Post

How to get URL link on X (Twitter) App

On the Twitter thread, click on or icon on the bottom
Click again on or Share Via icon
Click on Copy Link to Tweet
Paste it above and click "Unroll Thread"!
More info at Twitter Help

Tivadar Danka

Nov 3, 2022 • 18 tweets • 6 min read • Read on X

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.

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.

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.

The wise man says "hold my camel", and solves the problem by lending one to the boys.

Now the stable has 18. The eldest son takes 9 home, while the middle and smallest son leaves with 6 and 2, as their father wished.

The wise man takes his camel back, and everybody is happy.

Thus, the camel principle is born: adding and subtracting the same quantity doesn't change the equality, but can help in the computation.

In mathematics, you cannot live without this principle.

I'll show you two examples.

The first one is the quadratic equation.

Its solution formula is one of the few things that everybody remembers from high school. Even if they are woken up in the middle of the night.

This formula is derived from the camel principle. Let me show you how!

After factoring out 𝑎 from the equation, we notice that the famous identity

(α + β)² = α² + 2αβ + β²

might help to factor the quadratic equation into a product.

To achieve that, we apply the camel principle!

After adding and subtracting the same quantity, the terms with 𝑥 factor into a product.

This leads straight to the solution formula.

There is an alternative version of the camel principle, performing a similar feat: multiplying and dividing with the same quantity.

This doesn't change the equality either.

To illustrate, let's look at derivatives, the main engine behind mathematics, physics, and optimization.

(And tons of other fields that allowed technology to get where it is now.)

How would you calculate the derivative of a composite function?

This is a quintessential question. Without this, you don't have backpropagation, gradient descent, and thus neural networks.

(At least until someone invents a clever alternative. But that'll take a while.)

You guessed right: the camel principle!

(At least, the second version, where you multiply and divide with the same quantity.)

After the camel principle is applied, the limit can be carried out termwise.

(For those with the eagle's eyes: yes, the denominator can be zero. You can epsilon-delta your way out of that, but I won't do it here.)

And thus, we have the chain rule, one massive pillar of science and technology.

This is what we use to perform backpropagation, enabling us to train our neural networks in a reasonable time.

The lesson here: tiny mathematical curios such as the camel principle are often dismissed as "lacking any applications".

However, such short-sightedness frequently leads astray.

By understanding atoms, you are able to build skyscrapers.

https://twitter.com/TivadarDanka/status/1588131890040434688

If you have enjoyed this explanation, share it with your friends and give me a follow! I regularly post deep-dive explainers such as this.

Understanding mathematics will make you a better engineer, and I want to help you with that.

https://twitter.com/TivadarDanka/status/1588131890040434688

• • •

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

This Thread may be Removed Anytime!

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

More from @TivadarDanka

Tivadar Danka

@TivadarDanka

Dec 4, 2023

Understanding graph theory will seriously enhance your engineering skills; you must absolutely be familiar with them.

Here's a graph theory quickstart, in collaboration with Alejandro Piad Morffis.

Read on:

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.

As distinct as these things seem to be, they share common properties.

For example, the meaning of “distance” is different for

• physical networks,
• information netorks,
• orf social networks,

but in all cases, there is a sense in which some objects are “close” or “far”.

Read 15 tweets

Tivadar Danka

@TivadarDanka

Sep 13, 2023

Neural networks are stunningly powerful.

This is old news: deep learning is state-of-the-art in many fields, like computer vision and natural language processing. (But not everywhere.)

Why are neural networks so effective? I'll explain.

First, let's formulate the classical supervised learning task!

Suppose that we have a dataset D, where xₖ is a data point and yₖ is the ground truth.

The task is simply to find a function g(x) for which

• g(xₖ) is approximately yₖ,
• and g(x) is computationally feasible.

To achieve this, we fix a parametrized family of functions. For instance, linear regression uses this function family:

Read 19 tweets

Tivadar Danka

@TivadarDanka

Sep 12, 2023

A question we never ask:

"How large that number in the Law of Large Numbers is?"

Sometimes, a thousand samples are large enough. Sometimes, even ten million samples fall short.

How do we know? I'll explain.

First things first: the law of large numbers (LLN).

Roughly speaking, it states that the averages of independent, identically distributed samples converge to the expected value, given that the number of samples grows to infinity.

We are going to dig deeper.

There are two kinds of LLN-s: weak and strong.

The weak law makes a probabilistic statement about the sample averages: it implies that the probability of "the sample average falling farther from the expected value than ε" goes to zero for any ε.

Let's unpack this.

Read 15 tweets

Tivadar Danka

@TivadarDanka

Aug 24, 2023

With the power of mathematical induction, I'll prove that everyone has the same eye color.

Don't believe me? Read on.

(And see if you can spot the sleight of hand.)

To formalize the problem, define the proposition Aₙ by

Aₙ = "in a set of n people, everyone has the same eye color".

If n equals the human population of planet Earth, we get the original statement. We'll prove that Aₙ is true via induction.

Proof by induction works like climbing an infinite staircase.

First, we'll show A₁. Then, we'll show that if Aₙ is true, then Aₙ₊₁ is true as well.

This way, Aₙ is true for any positive integer via the chain of implications

A₁ → A₂ → ... → Aₙ.

Read 13 tweets

Tivadar Danka

@TivadarDanka

Aug 21, 2023

The single biggest argument about statistics: is probability frequentist or Bayesian? It's neither, and I'll explain why.

Buckle up. Deep-dive explanation incoming.

First, let's look at what is probability.

Probability quantitatively measures the likelihood of events, like rolling six with a dice. It's a number between zero and one. This is independent of interpretation; it’s a rule set in stone.

In the language of probability theory, the events are formalized by sets within an event space.

(The event space is also a set, usually denoted by Ω.)

Read 34 tweets

Tivadar Danka

@TivadarDanka

Aug 8, 2023

The Japanese multiplication method makes everybody feel "I wish they taught math like this in school."

It's not just a cute visual tool: it illuminates how and why long multiplication works.

Here is the full story.

First, the Japanese multiplication method.

The first operand (21 in our case) is represented by two groups of lines: two lines in the first (1st digit), and one in the second (2nd digit).

One group for each digit.