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If someone hands you a blank board representing the complex numbers, and offers to tell you either the sum or the product of any two places you put your fingers, you can work out most of the board rather quickly. There remains which way to flip the board, which way is up? +i and -i both square to -1.
This symmetry is the camel's nose under the tent of Galois theory, described in 1831 by Évariste Galois before he died in a duel at age twenty. This is one of the most amazing confluences of ideas in mathematics. It for example explains why we have the quadratic formula, and formulas solving degree 3 and 4 polynomials, but no general formula for degree 5. The symmetry of the complex plane is a toggle switch which corresponds to a square root. The symmetries of degree 3 and 4 polynomials are more involved, but can all be again translated to various square roots, cube roots... Degree 5 can exhibit an alien group of symmetries that defies such a translation.
The Greeks couldn't trisect an angle using a ruler and compass. Turns out the quantity they needed exists, but couldn't be described in their notation.
Integrating a bell curve from statistics doesn't have a closed form in the notation we study in calculus, but the function exists. Statisticians just said "oh, that function" and gave it a new name.
Roots of a degree 5 polynomial exist, but again can't be described in the primitive notation of square roots, cube roots... One needs to make peace with the new "simple group" that Galois found.
This is arguably the most mind blowing thing one learns in an undergraduate math education.
It really doesn't. The imaginary axis is symmetrical and it makes sense to make it horizontal - there's no intrinsic physical difference between left and right. But it makes sense to map the real axis +ve/-ve onto (gravitational) up/down. That's why the Mandelbrot set looks so much more beautiful with that orientation (as it's displayed at https://fractal.institute/introduction-to-fractals/how-to-ge...) - it's a Buddha!
I find that the easiest intuitive on-ramp to complex arithmetic is to start with compass headings: "Oh that nice coffee shop? Go two blocks north and then a block east." Numbers come with any direction on the compass, not just "east" and "west". It turns out that it is pretty easy to intuitively justify multiplication by a scalar and addition of complex numbers, but multiplication is harder. A great way to get a feel for multiplication is to consider the equation "(x+1)(x-1) = x*2 - 1". Then, substitute "i" for x. The left-hand side is (intuitively) on a circle of radius 2 centered at the origin, and the right-hand side is on a circle of radius 1, where the circle is shifted horizontally so that its center is on the real line at -1. There's only one place these two circles meet: -2 on the real line.
1. Bombelli built "an engine", but then suddenly it's temporarily "a machete" to cut through the jungle, instead of an engine to go up slopes, or even an engine to power a bulldozer.
2. The real-numbers are an "oil well", and somehow it's important to inform the reader that its depths "belongs to the country because it is measured by the country's own rulers." Does this mean it "belongs" because of a fiat declaration by unnamed decision-makers that are never mentioned ever again? Or does that mean handheld rulers, which... are fixed integer markings? For rationals? What?
As I was looking for more examples to add, I noticed this at the end, which might explain it:
> This article was written with the assistance of Gemini 3 and Claude Opus 4.5.
________________________
Skimming anew with heightened suspicion, I found an algebra error, which is kinda-amusing in a piece about math that is substantially more complex--in both senses, hah.
Specifically, these parts:
> The race begins in 1572 [with Bombelli] [...] And between 1799 and 1831, three men [...] finally drew the picture.
> [Bombelli] encoded rotation 176 years before anyone drew the picture.
1572 to 1799-1831 is a range of 227-259 years, not 176.
That said, 176 does match the time in-between Bombelli and Euler, however the shifting metaphors don't match, because Euler never "drew a picture."
Instead, Euler *checks notes* re-engineered the DNA of imaginary numbers so that they could be citizens in the land of mathematics. (Talk about a dystopian immigration policy!)
Notably, that "land" is separate from the other "land" made out of rational-numbers with rulers that own deep oil-wells made of of real-numbers etc.
I posted it 4 days ago here but it got zero attention. Surprisingly, the recovery process from Yesterday's outage of HN, had it reposted, and I was surprised to see it this morning in the front page.
here is the story of this post
I have just started studying math in October, and my first course is linear algebra.
I have read too many introductions to complex numbers that follow the same script:
"Mathematicians needed to solve x^2 = -1, so they invented i, and despite calling it imaginary, it turned out to be useful..."
Then comes the complex plane, and everyone nods along, pretending they understand why we’re drawing circles when we started with algebra.
I never bought it. Something felt wrong.
So, last week I took a break from my lecture/recitation routines to write down everything I know about the topic, fill in the gaps, and search for the real answers.
While I was working with the LLM to answer the questions to myself, at the end of the day, it felt like sharing it might be beneficial, so that took another two days of me fighting the LLM to control it in place and have it focused on the historical facts and chronological order of events.
When my search led to Cardano's actual book, and pages in discussion, I was so thrilled, naively thinking others will find it useful as well. Apparently, everyone want to start an "AI-STARTUP", but refusing to get involved even in reading if AI was involved in the process.
I am open and clear about the use of AI and had no intention of claiming "discoveries" whatsoever.
This is in fact my first "math" related post I put out online, and I get the criticism with open arms, as long as they related to the math and history facts (and there are issues spotted which I may take the time to correct).
The Oil Well analogy (and other spicy terms) is not an AI's but mine, see, at a certain point, I was drinking coffee in my balcony, here in Abu Dhabi, over looking the sandy horizons, and while thinking about a discovery of new layer of numbers, the association with the Oil wells was inevitable.
here is a comment I have written by hand, no AI/LLM involved whatsoever.
thank you for reading.
It was common knowledge you had to work with filtering and intermediate frequencies because negative frequencies would be reflected along the 0 Hz axis and overlaid on the positive frequencies but mirrored. With complex numbers you suddenly were able to downconvert signals directly to the baseband and keep negative frequencies separate.
My son asked me:
“what’s the derivative of e^x”?
I replied “it is e^x?”, inquisitively as it had been a while. Once he confirmed, I started envisioning circles and vectors moving and said “oh, that is another reason why Euler’s formula is true”.
I’m just a nerd, not a mathematician, so I didn’t completely grok my flash of insight. But simple enough to ask an LLM, and Claude built the system of differential equations showing it:
https://claude.ai/share/33033109-a0fa-4c7b-822d-ca897c442cf2
The rotation pops out from the binomial multiplication of real + imaginary pairs: (a + ib) (c + id).
And the way that is done is nothing new: it follows the same old FOIL rule: ac + iad + ibc + i^2bd. Where i^2 gives us -1 so we get ac - bd + i(ad + bc).
When you plot a few points on the complex plane and try multiplying them with the above, you will soon discover that their arguments (i.e. angles) are adding together, and you can then prove that with some basic trig.
For instance if you multiply together any points that are both 30 degrees off the real axis, you get a point that is 60 degrees, and so on.
You can completely removce the "i" from the picture and just have the multiplication rule as <a, b> x <c, d> = <ac - bc, ad + bc>, and then see what happens when you convert to polar coordinates.
You can connect that to linear agebra, in which rotation of a vector is achieved by multiplication by a suitable matrix (and in connection to the above, such a rotation matrix will have some complex eigenvalues; real ones correspond to scaling/shearing).