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!
🧵 👇🏽
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!
🧵 👇🏽
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.
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?
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.
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.
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.
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.)
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.
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.
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.
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!
I regularly post simple explanations of seemingly complicated concepts in machine learning, make sure you don't miss out on the next one!
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