There have been a number of Scientific discoveries that seemed to be purely scientific curiosities that later turned out to be incredibly useful. Hertz famously commented about the discovery of radio waves: “I do not think that the wireless waves I have discovered will have any practical application.”
Are there examples like this in math as well? What is the most interesting “pure math” discovery that proved to be useful in solving a real-world problem?
That’s not quite accurate because the two numbers have a relationship with each other. i^2 = - 1, so any time you square a complex number or multiply two complex numbers, some of the value jumps from one dimension to the other.
It’s like a vector, where sure, certain operations can be treated as if the dimensions of the vector are distinct, like a translation or scale. But other operations can have one dimension affecting the other, like rotation.
uh… broski… you do realize a vector of two real numbers can be rotated… right? Please give me a single example for a supposed impossible operation to do on a vector of two real numbers that you can do on complex numbers. I can just define v² where v is a vector (a,b) as (a,b)²=(a²-b²,2ab). Okay, now I’ve succeeded in reproducing your supposedly mathematically impossible operation. Give me another one.