how the fuck did we go from throwing sharp sticks to utterly deranged sentences like that? More importantly why do utterly deranged sentences like that accurately describe our universe and what is the next ludicrous math concept we’re going to discover is integral to the function of the universe?

You need 3Blue1Brown in your life.

Funny how imaginary numbers were invented to solve cube roots, but the most common give example (& definition) uses square root.

Complex numbers ≠ Imaginary numbers

the name seems to be an unfortunate choice that stems from their historical usage as “a means to an end”. i.e, they were first used as part of a method to find some solutions to cubic equations. this method would require algebraic manipulations of complex numbers, but the ultimate goal was to discover a real root. the complex roots would be discarded once a real root was found (if it existed).

the wikipedia article attributes the name to Descartes:

… sometimes only imaginary, that is one can imagine as many as I said in each equation, but sometimes there exists no quantity that matches that which we imagine.

which i think helps to highlight how skeptical the people at that time were about the existence of the “imaginary” numbers.

source: memories of my first complex analysis class, and https://en.wikipedia.org/wiki/Complex_number#History

i’d strongly recommend reading the history section of that wikipedia page to anyone interested in the topic, it has some pretty fun history

The Italian Bombelli in 1572 seemed to toy with both concepts but called imaginary numbers “quantità silvestri” (silvestri meaning ‘wild’) and complex numbers “numeri complessi”. Interesting the imaginary is a quantity and the complex is a number, but maybe old Italian didn’t have that distinction.

I suppose Descartes would agree with you, he first coined the term “imaginary” because he didn’t think they’d serve much purpose. Euler made use of them and continued using the term. Complex number is a complex - a number with a real and an imaginary component.

Are they as real as real numbers though?

Unless you're referring to the set of Real Numbers, in which case they're not real.

Why stop there? They're just as real as any number.

My calculator says they are an "error".

Serious question because I am math-challenged.

What things are we able to quantify by finding the square root of a negative number aside from square roots of negative numbers?

I'm not sure I agree with this... it's like if no one is around to hear a tree fall, does it make a sound?

Yes, this reply, I was gonna make a joke along the lines that I'm sure to someone saying shit like this (that we invented numbers, imaginary or not) is like the meme forcing Italians to look at pineapple pizza being made ...

... but I would lie if this post didn't disturb me a bit to - and the "Talk nerdy to me" part just makes it worse. Nerd stuff has to have a certain level of precision, nerd kinks are exact.

And are a very elegant solution at that.

Ah but that's man's greatest folly.

By answering one question we create a hundred more.

Yup. When you have a circuit that is not purely resistive the inductive or capacitive load causes the voltage and current to not be in phase. It looks like ohms law is being violated. However the missing part of the energy is in the imaginary component to be returned latter.

Ever since I went down a particularly nasty rabbit hole and came out with a tenuous grasp on quaternions, imaginary numbers started feeling very simple, familiar and logical.

I used that text as a little test for the AIs at DDG. Here's from Claude 3 Haiku.

Here is the translation to English:

Complex Numbers and Polynomials

Something that hardly anyone knows is that complex numbers, in addition to their application in Complex AC Calculations (known to HTL students), also have a second, important use.

In mathematics, there is often the problem of finding a number x for a given function f(x) such that f(x) = 0 holds.

This is often the case for polynomials, as many important mathematical functions can be well approximated by polynomials.

The Fundamental Theorem of Algebra states that if we allow x to be a complex number, then there is always a solution to the problem such that f(x) = 0.

Restriction: but the polynomial must not consist of only a constant term, i.e., f(x) = 5 would be inadmissible. The x must appear at least once in the polynomial. Otherwise, there are really no restrictions :-)

PS: In the example given in the image (f(x) = x^2 + 2 = 0), a solution would be approximately x = √2 i ≈ 1.4 i.

Then I asked: "Please make it understandable to a general audience and phrase it like a pop-science author would."

Here is the translation in a more accessible, pop-science style:

The Hidden Power of Complex Numbers

Most people think complex numbers are just some obscure math concept, useful only for electrical engineers. But these mysterious numbers actually have a hidden superpower - they can solve all sorts of important math problems.

You see, in math, we often want to find a number x that makes a function f(x) equal to zero. This comes up a lot when working with polynomials - those nice, smooth equations made up of powers of x.

The problem is, sometimes there just doesn't seem to be any real number x that does the trick. That's where complex numbers swoop in to save the day!

The Fundamental Theorem of Algebra proves that as long as we allow x to be a complex number (a number with both a real and imaginary part), then there will always be a solution that makes f(x) equal to zero. No exceptions!

So complex numbers unlock the ability to solve all kinds of polynomial equations that would be impossible with just regular old real numbers. It's like they have a secret mathematical superpower.

Of course, there is one small catch - the polynomial can't be a simple constant, like f(x) = 5. It has to actually contain the variable x somewhere. But other than that, complex numbers can tackle any polynomial problem you throw at them.

So the next time you see an imaginary number like √(-1), remember - it's not just some weird math trick. It's a powerful tool for solving important equations. Complex numbers may seem strange, but they're secretly the heroes of the math world!

For real man, Imagine Dragons