number in what is probably the most important shape in our universe - the lowest energy conformation of any bulk matter. You blow a bubble, it becomes a sphere, you spray water droplets, they become spheres, if you create fire in zero-gravity you obtain a sphere. And pi is always there.

Finally we arrive at the most fundamental of all numbers, 1 and 0. 1 being the identity that keeps all mathematics from falling apart, 0 being the absolute opposite. The ying and yang. All things are 1. Nothing is 0.

So - in conclusion we have developed an equation that relates natural growth, the imaginary number that is crucial to physics, the ratio of the most important geometries in our universe, and the fundamentals of existence and non-existence themselves. An incredible feat. Thank you for reading. I hope this inspires you all to learn some mafz.
 

IJesusChrist

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Nothing. It is simply a mathematical equation that was stumbled upon years ago that has no apparent meaning, but links the 5 (arguably) most important numbers in our universe.
 

IJesusChrist

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The way I learned it was e^pi*i = -1 , which has no 'cancellations'... :p you can look at the same math equation a thousand times and get a thousand approaches to its meaning.
 

IJesusChrist

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22 Juil 2008
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Einstein once said

"How can the human invention of math be so profound in nature as to predict nearly all action and reaction?"

Whether or not you believe math is invented or not, it has the capability to opening doors to new areas of though and realizations.
 
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