The Number That Turned Sideways
zuriby.github.io
zuriby.github.io
Cambridge, MA but still ... unexpected.
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.
However, I have questions: "Turns out the quantity they needed exists, but couldn't be described in their notation" What is this about? Sounds interesting.
"Statisticians just said "oh, that function" and gave it a new name." What is this?
I never understood there is a relationship between quadratic equations and some kind of underlying mathematic geometric symmetry. Is there a good intro to this? I only memorized how to solve them.
And the existential question. Is there a good way to teach this stuff?
I avoided math like the plague until my PhD program. Real analysis was a program requirement so I had to quickly teach myself calculus and get up to speed—and I found I really, really liked it. These high level questions are just so interesting and beyond the rote calculation I thought math was.
I hope I can give my daughter a glimpse of the interesting parts before the school system manages to kill her interest altogether (and I would welcome tips to that end if anyone has them).
In a polynomial equation, the coefficients can be written as symmetric functions of the roots: https://en.wikipedia.org/wiki/Vieta%27s_formulas - symmetric means it doesn't matter how you label the roots, because it would not make sense if you could say "r1 is 3, r2 is 7" and get a different set of coefficients compared to "r1 is 7, r2 is 3".
Since the coefficients are symmetric functions of the roots, that means that you can't write the roots as a function of the coefficients - there's no way to break that symmetry. This is where root extraction comes in - it's not a function. A function has to return 1 answer for a given input, but root extraction gives you N answers for the nth root of a given input. So that's how we're able to "choose" roots - consider the expression (r1 - r2) for a quadratic equation. That's not symmetric (the answer depends on which one we label as r1 and which we label as r2), so we can't write that expression as a function of the coefficients. But what about (r1 - r2)^2? That expression IS symmetric - you get the same answer regardless of how you label the roots. If we expand that out we get r1^2 - 2r1r2 + r2^2, which is symmetric, which means we can write it as a function of the coefficients. So we've come up with an expression whose square root depends on the way we've labeled the roots (using Vieta's formulas you can show it's b^2-4c, which you might recognize from the quadratic equation).
Galois theory is used to show that root extraction can only break certain types of symmetries, and that fifth degree polynomials can exhibit root symmetries that are not breakable by radicals.
The Greeks "notation" was a diagram full of points labeled by letters (Α, Β, Γ, ...) with various lines connecting them and a list of steps to do with an unmarked ruler and a compass, some of which added new points. But those tools alone can't be used to describe cube roots of arbitrary numbers (or, equivalently, trisections of arbitrary angles).
> What is this [statisticians' function]?
The integral of the bell curve (normal distribution) is called its cumulative distribution function (CDF). The CDF of the normal distribution is closely related to a special function called the "error function" erf(x).
> Is there a good intro to [the symmetry interpretation of the quadratic formula]?
There's some discussion at Wikipedia's article about the quadratic formula: https://en.wikipedia.org/wiki/Quadratic_formula#By_Lagrange_...
There are hierarchies of numbers (quantities) in mathematics, just as there are hierarchies of patterns (formal languages) in computer science, based on how difficult these objects are to describe. The most widely accepted hierarchy is actually the same in math and CS: rational, algebraic, transcendental.
In math, a rational number is one that can be described by dividing two integers. In CS, a rational pattern is one that can be described by a regular expression (regex). This is still "division": Even when we can't do 1-x or 1/x, we can recognize the pattern 1/(1-x) = 1 + x + x^2 + x^3... as "zero or more occurrences of x", written in a regex as x*.
In math, an algebraic number is one can be found as a root of a polynomial with integer coefficients. The square root of 2 is the poster child, solving x^2 - 2 = 0, and "baby's first proof" in mathematics is showing that this is not a fraction of two integers.
In CS, an algebraic pattern is one that can be described using a stack machine. Correctly nested parentheses (()(())) is the poster child here; we throw plates on a stack to keep track of how deep we are. The grammars of most programming languages are algebraic: If the square root of math is like nested parentheses, then roots of higher degree polynomials are like more complicated nested expressions such as "if then else" statements. One needs lots of colors of plates, but same idea.
In math, everything else (e, Pi, ...) is called trancendental. CS has more grades of eggs, but same idea.
One way to organize this is to take a number x and look at all expressions combining powers of x. If x^3 = 2, or more generally if x is the root of any polynomial, then the list of powers wraps around on itself, and one is looking at a finite dimensional space of expressions. If x is transcendental, then the space of expressions is infinite.
So where were the Greeks in all this? Figuring out where two lines meet is linear algebra, but figuring out where a line meets a circle uses the quadratic formula, square roots. It turns out that their methods could reach some but not all algebraic numbers. They knew how to repeatedly double the dimension of the space of expressions they were looking at, but for example they couldn't triple this space. The cube root of 2 is one of the simplest numbers beyond their reach. And "squaring the circle" ? Yup, Pi is transcendental. Way out of their reach.
When you have a hammer you see nails. When you have a circle you see doubling.
Yes, this is all Galois theory.
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.
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).
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.
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
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.
And "real" numbers introduce tons of wackiness, much more than gets introduced when you expand "real" numbers to "complex" numbers.