Creative abuse of rules can lead to game-changing discoveries.
In high school, you learned that -1 has no square roots. Yet, by ignoring this, you'll soon discover something that changed mathematics forever: complex numbers.
Follow along, and you'll see how!
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Let's start with a very simple equation:
𝑥² + 1 = 0
Can we solve this? Not at first glance, since the left side of the equation is always larger than one. This is equivalent to solving
𝑥² = -1,
which is (apparently) not possible.
But let's disregard this and imagine a number whose square is -1.
Let's appropriately name it the 𝑖𝑚𝑎𝑔𝑖𝑛𝑎𝑟𝑦 𝑛𝑢𝑚𝑏𝑒𝑟 and denote it with 𝑖.
So, 𝑖² = -1.
Now that we have this strange entity, what can we do?
Can we add or multiply 𝑖 with real numbers?
Sure, why not! We can compose a new number by taking the linear combination of 𝑖 with real numbers. So, we can form new ones by
𝑧 = 𝑎 + 𝑏𝑖,
where 𝑎 and 𝑏 are real numbers. Let's call 𝑎 the real part, 𝑏 the imaginary part.
We can perform addition and multiplication with these composite numbers by following elemental algebraic rules.
Let's name our creation the 𝑐𝑜𝑚𝑝𝑙𝑒𝑥 𝑛𝑢𝑚𝑏𝑒𝑟𝑠.
In the literal sense, there is nothing imaginary about complex numbers. The definition is, although quite mysterious, perfectly valid.
Also, note that the set of real numbers is a subset of complex numbers.
To make reasoning about complex numbers easier, we can represent them as vectors in the Cartesian plane.
Every 𝑧 = 𝑎 + 𝑏𝑖 can be represented as the (𝑎, 𝑏) vector.
The x and y axes are called real and imaginary axes.
Why are complex numbers important?
Now, not only 𝑥² + 1 = 0 has solutions, but every nontrivial polynomial equation with complex coefficients has at least one solution in the set of complex numbers.
(Polynomial equations are the ones like below.)
Believe it or not, this seemingly innocent fact makes complex numbers an ideal mathematical structure for many purposes.
For instance, complex matrices always have eigenvalues, which are not true for their real counterparts. This plays a significant role in linear algebra.
Without complex numbers, the Fourier transform wouldn't exist either.
We would be unable to adequately study certain mechanical systems. Even simple ones like pendulums.
As I mentioned in the beginning, creative abuse of rules can lead to game-changing discoveries.
Complex numbers arose when someone dared to challenge long-standing views and went outside the box.
Complex numbers changed science forever. The rest is history.
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I regularly post simple explanations of seemingly complicated concepts in machine learning, make sure you don't miss out on the next one!
What you see below is a 2D representation of the MNIST dataset.
It was produced by t-SNE, a completely unsupervised algorithm. The labels were unknown to it, yet it almost perfectly separates the classes. The result is amazing.
This is how the magic is done!
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Even though real-life datasets can have several thousand features, often the data itself lies on a lower-dimensional manifold.
Dimensionality reduction aims to find these manifolds to simplify data processing down the line.
So, we have data points 𝑥ᵢ in a high-dimensional space, looking for lower dimensional representations 𝑦ᵢ.
We want the 𝑦ᵢ-s to preserve as many properties of the original as possible.
For instance, if 𝑥ᵢ is close to 𝑥ⱼ, we want 𝑦ᵢ to be close to 𝑦ⱼ as well.
There is a mathematical formula so beautiful that it is almost unbelievable.
Euler's identity combines the famous numbers 𝑒, 𝑖, π, 0, and 1 in a single constellation. At first sight, most people doubt that it is true. Surprisingly, it is.
This is why.
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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.
One of the biggest misconceptions regarding education is that its main purpose is to give knowledge you can immediately use.
It is not.
The best thing education can give you is the mental agility to obtain knowledge at the speed of light.
Let's unpack this idea a bit!
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Consider a course where you build a custom neural network framework with NumPy.
This is hardly usable in practice: working with a custom library is insane.
However, if you know how they are built, you only need to learn the interface to master an actual framework!
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By understanding how the framework is built and how the underlying algorithms work, you'll be able to do much more: experiment with custom optimizers, implement your own layers, etc.
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