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?
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.