Tivadar Danka Profile picture
May 11, 2021 β€’ 9 tweets β€’ 3 min read β€’ Read on X
There is a mathematical formula so beautiful that it is almost unbelievable.

Euler's identity combines the famous numbers 𝑒, 𝑖, Ο€, 0, and 1 in a single constellation. At first sight, most people doubt that it is true. Surprisingly, it is.

This is why.

🧡 πŸ‘‡πŸ½
Let's talk about the famous exponential function 𝑒ˣ first.

Have you ever thought about how is this calculated in practice? After all, raising an irrational number to any power is not trivial.

It turns out that the function can be written as an infinite sum!
In fact, this can be done with many other functions.

For those that are differentiable infinitely many times, there is a recipe to find the infinite sum form. This form is called the Taylor expansion.

It does not always yield the original function, but it works for 𝑒ˣ.
Taylor expansions are advantageous for two reasons.

First, we can approximate functions by cutting of the sum at some N.

Second, we can simply extend functions to the complex plane with this formula!
The exponential function is not the only one that can be written as a Taylor series.

We can also do this with the trigonometric functions sine and cosine.

(Feel free to check this by hand using the general Taylor expansion formula.)
By plugging in 𝑖𝑧 into the exponential function, we discover that the complex exponential function can be written in terms of trigonometric functions!

(We use that 𝑖² = -1.)
In the special case 𝑧 = Ο€, we obtain the famous formula called Euler's identity.

This is how the magic happens.
When asked, Euler's identity often comes up among mathematicians as the most beautiful formula ever.

It is not only amazing because it connects together a bunch of famous constants, but because it establishes a connection between the exponential and trigonometric functions.
If you enjoyed this explanation, consider following me and hitting a like/retweet on the first tweet of the thread!

I regularly post simple explanations of mathematical concepts in machine learning, make sure you don't miss out on the next one!

β€’ β€’ β€’

Missing some Tweet in this thread? You can try to force a refresh
γ€€

Keep Current with Tivadar Danka

Tivadar Danka Profile picture

Stay in touch and get notified when new unrolls are available from this author!

Read all threads

This Thread may be Removed Anytime!

PDF

Twitter may remove this content at anytime! Save it as PDF for later use!

Try unrolling a thread yourself!

how to unroll video
  1. Follow @ThreadReaderApp to mention us!

  2. From a Twitter thread mention us with a keyword "unroll"
@threadreaderapp unroll

Practice here first or read more on our help page!

More from @TivadarDanka

Oct 14
In machine learning, we use the dot product every day.

However, its definition is far from revealing. For instance, what does it have to do with similarity?

There is a beautiful geometric explanation behind: Image
By definition, the dot product (or inner product) of two vectors is defined by the sum of coordinate products. Image
To peek behind the curtain, there are three key properties that we have to understand.

First, the dot product is linear in both variables. This property is called bilinearity. Image
Read 15 tweets
Oct 13
Matrix factorizations are the pinnacle results of linear algebra.

From theory to applications, they are behind many theorems, algorithms, and methods. However, it is easy to get lost in the vast jungle of decompositions.

This is how to make sense of them. Image
We are going to study three matrix factorizations:

1. the LU decomposition,
2. the QR decomposition,
3. and the Singular Value Decomposition (SVD).

First, we'll take a look at LU.
1. The LU decomposition.

Let's start at the very beginning: linear equation systems.

Linear equations are surprisingly effective in modeling real-life phenomena: economic processes, biochemical systems, etc. Image
Read 18 tweets
Oct 11
Behold one of the mightiest tools in mathematics: the camel principle.

I am dead serious. Deep down, this tiny rule is the cog in many methods. Ones that you use every day.

Here is what it is, how it works, and why it is essential: Image
First, the story:

The old Arab passes away, leaving half of his fortune to his eldest son, third to his middle son, and ninth to his smallest.

Upon opening the stable, they realize that the old man had 17 camels. Image
This is a problem, as they cannot split 17 camels into 1/2, 1/3, and 1/9 without cutting some in half.

So, they turn to the wise neighbor for advice. Image
Read 18 tweets
Oct 9
Matrix multiplication is not easy to understand.

Even looking at the definition used to make me sweat, let alone trying to comprehend the pattern. Yet, there is a stunningly simple explanation behind it.

Let's pull back the curtain! Image
First, the raw definition.

This is how the product of A and B is given. Not the easiest (or most pleasant) to look at.

We are going to unwrap this. Image
Here is a quick visualization before the technical details.

The element in the i-th row and j-th column of AB is the dot product of A's i-th row and B's j-th column. Image
Read 16 tweets
Oct 8
Graph theory will seriously enhance your engineering skills.

Here's why you must be familiar with graphs: Image
What do the internet, your brain, the entire list of people you’ve ever met, and the city you live in have in common?

These are all radically different concepts, but they share a common trait.

They are all networks that establish relationships between objects. Image
As distinct as these things seem to be, they share common properties.

For example, the meaning of β€œdistance” is different for

β€’ Social networks
β€’ Physical networks
β€’ Information networks

But in all cases, there is a sense in which some objects are β€œclose” or β€œfar”. Image
Read 14 tweets
Oct 7
One of the coolest ideas in mathematics is the estimation of a shape's area by throwing random points at it.

Don't believe this works? Check out the animation below, where I show the method on the unit circle. (Whose area equals to Ο€.)

Here is what's behind the magic:
Let's make this method precise!

The first step is to enclose our shape S in a square.

You can imagine this as a rectangular dartboard. Image
Now, we select random points from the board and count how many hit the target.

Again, you can imagine this as closing your eyes, doing a 360Β° spin, then launching a dart.

(Suppose that you always hit the board. Yes, I know. But in math, reality doesn't limit imagination.) Image
Read 14 tweets

Did Thread Reader help you today?

Support us! We are indie developers!


This site is made by just two indie developers on a laptop doing marketing, support and development! Read more about the story.

Become a Premium Member ($3/month or $30/year) and get exclusive features!

Become Premium

Don't want to be a Premium member but still want to support us?

Make a small donation by buying us coffee ($5) or help with server cost ($10)

Donate via Paypal

Or Donate anonymously using crypto!

Ethereum

0xfe58350B80634f60Fa6Dc149a72b4DFbc17D341E copy

Bitcoin

3ATGMxNzCUFzxpMCHL5sWSt4DVtS8UqXpi copy

Thank you for your support!

Follow Us!

:(